\section{\label{sec:intro}Introduction}

Dispersive shock waves~(DSWs) are widespread nonlinear phenomena in physics, from hydrodynamics~\cite{1966Peregrine,1988Smyth,2016Maiden} to acoustics~\cite{1970Taylor}, from Bose-Einstein condensates~\cite{2004Damski,2004Kamchatnov,2004Perez,2005Simula,2006Hoefer,2008Chang} to plasma physics~\cite{2008Romagnani}, and from quantum liquids~\cite{2006Bettelheim} to optics~\cite{1967Akhmanov,1989Rothenberg,2007El,2007HoeferPhysD,2007Ghofraniha,2007Wan,2008Conti,2012Ghofraniha,2013Garnier,2013Gentilini,2014Gentilini,2014Smith1,2015GentiliniPRA,2015GentiliniSciRep,2015Xu,2016Braidotti,2016Wetzel,2016XuMussot,2016XuGarnier,2017Zannotti,2019Gautam,2019Marcucci,2019MarcucciReview}.

In optics and photonics, when light propagates in a medium whose refractive index depending on the beam intensity, (e.g., light experiences the Kerr effect~\cite{2008Boyd}), the interplay between diffraction (or dispersion) and nonlinearity can lead to steep gradients in the phase profile, and, in some cases, to a wave breaking~\cite{2019MarcucciReview}.
Such discontinuity is regularized by very rapid oscillations, both in phase chirp and in intensity outlines, called undular bores~\cite{2007Wan}.
If we slightly mutate this third-order nonlinearity, making it nonlocal (or noninstantaneous), then the phenomenology changes very little. When the phase chirp reaches the discontinuity and starts to oscillate, the intensity does not develop undular bores, but rather the annular collapse singularity~(ACS) (or M-shaped singularity if the beam is $1+1$-dimensional)~\cite{2015Xu,2016Braidotti,2018Xu,2019MarcucciReview}. Spatial collapse-DSWs occur in thermal media, where the radiation-matter interaction is led by a thermo-optic effect, and the refractive index perturbation depends on the whole intensity profile~\cite{2008Boyd,2019MarcucciReview}.

Theoretically, such modification to the Kerr nonlinearity has significant consequences. Laser beam propagation in a standard Kerr medium is ruled by the nonlinear Schr\"odinger equation~(NLSE), exactly solvable by the inverse scattering transform method~\cite{1967Gardner,1972Zacharov,2015Fibich}. However, the NLSE with a nonlocal potential cannot be solved by inverse scattering transform (despite some recent progress in two-dimensional~(2D) media~\cite{2017Hokiris,2019Hokiris}), but only through the Whitham modulation and the hydrodynamic approximation~\cite{1999Whitham}. Moreover, the dynamics beyond the shock appears to be intrinsically irreversible, as recently shown by applying the time asymmetric quantum mechanics~\cite{1981Bohm,1989Bohm,1998Bohm,1999Bohm,2002Chruscinski,2002Delamadrid,2003Gadella,2003Chruscinski,2004Chruscinski,2004Civitarese,2016Celeghini,2016Marcucci} to the description of DSWs in highly nonlocal approximation~\cite{2015GentiliniPRA,2015GentiliniSciRep,2016Braidotti,2017Marcucci,2019MarcucciReview}.

In highly nonlocal Kerr media, the ACS is modeled by the simplest Hamiltonian of time asymmetric quantum mechanics: the reversed harmonic oscillator~(RHO). This Hamiltonian has a real complete continuous spectrum, which corresponds to a basis of eigenfunctions that have simple poles in their analytical prolongations to the complex plane. Since the RHO is a harmonic oscillator~(HO) with a pure imaginary frequency, starting from the HO complete point spectrum, it turns out that the RHO point spectrum is the set of the mentioned simple poles, and the related eigenfunctions are non-normalizable eigenvectors belonging to a rigged Hilbert space, called Gamow vectors~(GVs)~\cite{1928GamowDE,1928GamowENG}.
Light propagation beyond the collapse is then expressed as a superposition of GVs, which exponentially decay with quantized decay rates. Such laser beam evolution is the outcome of a phenomenon, the shock, that is intrinsically irreversible: in the absence of absorption and interaction with an external thermal bath, the dynamics cannot be inverted, i.e., it is time asymmetric.
Can this theoretical model be used to describe much more complex scenarios? To answer this question, we need to access regimes with much stronger nonlinearity

In recent experiments, it is showed that M-Cresol/Nylon solutions exhibit an isotropic giant self-defocusing nonlinearity, tunable by varying the nylon concentration~\cite{2014Smith}.
M-Cresol/Nylon is a thermal chemical mixture, consisting of an organic solvent (m-cresol) and a synthetic polymeric solute (nylon).
The nonlinear Kerr coefficient $n_2$ of pure m-cresol is $-9\times10^{-8} cm^2/W$, but it was found that, in such mixtures, $n_2=-1.6\times10^{-5} cm^2/W$ for a nylon mass concentration of $3.5\%$, higher than other thermal nonlinear materials in which ACSs have been observed~\cite{2014Smith1,2019MarcucciReview}.

In this Letter, we report on our theoretical discovery and experimental evidence of optical DSWs with an \textit{anisotropic zero-singularity}~(ZS) (i.e., a gap in the intensity profile along only one direction) in M-Cresol/Nylon solutions. Fixing $z$ as the longitudinal and $x,y$ as the transverse directions, we consider an initial beam which is even in the $y$ direction, and odd along the $x$ direction. This initial condition causes a new phenomenon: the shock does develop an annular collapse, but around the ZS it presents an abrupt intensity discontinuity.
We theoretically analyze this anisotropic wave breaking. We model the beam propagation beyond the shock point by time asymmetric quantum mechanics and uncover the mechanism that determines how such an abrupt intensity discontinuity is generated. We numerically simulate these results and find remarkable agreement between experimental outcomes and theoretical predictions.

For a laser beam propagating in a thermal medium with refractive index $n=n_0+\Delta n[|A|^2](\mathbf{R})$, where $\mathbf{R}=(\mathbf{R_{\perp}},Z)=(X,Y,Z)$, the NLSE describes the evolution of the envelope $A(\mathbf{R})$ of the monochromatic field $\mathbf{E}(\mathbf{R})=\hat{\mathbf{E}}_0 A(\mathbf{R})e^{\imath k Z}$, and it reads as follows:
\begin{equation}
2\imath k \partial_Z A+\nabla_{\mathbf{R_{\perp}}}^2 A+2k^2 \frac{\Delta n[|A|^2]}{n_0} A=-\imath\frac{k}{ L_{loss}}A,
\label{eq:NLSE}
\end{equation}
where $\nabla_{\mathbf{R_{\perp}}}^2=\partial_X^2+\partial_Y^2$, $k=\frac{2\pi n_0}{\lambda}$ is the wavenumber, $\lambda$ is the wavelength, and $L_{loss}$ is the linear loss length. By defining $I=|A|^2$ the intensity, $\bar{P}(Z)=\int\int\mathrm{d}\mathbf{R_{\perp}}I(\mathbf{R})$ the power, $L_d=k W_0^2$ the diffraction length, with $W_0$ the initial beam waist, and $\alpha=\frac{L_d}{L_{loss}}$, it turns out that $\bar{P}$ is not conserved only if $\alpha\neq0$. Indeed, if $\alpha\sim0$, then $\partial_Z \bar{P}\sim0$~\cite{1991Lisak}.

In low absorption regime, the refractive index perturbation in Eq.~(\ref{eq:NLSE}) is~\cite{2019MarcucciReview}
\begin{equation}
\Delta n[|A|^2](\mathbf{R_{\perp}})=n_2 \int\int\mathrm{d}\mathbf{R_{\perp}'}K(\mathbf{R_{\perp}}-\mathbf{R_{\perp}'})I(\mathbf{R_{\perp}'}),
\label{eq:n1}
\end{equation}
with $n_2$ the Kerr coefficient and $K(\mathbf{R_{\perp}})$ is the kernel function describing the nonlocal nonlinearity, normalized such that $\int\int\mathrm{d}\mathbf{R_{\perp}}K(\mathbf{R_{\perp}})=1$.
For $K(\mathbf{R_{\perp}})=\delta(\mathbf{R_{\perp}})$ we attain the well-known local Kerr effect, i.e., $n=n_0+n_2I$~\cite{2008Boyd}. In our nonlocal case, where the laser beam produces a thermo-optic effect that generates an isotropic variation of the medium density distribution, the response function is
\begin{equation}
K(X,Y)=\tilde{K}(X)\tilde{K}(Y),\;\; \tilde{K}(X)=\frac{e^{-\frac{|X|}{L_{nloc}}}}{2L_{nloc}}.
\label{eq:kernel}
\end{equation}
Here $L_{nloc}$ is the nonlocality length~\cite{1997Snyder,2003Conti,2007Minovich,2007Ghofraniha,2019Marcucci}.
We rescale Eq.~(\ref{eq:NLSE}), with $\alpha\sim0$, by defining the dimensionless variables $x=X/W_0$, $y=Y/W_0$ and $z=Z/L_d$, and obtain
\begin{equation}
  \imath \partial_z \psi+\frac{1}{2}\nabla_{\mathbf{r_{\perp}}}^2\psi+\chi P K_0*|\psi|^2 \psi=0,
\label{eq:NLSEnorm}
\end{equation}
where $\mathbf{r}=(\mathbf{r_{\perp}},z)=(x,y,z)$, $\nabla_{\mathbf{r_{\perp}}}^2=\partial_x^2+\partial_y^2$, $\psi(\mathbf{r})=\frac{W_0}{\sqrt{\bar{P}}}A(\mathbf{R})$, $\chi=\frac{n_2}{|n_2|}$ and $P=\frac{\bar{P}}{P_{REF}}$ with $P_{REF}=\frac{\lambda^2}{4\pi^2 n_0 |n_2|}$.
The asterisk in Eq. (\ref{eq:NLSEnorm}) stands for the convolution product, while $K_0(x,y)=\tilde{K_0}(x)\tilde{K_0}(y)$ with $\tilde{K_0}(x)=W_0 \tilde{K}(X)=\frac{e^{-\frac{|x|}{\sigma}}}{2\sigma}$, $\tilde{K_0}(y)$ of the same form, and $\sigma=\frac{L_{nloc}}{W_0}$ the nonlocality degree.

In highly nonlocal approximation ($\sigma>>1$), once the initial conditions are fixed, $|\psi|^2$ mimics a delta function (or a narrow superposition of delta functions), and the nonlocal potential looses its $I$-dependence, becoming a simple function of the transverse coordinates~\cite{1997Snyder,2012Folli}:
\begin{equation}
\begin{array}{l}
K_0*|\psi|^2\simeq \kappa(\mathbf{r_{\perp}})\simeq\kappa(\mathbf{0})+\left(\partial_x\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)x+\left(\partial_y\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)y+\\ \\
+\frac{1}{2}\left(\partial_x^2\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)x^2+\left(\partial_x\partial_y\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)xy+\frac{1}{2}\left(\partial_y^2\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)y^2,
\end{array}
\label{eq:nonlocpot}
\end{equation}
after a Taylor second-order expansion.
This approximation maps the NLSE in Eq.~(\ref{eq:NLSEnorm}) into a linear Schr\"odinger equation $\imath \partial_z\psi(\mathbf{r})=\hat{H}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})\psi(\mathbf{r})$, with $\hat{H}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\frac{1}{2}\hat{\mathbf{p_{\perp}}}^2+\hat V(\mathbf{r_{\perp}})$ the Hamiltonian, $\mathbf{\hat p_{\perp}}=(\hat p_x, \hat p_y)=(-\imath\partial_x, -\imath\partial_y)$ the transverse momentum and $\hat V(\mathbf{r_{\perp}})=-\chi P\kappa(\mathbf{r_{\perp}})\mathbb{1}$ the multiplicative potential ($\mathbb{1}$ is the identity operator). Let us consider the initial condition
\begin{equation}
\psi_{\mathrm{ISO}}(\mathbf{r_{\perp}})=\psi_{even}(x)\psi_{even}(y),\;\;\psi_{even}(x)=\frac 1{\sqrt[4]{\pi}}e^{-\frac{x^2}2},
\label{eq:isoini}
\end{equation}
and $\psi_{even}(y)$ of the same form.
The shape of $\kappa(\mathbf{r_{\perp}})$ depends on $\psi_{\mathrm{ISO}}(\mathbf{r_{\perp}})$. Indeed, since $\psi_{\mathrm{ISO}}$ is an even, separable function, all the first derivatives in Eq.~(\ref{eq:isoini}) vanish, hence $\kappa(\mathbf{r_{\perp}})=\kappa_0^2-\frac{1}{2}\kappa_2^2\left|\mathbf{r_{\perp}}\right|^2$, where $\kappa_0^2=\frac1{4\sigma^2}$ and $\kappa_2^2=\frac{1}{2\sqrt{\pi}\sigma^3}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{figISO.eps}
\end{center}
\caption{Solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:isoini}), for $P=4\times10^6$ and $\sigma=120$, in arbitrary units: (a) shows the intensity transverse profile at $z=0$, (b) exhibits the symmetric response function derived from Eq.~(\ref{eq:isoini}), and (c) reports the intensity longitudinal outline, here on the plane $(x,z)$, equal to one on the plane $(y,z)$.
\label{fig:iso}}
\end{figure}

In the defocusing case ($n_2<0$), the transverse profile of the solution of Eq.~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:isoini}) is shown in Fig.~\ref{fig:iso}(a). Figure~\ref{fig:iso}(b) exhibits the central part of the symmetric response function $K_0(x,y)$, while the longitudinal profile on $x,z$ (same of $y,z$) is reported in Fig.~\ref{fig:iso}(c). The corresponding Hamiltonian reads
$\hat{H}=P\kappa_0^2 +\hat{H}_{\mathrm{RHO}}(p_x,x)+\hat{H}_{\mathrm{RHO}}(p_y,y)$,
where
\begin{equation}
\hat{H}_{\mathrm{RHO}}(p_x,x)=\frac 12\hat{p_x}^2-\frac{\gamma^2}{2}\hat{x}^2
\label{eq:RHO}
\end{equation}
is the 1D-RHO Hamiltonian of frequency $\gamma=\sqrt{P}\kappa_2$.
Once moved to $\phi(\mathbf{r})=e^{\imath P\kappa_0^2}\psi(\mathbf{r})$, the Schr\"odinger equation becomes
$\imath \partial_z\phi(\mathbf{r})=\left[\hat{H}_{\mathrm{RHO}}(p_x,x)+\hat{H}_{\mathrm{RHO}}(p_y,y)\right]\phi(\mathbf{r})$, which is completely separable.
In bra-ket notation
\begin{equation}
\begin{array}{l}
\imath \frac{\mathrm{d}}{\mathrm{d}z}|\phi(z)\rangle=\hat{H}_{\mathrm{ISO}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})|\phi(z)\rangle,\\ \\
\hat{H}_{\mathrm{ISO}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\hat{H}_{\mathrm{RHO}}(p_x,x)\otimes\mathbb{1}_y+\mathbb{1}_x\otimes\hat{H}_{\mathrm{RHO}}(p_y,y),\\ \\
|\phi(z)\rangle=|\phi_{even}(z)\rangle_x\otimes|\phi_{even}(z)\rangle_y,
\end{array}
\label{eq:iso}
\end{equation}
with $\otimes$ the tensorial product, no more explicitly written hereafter. The solution of Eq.~(\ref{eq:iso}) lives in a tensorial product between two 1D rigged Hilbert spaces. Indeed, if we consider the evolution operator $\hat{U}(z)=e^{-\imath\hat{H}z}$ such that $|\phi(z)\rangle=\hat{U}(z)|\phi(0)\rangle$, for Eq.~(\ref{eq:iso}) $|\phi(z)\rangle=e^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_xe^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_y$. The representation of $|\phi_{even}(z)\rangle_{x,y}=e^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_{x,y}$ in terms of GVs was already studied and was also already demonstrated to describe 1D DSWs in thermal media~\cite{2015GentiliniPRA,2015GentiliniSciRep,2016Braidotti,2017Marcucci,2019MarcucciReview}. It is $|\phi_{even}(z)\rangle_{x,y}=|\phi_N^G(z)\rangle+|\phi_N^{BG}(z)\rangle$, with
\begin{equation}
|\phi_N^G(z)\rangle=\sum_{n=0}^N e^{-\frac{\gamma}2(2n+1)z}|\mathfrak{f}_n^-\rangle\langle \mathfrak{f}_n^+|\psi_{even}\rangle
\label{eq:GV}
\end{equation}
the decaying superposition of Gamow states $|\mathfrak{f}_n^-\rangle$, corresponding to the energy levels $E_n^{RHO}=\imath\frac{\gamma}2(2n+1)$, and $|\phi_N^{BG}(z)\rangle$ the background function, both belonging to the same 1D rigged Hilbert space.

Our experiments are performed in an isotropic medium with asymmetric initial conditions for the beam, designed through a phase mask, to attain an anisotropic light propagation. The setup is illustrated in Fig.~\ref{fig:exp}(a). A laser beam with wavelength $\lambda=532$nm passes through two collimating lenses (L1 and L2), and then through a beam splitter (BS1), which divides the beam into two arms, one used for the nonlinear experiment, and the other for getting a reference beam for interference measurements. The beam outcoming from the first arm is transformed into an asymmetric input by a phase mask, and then is focused (via L3) onto the facet of a $2$mm-long cuvette, which contains a M-Cresol/Nylon solution  (with $3.5\%$ Nylon). The output is imaged (via L4 and BS2) onto a CCD-camera.

Figure~\ref{fig:exp}(b) reports the input beam (zoom-in intensity and phase patterns at initial power $\bar{P}=2$mW and waist $W_0=15.8\mu$m) and its outputs at different initial powers. The input beam presents a phase discontinuity of $\pi$ along $x=0$.
The output beam exhibits diffraction at a low power, but evolves into an overall shock pattern with two parts at high powers. It's important to note that, while the whole pattern expands, the gap between two parts remains constant. This represents the first realization of what we define as \textit{anisotropic DSWs}: ACSs with an initial ZS, which generates two barriers of light intensity around a constant gap in the middle of the beam. Despite the medium isotropy, the oddity of the initial condition generates an anisotropic final transverse profile.

\begin{figure}[h!]
\begin{center}
\textbf{(a)
\raisebox{-1\height}{\includegraphics[width=0.95\linewidth]{Setup.eps}}
\\\vspace{3mm}
\textbf{(b)}\raisebox{-0.85\height}{\includegraphics[width=\linewidth]{anisotropic.png}}
\end{center}
\caption{
(a) Experimental setup. A $\lambda=532$nm CW-laser beam is collimated through two lenses (L1 and L2). A beam splitter (BS1) divides the beam in two arms. the first is made asymmetric by a phase mask and propagates in a $2$mm-long cuvette filled with M-Cresol/Nylon $3.5\%$-solution. The second is a reference beam for interference measurements. The output is imaged (via L4 and BS2) onto a CCD-camera.
(b) Input and outputs observed. The phase mask in (a) generates a $\pi$ discontinuity in the input phase, here reported with initial power $\bar{P}=2$mW and waist $W_0=15.8\mu$m, together with the intensity profile. Several output at different initial powers are shown. The low power cannot distinguish nonlinear effects from diffraction, but the higher the power is, the stronger the nonlinear effects are, leading to the formation of anisotropic DSWs.
\label{fig:exp}}
\end{figure}

We model the initial asymmetric beam-shape as follows:
\begin{equation}
\psi_{\mathrm{ANI}}(\mathbf{r_{\perp}})=\psi_{odd}(x)\psi_{even}(y),\;\;\psi_{odd}(x)=-\frac{\sqrt{2}}{\sqrt[4]{\pi}}xe^{-\frac{x^2}2},
\label{eq:aniini}
\end{equation}
$\psi_{even}(y)=\frac 1{\sqrt[4]{\pi}}e^{-\frac{y^2}2}$ as in Eq.~(\ref{eq:isoini}).
In this case, Eq.~(\ref{eq:nonlocpot}) is reduced to $\kappa(\mathbf{r_{\perp}})=\kappa_0^2+\frac{1}{2}\kappa_1^2 x^2-\frac{1}{2}\kappa_2^2 y^2$, with $\kappa_0^2=\frac1{4\sigma^2}$, $\kappa_1^2=\frac{1}{4\sigma^4}$ and $\kappa_2^2=\frac{1}{2\sqrt{\pi}\sigma^3}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{figANI.eps}
\end{center}
\caption{
Solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:aniini}), for $P=4\times10^6$ and $\sigma=120$, in arbitrary units: (a) shows the intensity transverse profile at $z=1$, (b) exhibits the asymmetric response function derived from Eq.~(\ref{eq:isoini}), and (c) reports the intensity longitudinal outline on the plane $(x,z)$, with the zero-singularity.
\label{fig:ani}}
\end{figure}

The anisotropy appears evident: not only the initial condition presents a ZS, but also the response function has two different behaviors along $x,y$ directions. Numerical simulations are illustrated in Fig.~\ref{fig:ani}. Figure~\ref{fig:ani}(a) shows the anisotropic DSWs, solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:aniini}). Figure~\ref{fig:ani}(b) gives numerical proof of the response function anisotropy: the $(x,y)$-plane origin corresponds to a saddle point, with a locally increasing profile along $x>0$, $y<0$ and a locally decreasing outline along $x<0$, $y>0$. Figure~\ref{fig:ani}(c) reports the intensity-ZS in a neighborhood of $x=0$ during propagation.

The presence of the saddle point in the response function has direct consequences through highly nonlocal approximation in mapping the NLSE in the quantum-like linear Schr\"odinger equation. From the expression of $\kappa(\mathbf{r_{\perp}})$ above, for $\phi(\mathbf{r})=e^{\imath P\kappa_0^2}\psi(\mathbf{r})$ we obtain
\begin{equation}
\begin{array}{l}
\imath \frac{\mathrm{d}}{\mathrm{d}z}|\phi(z)\rangle=\hat{H}_{\mathrm{ANI}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})|\phi(z)\rangle,\\ \\
\hat{H}_{\mathrm{ANI}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\hat{H}_{\mathrm{HO}}(p_x,x)\mathbb{1}_y+\mathbb{1}_x\hat{H}_{\mathrm{RHO}}(p_y,y),\\ \\
|\phi(z)\rangle=|\phi_{odd}(z)\rangle_x|\phi_{even}(z)\rangle_y,
\end{array}
\label{eq:ani}
\end{equation}
where $\hat{H}_{\mathrm{HO}}(p_x,x)=\frac 12\hat{p_x}^2+\frac{\omega^2}{2}\hat{x}^2$ is the one-dimensional harmonic oscillator Hamiltonian with $\omega=\sqrt{P}\kappa_{odd}$, and $\hat{H}_{\mathrm{RHO}}$ is the one-dimensional RHO Hamiltonian in Eq.~(\ref{eq:RHO}).
The solution of Eq.~(\ref{eq:ani}) is the tensorial product of $|\phi_{odd}(z)\rangle_{x}=\sum_{n=0}^{+\infty} e^{\imath\frac{\omega}2(2n+1)z}|\Psi_n^{HO}\rangle\langle\Psi_n^{HO}|\psi_{odd}\rangle$, where $|\Psi_n^{HO}\rangle$ are $\hat{H}_{\mathrm{HO}}$-eigenstates corresponding to the energy levels $E_n^{HO}=\frac{\omega}2(2n+1)$~\cite{2016Marcucci}, and $|\phi_{even}(z)\rangle_{y}=|\phi_N^G(z)\rangle+|\phi_N^{BG}(z)\rangle$, explicitly written in Eq.~(\ref{eq:GV}).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{GVSign.eps}
\end{center}
\caption{
GVs signature. From Fig.~\ref{fig:ani}, in the same conditions, (a) is the $y,z$ profile at fixed $x=2.29$. Intensity along the pink line, i.e., $|\phi(x=2.29,y=2.29,z)|^2=\langle\phi_{even}(z)|\phi_{even}(z)\rangle_{y}=\sum_{n=0}^Ne^{-\Gamma_n z}\left|\langle\mathfrak{f}_n^+|\phi_{even}\rangle\right|^2$, is in (b) [$\Gamma_n=\gamma(2n+1)$ quantized decay rates].  The continuous lines represent the two first exponential functions of the summation above that fit the decaying part: the red line is the fundamental GV, with decay rate $\Gamma_0$, and the blue line is the first excited GV, with decay rate $\Gamma_2$.
\label{fig:GV}}
\end{figure}

Evidence of the presence of GVs is given in Fig.~\ref{fig:GV}. By defining $\Gamma_n=\gamma(2n+1)$, we look for the first two quantized decay rates $\Gamma_{0,2}$ (the even Gaussian initial function only leads to even energy levels) in the longitudinal propagation in $y$-direction. Indeed, if one computes the intensity of the $y$-part, one finds $\langle\phi_{even}(z)|\phi_{even}(z)\rangle_{y}\stackrel{N>>0}{\simeq}\langle\phi_N^G(z)|\phi_N^G(z)\rangle=\sum_{n=0}^Ne^{-\Gamma_n z}\left|\langle\mathfrak{f}_n^+|\phi_{even}\rangle\right|^2$. Figure~\ref{fig:GV}(a) shows the theoretical section of the nonlinear sample where we seek decaying states. We fix $x=2.29$, a little distant from the shock-gap, and report the corresponding intensity in $y,z$ plane. The pink line is equivalent to $x=y=2.29$. Figure~\ref{fig:GV}(b) exhibits $|\phi(x=2.29,y=2.29,z)|^2$, exponentially decaying. Two exponential fits demonstrate the GV occurrence: the fundamental Gamow state represents the plateau with decay rate $\Gamma_0=1.51$, whereas the first excited one interpolates the peak, with decay rate $\Gamma_2=7.51$.
We stress that the rule $\frac{\Gamma_2}{\Gamma_0}=5$ is respected.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{GapExp.eps}
\end{center}
\caption{
Observation of the shock-gap versus the initial power $\bar{P}$. The red line shows the theoretical fit with a functions $\propto\frac{1}{\sqrt{\bar{P}}}$.
\label{fig:gap}}
\end{figure}

The analysis of the barriers, due to the HO component, and the corresponding shock-gap is also examinated. Since $\hat{H}_{\mathrm{HO}}$ has potential $\hat{V}_{\mathrm{HO}}(x)=\frac{\omega^2}2\hat{x}^2$, we expect a shock-gap with the same behavior of the potential width $\Delta x\propto\frac1{\sqrt{\omega}}=\frac{2\sigma^2}{\sqrt{P}}$, that is, a shock-gap that varies among different experiments by changing the initial power but, in a single observation with a given $P$, does not change by varying $z$ [as shown in Fig.~\ref{fig:ani}(c)].
This is experimentally proven and reported in Fig.~\ref{fig:gap}, which shows measurements of the shock-gap at variance of initial power $\bar{P}$. A theoretical fit with a function $\propto\frac{1}{\sqrt{\bar{P}}}$ is drawn in the red line. The agreement between observations and numerical simulations confirms the theoretical statement.

We have proved that the interplay of a trapping (harmonic oscillator) and an antitrapping (reversed harmonic oscillator) potential generates a novel kind of dispersive shock waves, with the simultaneous presence of annular collapse singularities and a shock-gap enclosed by very intense light barriers.
The use of a thermal medium with a giant Kerr coefficient, the M-Cresol/Nylon solution with $3.5\%$ of nylon concentration, let us access an extremely-nonlinear highly-nonlocal regime and perform accurate experiments with negligible loss.
We modeled the outcoming dynamics through an advanced theoretical description in rigged Hilbert spaces, by means of time asymmetric quantum mechanics, proving its intrinsic irreversibility.
Our results not only confirm previous studies on the giant nonlinear response of M-Cresol/Nylon, but also disclose fundamental insights on propagation of dispersive shock waves with a singular initial intensity profile.
We believe that this work can be a further step towards a complete description of optical nonlinear phenomena, where inverse scattering transform, Whitham modulation, hydrodynamic approximation and time asymmetric quantum mechanics cooperate in establishing one uniform theory of dispersive shock waves.

We are pleased to acknowledge support from the QuantERA ERA-NET Co-fund 731473 (Project QUOMPLEX), H2020 project grant number 820392, Sapienza Ateneo, PRIN 2015 NEMO, PRIN 2017 PELM, Joint Bilateral Scientic Cooperation CNR-Italy/RFBR-Russia 2018-2020, the NSF Award DMR-1308084, and the National Key R\&D Program of China (2017YFA0303800).


\section{Introduction}

\noindent
Let $(M,g)$ be a Lorentzian spacetime and $(N,h)$ a Riemannian manifold. In this paper, we study wave maps $u:(M,g) \longrightarrow (N,h)$, that is, critical points of the geometric action functional
\begin{align*}
S_{g}[u]:=\frac{1}{2} \int _{M} |d_{g} u|^{2}~d \mu_{g}.
\end{align*}
Here,
\begin{align*}
|d_{g}u(x)|^{2} \equiv  |d_{g}u (x)|_{ T^{\star}_{x}M \otimes T_{u(x)} N  }^{2}:= \text{tr}_{g} \left (  u^{\star} \left( h \right) \right)
\end{align*}
is the trace (with respect to $g$) of the pullback metric on $(M,g)$ via the map $u$. The integral is understood with respect to the standard measure $d\mu _{g}$ on the domain manifold. In local coordinates $(x_{\mu})$ on $(M,g)$, this expression reads
\begin{align*}
S_{g}[u]= \int_{M} g ^{\mu \nu} (\partial _{\mu}  u^{a}) (\partial _{\nu}  u^{b}) h_{a b} \circ u~ d \mu_{g}
\end{align*}
where the Einstein summation convention is used. The Euler-Lagrange equations associated to this functional are
\begin{align} \label{WM}
\Box _{g} u^{a} + g ^{\mu \nu} (\Gamma ^{a}_{b c} \circ u) (\partial _{\mu}  u^{b} ) (\partial _{\nu}  u^{c}) =0
\end{align}
and they constitute a system of semi-linear wave equations. Here, $\Box _{g}$ is the Laplace-Beltrami operator on $(M,g)$
\begin{align*}
\Box_{g} := \frac{1}{|g|} \partial _{\mu} (g^{\mu \nu} |g| \partial _{\nu}),\quad |g|:=\sqrt{ \left| \text{det}(g_{\mu \nu}) \right| }
\end{align*}
and $\Gamma ^{a}_{b c}$ are the Christoffel symbols associated to the metric $h$ on the target manifold. Eq.~\eqref{WM} is called the wave maps equation (known in the physics literature as non-linear $\sigma$ model) and is the analog of harmonic maps between Riemannian manifolds in the case where the domain is a Lorentzian manifold instead. For more details, we refer the reader to \cite{Ren08} and \cite{Str97}.

\subsection{Intuition} Recently, the wave maps equation has attracted a lot of interest. On the one hand, the wave maps equation is a rich source for understanding nonlinear geometric equations since it is a nonlinear generalization of the standard wave equation on Minkowski space. In addition, the wave maps equation has a pure geometric interpretation: it generalizes the notion of geodesic curves. Notice that, if $M = (\alpha, \beta)$ is an open interval and $(N,h)$ any curved Riemannian manifold, the wave maps equation is the geodesic equation
\begin{align*}
\frac{d^2 u^{a}}{dt ^2} (t) + (\Gamma ^{a}_{b c} \circ u(t))  \frac {d u^{b} }{dt} (t) \frac{d  u^{c} }{dt} (t)=0.
\end{align*}
On the other hand, the Cauchy problem for the wave maps system provides an attractive toy-model for more complicated relativistic field equations. Specifically, wave maps contain many features of the more complex Einstein equations but are simple enough to be accessible for rigorous mathematical analysis.
Further details on the correlation between the wave maps system and the Einstein equations can be found in \cite{Mis78, Mon89, Wei90, Kla97}.

Being a time evolution equation, the fundamental problem is the Cauchy problem: given specified smooth initial data, does there exist a unique smooth solution to the wave maps equation with this initial data? Furthermore, does the solution exist for all times? On the other hand, if the solution only exists up to some finite time $T$, how does the solution blow up as $t$ approaches $T$?
The investigation of questions of global existence and formation of singularities for the wave maps equation can give insight into the analogous, but much more difficult, problems in general relativity.
\subsection{Equivariant wave maps} Now, we turn our attention to the Cauchy problem in the case where the domain is the Minkowski spacetime $(\mathbb{R}^{1+d},g)$ and the target manifold is the sphere $(\mathbb{S}^{d},h)$ for $d \geq 3$. Hence, we pick $g =$diag$(-1,1,\dots,1)$ and $h$ to be the standard metric on the sphere. Furthermore, we choose standard spherical coordinates on Minkowski space and hyper-spherical coordinates on the sphere. The respective metrics are given by
\begin{align*}
g = - dt^2 + dr^2 + r^2 d \omega ^2, \quad h= d \Psi ^2 + \sin^2(\Psi) d \Omega ^2,
\end{align*}
where $d\omega ^2$ and $d \Omega ^2$ are the standard metrics on $\mathbb{S}^{d-1}$.  Moreover, a map $u:(\mathbb{R}^{1+d},g) \longrightarrow (\mathbb{S}^{d},h)$ can be written as
\begin{align*}
u (t,r,\omega) = \big( \Psi  (t,r,\omega ), \Omega (t,r,\omega) \big).
\end{align*}
We restrict our attention to the special subclass known as 1-equivariant or co-rotational, that is
\begin{align*}
\Psi  (t,r,\omega ) \equiv \psi (t,r),~~~ \Omega (t,r, \omega ) = \omega.
\end{align*}
 Under this ansatz, the wave maps system for functions $u:(\mathbb{R}^{1+d},g) \longrightarrow (\mathbb{S}^{d},h)$ reduces to the single semi-linear wave equation
 \begin{equation}
 \label{eq:main} \psi_{tt}-\psi_{rr}-\frac{d-1}{r}\psi_r+\frac{d-1}{2}\frac{\sin(2\psi)}{r^2}=0.
 \end{equation}
 By finite speed of propagation and radial symmetry it is natural to
study this equation in backward light-cones with vertex $(T,0)$, that is
\begin{align*}
C_{T} :=\left \{  (t,r) : 0<t<T,~0 \leq r \leq T-t \right \}
\end{align*}
where $T>0$.
Consequently, we consider the Cauchy problem \\
\begin{align} \label{cauchy}
     \begin{cases}
       \psi _{tt} (t,r) - \Delta ^{  \text{rad}  }_{r,d} \psi (t,r)  = - \frac{d-1}{2} \frac{ \sin ( 2 \psi (t,r) ) }{r^2}, &\quad \text{in } C_{T} \\
       \psi (0,r)= f(r),~~~ \psi _{t} (0,r)= g(r), &\quad \text{on } \{ t=0 \} \times [0,+\infty)  \
     \end{cases}
\end{align}
where $\Delta ^{  \text{rad}  }_{r,d} $ stands for the radial Laplacian
\begin{align*}
\Delta ^{  \text{rad}  }_{r,d} \psi (t,r) :=  \psi _{rr} (t,r) + \frac{d-1}{r} \psi _{r} (t,r).
\end{align*}
To ensure regularity of solutions, equations $\eqref{cauchy}$ must be supplemented by the boundary condition
\begin{align} \label{reg}
\psi (t,0)=0,\quad  \text{~for all~} t \in (0,T).
\end{align}

\subsection{Self-similar solutions}
A basic question for the Cauchy problem $\eqref{cauchy}$ is whether solutions starting from smooth initial data
\begin{align*}
(f,g)=\left( \psi (0, \cdot), \partial _{t} \psi (0, \cdot) \right)
\end{align*}
can become singular in the future. Note that Eq.~\eqref{eq:main} has the conserved energy
\begin{align*}
E[\psi]:=\int_{0}^{\infty} \left( \psi _{t}^2 + \psi _{r}^2 + (d-1)\frac{\sin ^2(\psi) }{r^2} \right) r^2 dr .
\end{align*}
However, the energy cannot be used to control the evolution since Eq.~\eqref{cauchy} is not well-posed at energy regularity, cf.~\cite{ShaTah94}.
Indeed, Eq.~\eqref{eq:main} is invariant under dilations
\begin{align} \label{dilation}
\psi _{\lambda} (t,r):=\psi \left( \frac{t}{\lambda},\frac{r}{\lambda} \right),~ \lambda >0
\end{align}
and the critical Sobolev space for the pair $(\psi(t,\cdot),\partial_t \psi(t,\cdot))$ is $\dot H^{\frac{d}{2}}\times \dot H^{\frac{d}{2}-1}$.
Consequently, Eq.~\eqref{eq:main} is energy-supercritical for $d\geq 3$.

In fact, due to the scaling $\eqref{dilation}$ and the supercritical character it is natural to expect self-similar solutions and indeed, it is well known that there exist smooth initial data which lead to solutions that blowup in finite time in a self-similar fashion. Specifically, Eq.~\eqref{eq:main} admits the self-similar solution
\begin{align*}
\psi ^{T} (t,r) := f_{0} \Big ( \frac{r}{T-t } \Big) = 2 \arctan \Bigg( \frac{r}{\sqrt{d-2}(T-t)  } \Bigg),\qquad T>0.
\end{align*}
This example is due to Shatah \cite{Sha88}, Turok-Spergel \cite{TurSpe90} for $d=3$, and  Bizo\'n-Biernat \cite{BizBie15} for $d \geq 4$ and provides an explicit example for singularity formation from smooth initial data. Indeed, the self-similar solution $\psi^T$ is perfectly smooth for all $0<t < T$ but breaks down at $t = T$ in the sense that
\begin{align*}
\partial_r\psi ^{T} (t,r)|_{r=0}
\simeq \frac{1}{T-t} \longrightarrow + \infty,~~~\text{as}~ t \longrightarrow T^{-}.
\end{align*}
We note in passing that for $d\in \{3,4,5,6\}$, $\psi^T$ is just one member of a countable family of self-similar solutions, see \cite{Biz00, BieBizMal16}.

\subsection{The main result}
By finite speed of propagation one can use $\psi ^{T}$ to construct smooth, compactly supported initial data which lead to a solution that blows up as $t \longrightarrow T$. Our main theorem is concerned with the asymptotic nonlinear stability of $\psi^T$. In other words, we prove the existence of an open set of radial data which lead to blowup via $\psi ^{T}$. In this sense, the blowup described by $\psi ^{T}$ is stable. To state our main result, we will need the notion of the blowup time at the origin.
From now on we use the abbreviation $\psi[t]=(\psi(t,\cdot),\partial_t \psi(t,\cdot))$.
\begin{definition}
Given initial data $(\psi_{0},\psi_{1})$, we define
\begin{align*}
T_{(\psi_{0},\psi_{1})} := \sup  \left\{
T >0 \middle|
  \begin{subarray}{c}
  \exists \text{~solution~} \psi : C_{T} \longrightarrow \mathbb{R} \text{~to~} \eqref{cauchy} \text{~in the sense of}   \\
   \text{~Definition~} \ref{def} \text{~with initial data~} \psi[0]=(\psi_{0},\psi_{1}) |_{\mathbb{B}_{T}^{d}}
  \end{subarray} \right\} \cup  \{0\}.
\end{align*}
In the case where $T_{(\psi_{0},\psi_{1})} < \infty$, we call $T \equiv T_{(\psi_{0},\psi_{1})}$ the blowup time at the origin.
\end{definition}
 We remark that the effective spatial dimension for the problem \eqref{cauchy} is $d+2$.
To see this, recall that, by regularity, we get the boundary condition $\eqref{reg}$. Therefore, it is natural to switch to the variable $\widehat \psi(t,r):=r^{-1}\psi (t,r)$.  Then \eqref{cauchy} transforms into
\begin{align*}
     \begin{cases}
        \widehat{\psi}  _{tt} (t,r) - \Delta ^{  \text{rad}  }_{r,d+2}  \widehat{\psi} (t,r)  = - \frac{d-1}{2} \frac{ \sin ( 2 r  \widehat{\psi}  (t,r) ) -2r \widehat{\psi} (t,r)}{r^3}, &\quad \text{in } C_{T} \\
        \widehat{\psi}  (0,r)= \frac{f(r)}{r},~~~  \widehat{\psi}  _{t} (0,r)= \frac{g(r)}{r}, &\quad \text{on } \{ t=0 \} \times [0,+\infty)  \
     \end{cases}
\end{align*}
Note that the nonlinearity is now generated by a smooth function and the radial Laplacian is in $d+2$ dimensions.

\begin{theorem}
Fix $T_{0}>0$ and $d\geq 3$ odd. Then there exist constants $M,\delta,\epsilon >0$ such that for any radial initial data
$\psi [0]$
satisfying
\begin{align*}
\Big \| |\cdot|^{-1} \Big( \psi[0] -\psi^{T_{0}}[0] \Big) \Big \|_{ H^{\frac{d+3}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2}) \times H^{\frac{d+1}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2}  )} \leq \frac{\delta}{M}
\end{align*}
the following statements hold:
\begin{enumerate}
\item $T \equiv T_{\psi[0]} \in [T_{0}-\delta,T_{0}+\delta]$,
\item the solution $\psi :C_{T} \longrightarrow \mathbb{R}$ satisfies
\begin{align*}
(T-t)^{k-\frac{d}{2}} \Big \|  |\cdot|^{-1} \Big( \psi (t,\cdot) - \psi ^{T} (t, \cdot) \Big) \Big \|_{ \dot{H}^{k}(\mathbb{B}^{d+2}_{T-t} ) }  &\leq\delta (T-t)^{\epsilon} \\
(T-t)^{\ell+1-\frac{d}{2}} \Big \|  |\cdot|^{-1} \Big( \partial_t \psi (t,\cdot) - \partial_t \psi ^{T} (t, \cdot) \Big) \Big \|_{ \dot{H}^{\ell}(\mathbb{B}^{d+2}_{T-t} ) }  &\leq \delta (T-t)^{\epsilon}
\end{align*}
for all $k=0,1,2, \dots, \frac{d+3}{2}$ and $\ell=0,1,2\dots,\frac{d+1}{2}$.
\end{enumerate}
\end{theorem}

\begin{remark}
Note that the normalizing factors on the left-hand sides appear naturally and reflect the behavior of the self-similar solution $\psi^T$ in the respective homogeneous Sobolev norms,
i.e.,
\begin{align*}
  \||\cdot|^{-1}\psi^T(t,\cdot)\|_{\dot H^k(\mathbb B^{d+2}_{T-t})}&=\left \||\cdot|^{-1} f_0 \left(\frac{|\cdot|}{T-t} \right)  \right \|_{ \dot{H}^{k}( \mathbb{B}^{d+2}_{T-t} ) } =(T-t)^{\frac{d}{2}-k} \||\cdot|^{-1} f_0 \left(|\cdot| \right)  \|_{ \dot{H}^{k}( \mathbb{B}^{d+2}_{1} ) }
  \end{align*}
  and
  \begin{align*}
\||\cdot|^{-1}\partial_t \psi^T(t,\cdot)\|_{\dot H^\ell(\mathbb B^{d+2}_{T-t})}&=(T-t)^{-2}\left \|f_0' \left(\frac{|\cdot|}{T-t} \right)  \right \|_{ \dot{H}^{\ell}( \mathbb{B}^{d+2}_{T-t} ) } \\
& =(T-t)^{\frac{d}{2}-\ell-1} \| f_0' \left(|\cdot| \right)  \|_{ \dot{H}^{\ell}( \mathbb{B}^{d+2}_{1} ) }  .
\end{align*}
\end{remark}

\subsection{Related results}
The question of singularity formation for the wave maps equation attracted a lot of interest in the recent past, in particular in the energy-critical case $d=2$.
Bizo\'n-Chmaj-Tabor \cite{BizChmTab01} were the first to provide numerical evidence for the existence of blowup for critical wave maps with $\mathbb S^2$ target.
Rigorous constructions of blowup solutions for this model are due to Krieger-Schlag-Tataru \cite{KriSchTat08}, Rodnianski-Sterbenz \cite{RodSte10}, and Rapha\"el-Rodnianski \cite{RapRod12}. Struwe \cite{Str03} showed that blowup for equivariant critical wave maps takes place via shrinking of a harmonic map.
This result was considerably generalized to the nonequivariant setting by Sterbenz-Tataru \cite{SteTat10a, SteTat10b}, see also Krieger-Schlag \cite{KriSch12} for a different approach to the large-data problem and e.g.~\cite{CotKenLawSch15a, CotKenLawSch15b, Cot15,
CanKri15, Sha16, LawOh16} for more recent results on blowup and large-data global existence.

The energy-supercritical regime $d\geq 3$ is less understood. The small-data theory at minimal regularity is due to Shatah-Tahvildar-Zadeh \cite{ShaTah94} in the equivariant setting whereas Tataru \cite{Tat98, Tat01} and Tao \cite{Tao01a, Tao01b} treat the general case, see also \cite{KlaRod01, ShaStr02, NahSteUhl03, Kri03, Tat05}.
Self-similar blowup solutions were found by Shatah \cite{Sha88}, Turok-Spergel \cite{TurSpe90}, Cazenave-Shatah-Tahvildar-Zadeh \cite{CazShaTah98}, and Bizo\'n-Biernat \cite{BizBie15}.
The stability of self-similar blowup was investigated numerically in \cite{BizChmTab00, BizBie15, BieBizMal16} and proved rigorously in \cite{Don11, DonSchAic12, CosDonXia16, CosDonGlo16} in the case $d=3$. Furthermore, Dodson-Lawrie \cite{DodLaw15} proved that solutions with bounded critical norm scatter.

Finally, concerning the method, we remark that our proof relies on the techniques developed in the series of papers \cite{Don11, DonSchAic12, DonSch12, DonSch14, Don14, DonSch16, DonSch16a}.
However, we would like to emphasize that the present paper is not just a straightforward continuation of these works. In fact, new interesting issues arise, e.g.~in the spectral theory part, see Proposition \ref{projection} below.

\section{Radial wave equation in similarity coordinates}
\label{sec:sim}

\noindent
To start our analysis, we rewrite the initial value problem $\eqref{cauchy}$ as an abstract Cauchy problem in a Hilbert space. First, we rescale the variable $\psi \equiv \psi (t,r)$ and switch to similarity coordinates. Then, we linearize around the rescaled blowup solution and derive the evolution problem satisfied by the perturbation.
\subsection{Rescaled variables}
We define
\begin{align*}
\chi _{1} (t,r) := \frac{T-t}{r} \psi (t,r),\qquad \chi _{2} (t,r) := \frac{(T-t)^2}{r} \psi _{t} (t,r).
\end{align*}
Using the fact that $\psi$ is a solution to \eqref{cauchy}, we get
\begin{align*}
 \partial _{t} \chi _{1} (t,r) = &  - \frac{1}{T-t} \chi _{1} (t,r) +  \frac{1}{T-t} \chi _{2} (t,r), \\
 \partial _{t} \chi _{2} (t,r) =  & - \frac{2}{T-t} \chi _{2} (t,r) +  (T-t) \Delta ^{ \text{rad} }_{r,d} \chi _{1} (t,r) + \frac{2(T-t)}{r} \partial _{r} \chi _{1} (t,r) \\
& + (d-1) \frac{T-t}{r^2} \chi _{1} (t,r) - \frac{d-1}{2} (T-t)^2 \frac{ \sin \big( \frac{2r}{T-t} \chi _{1} (t,r) \big) }{r^3}.
\end{align*}
We introduce similarity coordinates
\begin{align*}
\mu: C_{T} \longrightarrow  \mathcal{C} ,~~ (t,r) \longmapsto \mu (t, r) =(\tau,\rho) := \Big( \log \Big ( \frac{T}{T-t} \Big ), \frac{r}{T-t} \Big ),
\end{align*}
which map the backward light-cone $C_T$ to the cylinder $\mathcal{C}:=(0,+ \infty) \times  [0,1]$.
By the chain rule, the derivatives transform according to
\begin{align*}
\partial _{t} = \frac{e^{\tau}}{T} (\partial _{\tau} + \rho \partial _{\rho}),~ \partial _{r} = \frac{e^{\tau}}{T} \partial _{\rho},~ \partial ^{2}_{r} = \frac{e^{2 \tau}}{T^{2}} \partial ^{2}_{\rho},~ \Delta ^{ \text{rad} }_{r,d} = \frac{e^{2 \tau}}{T^{2}} \Delta ^{ \text{rad} }_{\rho,d}.
\end{align*}
Finally, setting
\begin{align*}
\psi _{j} (\tau, \rho) := \chi _{j} (t(\tau,\rho),r(\tau,\rho) ) = \chi _{j} (T(1-e^{-\tau}), T \rho e^{-\tau}),
\end{align*}
for $j=1,2$, we obtain the system
\begin{align} \label{free}
& \begin{pmatrix}
   \partial _{\tau}  \psi _{1}  (\tau,\rho)  \\
   \partial _{\tau}  \psi _{2} (\tau,\rho)
 \end{pmatrix}
  =
  \begin{pmatrix}
   - \psi _{1}  (\tau,\rho) + \psi _{2} (\tau,\rho)   - \rho \partial _{\rho} \psi _{1}  (\tau,\rho)    \\
   \Delta ^{ \text{rad} }_{\rho,d+2} \psi _{1} (\tau,\rho) - \rho \partial _{\rho} \psi _{2} (\tau,\rho)  - 2 \psi _{2} (\tau,\rho)
   \end{pmatrix}  \\ \nonumber
& \quad  \quad   \quad   \quad  \quad   \quad  -  \frac{d-1}{ 2 \rho ^3}
   \begin{pmatrix}
   0 \\
   \sin ( 2 \rho \psi _{1} (\tau,\rho)  ) -2 \rho \psi _{1} (\tau,\rho)
   \end{pmatrix},
\end{align}
for $(\tau,\rho) \in \mathcal{C}$.
 Note that the linear part is the free operator of the $(d+2)-$dimensional wave equation in similarity coordinates and the nonlinearity is perfectly smooth. Furthermore, the initial data transform according to
\begin{align*}
\begin{pmatrix}
   \psi _{1}   (0,\rho) \\
   \psi _{2}   (0,\rho)
 \end{pmatrix}
=   \frac{1}{\rho}
\begin{pmatrix}
 f(T\rho) \\
  T g (T\rho)
 \end{pmatrix}
=  \frac{1}{\rho}
\begin{pmatrix}
 \psi^{T_{0}}(0,T\rho) \\
 T\partial_0 \psi ^{T_{0}} (0,T\rho)
\end{pmatrix}+
\frac{1}{\rho}
\begin{pmatrix}
F(T\rho) \\
T G(T\rho)
\end{pmatrix},
\end{align*}
for all $\rho \in [0,1]$. Here, $T_{0}>0$ is a fixed parameter and
\begin{align*}
& \psi ^{T_{0}} (0,T\rho) =  2 \arctan \left(  \frac{T }{T_{0}} \frac{\rho}{ \sqrt{d-2} }    \right),\quad \rho \equiv \rho (t,r):=\frac{r}{T-t}, \\
& F:=f-\psi ^{T_{0}}(0,\cdot), \quad G:=g-\partial_0 \psi^{T_{0}}(0,\cdot).
\end{align*}
We emphasize that the only trace of the parameter $T$ is in the initial data.

\subsection{Perturbations of the rescaled blowup solution} We linearize around the rescaled blowup solution and use the initial value problem for $(\psi_{1},\psi_{2})^{T}$ to obtain an initial value problem for the perturbation as an abstract Cauchy problem in a Hilbert space. For notational convenience we set
\begin{align*}
 \mathbf{ \Psi } (\tau)  (\rho) :=
 \begin{pmatrix}
   \psi _{1} (\tau, \rho)  \\
   \psi _{2} (\tau, \rho)
 \end{pmatrix} .
\end{align*}
The blowup solution is given by
\begin{align*}
\mathbf{  \Psi } ^{ \text{res}  } (\tau) (\rho)
=
\begin{pmatrix}
   \frac{T-t}{r} \psi ^{T} (t,r)  \\
    \frac{(T-t)^2}{r} \psi _{t}^{T} (t,r)
 \end{pmatrix} \Bigg | _{
 (t,r)=\mu ^{-1} (\tau,\rho)
} =
\begin{pmatrix}
   \frac{1}{\rho} f_0 (\rho) \\
   f_0'(\rho)
 \end{pmatrix},
\end{align*}
i.e., it is static.
We linearize around $\mathbf{\Psi}^{\text{res}}$ by inserting the ansatz $\mathbf{ \Psi }=  \mathbf{  \Psi } ^{ \text{res} } + \mathbf{  \Phi } $ into \eqref{free}.
For brevity we write
\begin{align*}
\eta (x):= \sin(2x) - 2x,\quad x \in \mathbb{R}
\end{align*}
and use Taylor's theorem to expand the nonlinearity around
$\frac{1}{\rho}f_0(\rho)$. We get
\begin{align*}
\sin \left(  2 \rho \psi _{1}  \right) - 2 \rho \psi _{1} & = \eta \left( \rho \psi _{1} \right) = \eta \left( f_0 + \rho \phi _{1} \right) = \eta \left( f_0  \right) + \eta ^{\prime} \left( f_0  \right) \rho \phi _{1} + N(\rho \phi _{1}),
\end{align*}
where, by definition,
\begin{align*}
N (\rho \phi _{1} ) := \eta( f_0  + \rho \phi _{1}) - \eta ( f_0 ) - \eta ^{\prime} ( f_0 ) \rho \phi _{1}.
\end{align*}
We plug the ansatz and the Taylor expansion into Eq.~\eqref{free} which yields the abstract evolution equation
\begin{align} \label{Evolution}
\left\{
	\begin{array}{ll}
	\partial _{\tau} \mathbf{ \Phi } (\tau) = \widetilde{\mathbf{  L } } \big( \mathbf{ \Phi } (\tau) \big) + \mathbf{ N } \big(  \mathbf{ \Phi }( \tau ) \big ), & \mbox{for } \tau \in  (0,+\infty)  \\
	\mathbf{ \Phi } (0)= \mathbf{U}(\mathbf{v},T),
         \end{array}
\right.
\end{align}
for the perturbation
\begin{align*}
\mathbf{ \Phi } (\tau) (\rho)=
\begin{pmatrix}
    \phi _{1} (\tau, \rho)   \\
  \phi _{2} (\tau, \rho)
 \end{pmatrix}
 =
\begin{pmatrix}
    \psi _{1} (\tau, \rho) -\frac{1}{\rho} f_0 (\rho)   \\
  \psi _{2} (\tau, \rho) - f_0' (\rho)
 \end{pmatrix}
\end{align*}
where
\begin{align}
&\widetilde{  \mathbf{  L } }  := \widetilde{ \mathbf{  L  } }_{0}  + \mathbf{  L ^{\prime} }, \label{1} \\
& \widetilde{ \mathbf{  L  } }_{0} \mathbf u (\rho):=
\begin{pmatrix}
- \rho u_1'(\rho)-u_1(\rho)+ u_2(\rho)  \\
  \Delta _{\rho,d+2}^{ \text{rad} } u_1(\rho) - \rho u_2'(\rho) - 2 u_{2}(\rho)
 \end{pmatrix},  \label{2}  \\
&  \mathbf{  L ^{\prime} } \mathbf u(\rho):=
\begin{pmatrix}
   0 \\
   - \frac{d-1}{2} \frac{\eta' ( f_0(\rho) ) }{\rho ^2} u_{1}(\rho),
 \end{pmatrix}, \label{3} \\
&  \mathbf{  N }(\mathbf u) (\rho):=
 \begin{pmatrix}
   0 \\
   - \frac{d-1}{2} \frac{N( \rho u_{1}(\rho) )}{\rho ^3}
 \end{pmatrix},  \label{4}
\end{align}
for $\mathbf u=(u_1,u_2)$ and
\begin{align*}
 \eta'(f_0(\rho))=2  \cos (2f_0(\rho)) -2  = -16(d-2) \frac{\rho ^2}{ \left( \rho^2 +d-2 \right)^2  }.
\end{align*}
Furthermore, the initial data are given by
\begin{align}
\label{5}
\mathbf \Phi(0)(\rho)=\mathbf U(\mathbf v,T)(\rho)=
\left (\begin{array}{c}
\frac{1}{\rho}f_0(\frac{T}{T_0}\rho) \\
\frac{T^2}{T_0^2}f_0'(\frac{T}{T_0}\rho) \end{array} \right )
-\left ( \begin{array}{c}
\frac{1}{\rho}f_0(\rho) \\ f_0'(\rho) \end{array} \right )
+\mathbf V(\mathbf v,T)(\rho)
\end{align}
where
\[  \mathbf{ V } (\mathbf{v},T) (\rho):=
 \begin{pmatrix}
   \frac{1}{\rho} F (T \rho)   \\
   \frac{T}{\rho} G (T \rho)
 \end{pmatrix},~ \mathbf{v} :=
\begin{pmatrix}
   F   \\
   G
 \end{pmatrix} .
 \]

\subsection{Strong light-cone solutions}
To proceed, we need to define what it means to be a solution to the evolution problem $\eqref{Evolution}$. We introduce the Hilbert space
\begin{align*}
\mathcal{H} := H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} ) \times H_{\text{rad}}^{\frac{d+1}{2}} (\mathbb{B}^{d+2} ).
\end{align*}
Below we prove that the closure of the operator $\widetilde{\mathbf L}$, augmented with a suitable domain, generates a semigroup $\mathbf S(\tau)$ on $\mathcal H$.
This allows us to formulate $\eqref{Evolution}$ as an abstract integral equation via Duhamel's formula,
\begin{align} \label{Duhamel}
\Phi(\tau)= \mathbf{S}(\tau) \mathbf{U}(\mathbf{v},T)+ \int _{0}^{\tau} \mathbf{S}(\tau - s) \mathbf{N} \big( \Phi (s) \big)  ds.
\end{align}
Eq.~\eqref{Duhamel} yields a natural notion of strong solutions in light-cones.
\begin{definition} \label{def}
We say that $\psi:C_{T} \longrightarrow \mathbb{R}$ is a solution to $\eqref{cauchy}$ if the corresponding $\Phi:[0,\infty) \longrightarrow \mathcal{H}$ belongs to $C \big( [0,\infty);\mathcal{H} \big)$ and satisfies $\eqref{Duhamel}$ for all $\tau \ge 0$.
\end{definition}

\section{Proof of the theorem}
\subsection{Notation} Throughout we denote by $\sigma(\mathbf{L}),~\sigma_{p}(\mathbf{L})$ and $\sigma_{e}(\mathbf{L})$ the spectrum, point spectrum, and essential spectrum, respectively, of a linear operator $\mathbf{L}$. Furthermore, we write $\mathbf{R}_{\mathbf{L}}(\lambda):=\left(\lambda - \mathbf{L} \right)^{-1}$, $\lambda \in \rho(\mathbf{L})$, for the resolvent operator where $\rho (\mathbf{L}):=\mathbb{C} \setminus \sigma(\mathbf{L})$ stands for the resolvent set. As usual, $a \lesssim b$ means $a \leq cb$ for an absolute, strictly positive constant $c$ which may change from line to line. Similarly, we write $a \simeq b$ if $a \lesssim b$ and $b \lesssim a$.
\subsection{Functional setting}
In the following we consider radial Sobolev functions $\hat{u} : \mathbb{B}_{R}^{d+2} \rightarrow \mathbb{C}$, that is, $\hat{u} (\xi)=u(|\xi|)$ for all $\xi \in \mathbb{B}_{R}^{d+2}$ where
$u:(0,R) \rightarrow \mathbb{C}$. In particular, we define
\begin{align*}
u \in
 H_{\text{rad}}^{m} ( \mathbb{B}_{R}^{d+2} ) \iff \hat{u} \in H^{m}  (\mathbb{B}_{R}^{d+2} ) := W^{m,2}  (\mathbb{B}_{R}^{d+2} ).
 \end{align*}
The function space $H_{\text{rad}}^{m} (\mathbb{B}_R^{d+2} )$ becomes a Banach space endowed with the norm
\begin{align*}
\| u \|_{{H^{m}_{\text{rad}}} ( \mathbb{B}_{R}^{d+2} )} =  \| \hat{u}  \|_{ {H}^{m}  (\mathbb{B}_{R}^{d+2} )}.
\end{align*}
From now, we shall not distinguish between $u(|\cdot|)$ and $\hat{u}$. In addition, we introduce the Hilbert space
\begin{align} \label{H}
\mathcal{H} := H_{\text{rad}}^{m} (\mathbb{B}^{d+2} ) \times H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ),\quad m \equiv m_{d}:=\frac{d+3}{2}
\end{align}
associated with the induced norm
\begin{align*}
\| \mathbf{u} \|^{2}  =  \left \| (u_{1},u_{2})  \right \|^{2}  := \| u_{1} \|_{H_{\text{rad}}^{m} (\mathbb{B}^{d+2} )}^{2} + \| u_{2} \|_{H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) }^{2}.
\end{align*}
\subsection{Well-posedness of the linearized problem}
We start with the study of the linearized problem and we convince ourselves that it is well-posed. Recall that the linear operator is given by $\eqref{1}$. To proceed, we follow \cite{DonSch16} and define the domain of the free part by
\begin{align*}
\mathcal{D} (\widetilde{ \mathbf{L} }_{0})  := \Big \{ \mathbf{u} \in  C^{\infty}(0,1)^2 \cap \mathcal H: w_{2} \in C^{2}\left([0,1]\right),~w_{1} \in C^{3} \left([0,1] \right),~ w_{1}^{\prime \prime} (0)=0 \Big \},
\end{align*}
where, for all $\rho \in [0,1]$ and $j=1,2$,
\begin{align*}
w_{j} (\rho) := D_{d+2} u_{j} (\rho) := \Big (  \frac{1}{\rho} \frac{d}{d\rho}  \Big )^{ \frac{d-1}{2} } \big( \rho ^{d} u_{j}(\rho) \big) = \sum _{n=0}^{ \frac{d-1}{2} } c _{n} \rho^{n+1} u_{j}^{(n)} (\rho),
\end{align*}
for some strictly positive constants $c_{n}~ (n=0,1,\dots,\frac{d-1}{2})$. Note that the density of $C^{\infty} ( \overline{ \mathbb{B}^{d+2} } )$ in $H^{m}(\mathbb{B}^{d+2})$ implies the density of
\begin{align*}
\big( C_{\text{even}}^{\infty} [0,1] \big)^2 := \Big \{  \mathbf{u} \in \big( C^{\infty} [0,1] \big)^2:~~\mathbf{u}^{(2k+1)}(0)=0,~~k=0,1,2,\dots \Big \}  \subset \mathcal{D} ( \widetilde{ \mathbf{L} }_{0})
\end{align*}
in $\mathcal{H}$ which in turn proves the density of $\mathcal{D} ( \widetilde{ \mathbf{L} }_{0})$ in $\mathcal{H}$. In other words, $\overline{ \mathcal{D} ( \widetilde{ \mathbf{L} }_{0}) } = \mathcal{H}$ and $\widetilde{ \mathbf{L} }_{0}$ is densely defined.

\begin{prop} \label{growthestimatelinear}
The operator $\widetilde{ \mathbf{L} }_{0}: \mathcal{D}(\widetilde{ \mathbf{L} }_{0}) \subset \mathcal{H} \longrightarrow \mathcal{H} $ is closable and its closure $ \mathbf{L} _{0} : \mathcal{D}(\mathbf{L} _{0}) \subset \mathcal{H} \longrightarrow \mathcal{H} $ generates a strongly continuous one-parameter semigroup $( \mathbf{S}_{0}(\tau) )_{\tau \ge 0}$ of bounded operators on $\mathcal{H}$ satisfying the growth estimate
\begin{align} \label{growth}
\|  \mathbf{S} _{0} (\tau) \| \leq M e^{-\tau}
\end{align}
for all $\tau\geq 0$ and some constant $M\geq 1$.
In addition, the operator $\mathbf{L}:= \mathbf{L}_{0} + \mathbf{L}^{\prime}: \mathcal{D}(\mathbf{L}) \subset \mathcal{H} \longrightarrow \mathcal{H},~\mathcal{D}(\mathbf{L}) =\mathcal{D}(\mathbf{L}_{0})$, is the generator of a strongly continuous semigroup $( \mathbf{S}(\tau) )_{\tau \ge 0}$ on $\mathcal H$
and $\mathbf L': \mathcal H\to \mathcal H$ is compact.
\end{prop}
\begin{proof}
The fact that $\widetilde{ \mathbf{L} }_{0}$ is closable and its closure generates a semigroup satisfying the growth estimate $\eqref{growth}$ follows from Proposition 4.9 in \cite{DonSch16} by replacing $d$ in \cite{DonSch16} with $d+2$ and setting $p=3$.  It remains to apply the Bounded Perturbation Theorem to show that $\mathbf{L}:=\mathbf{L}_{0} + \mathbf{L}^{\prime}$ is the generator of a strongly continuous semigroup $( \mathbf{S}(\tau) )_{\tau \ge 0}$. In fact, we prove that $\mathbf{L}^{\prime}: \mathcal{H} \longrightarrow \mathcal{H}$, defined in $\eqref{3}$,  is compact. We pick an arbitrary sequence $(\mathbf{u}_{n})_{n \in \mathbb{N}} \subseteq \mathcal{H}$ that is uniformly bounded. By Lemma 4.2 in \cite{DonSch16}, $(D_{d+2} u_{1,n} )_{n \in \mathbb{N}}$ is uniformly bounded in $H^{2} (0,1)$ and the compactness of the Sobolev embedding $H^{2}(0,1) \xhookrightarrow{}  H^{1}(0,1)$ implies the existence of a subsequence, again denoted by $(D_{d+2} u_{1,n} )_{n \in \mathbb{N}}$, which is Cauchy in $H^{1}(0,1)$. Hence, for any $n,m \in \mathbb{N}$ sufficiently large, we get
\begin{align*}
\| \mathbf{L}^{\prime} \mathbf{u}_{n} - \mathbf{L}^{\prime}
  \mathbf{u}_{m} \| & \lesssim  \left \| \frac{ \eta'
\circ f_0 }{|\cdot|^2} \right \|_{W^{1,\infty}(0,1)} \| D_{d+2} u_{1,n} - D_{d+2} u_{1,m} \|_{H^{1}(0,1)} \\
&  \simeq  \left \| \frac{  1}{ (|\cdot|^{2} + d-2)^2  }\right \|_{W^{1,\infty}(0,1)} \|D_{d+2} u_{1,n} - D_{d+2} u_{1,m} \|_{H^{1}(0,1)} \\
& \simeq   \| D_{d+2} u_{1,n} - D_{d+2} u_{1,m} \|_{H^{1}(0,1)},
\end{align*}
which shows that $(\mathbf{L}^{\prime} \mathbf{u}_{n})_{n \in \mathbb{N}}$ is Cauchy in $\mathcal{H}$. This proves that $\mathbf{L}^{\prime}$ is compact.
\end{proof}
\subsection{The spectrum of the free operator} \label{O1}
We can use the previous decay estimate for the semigroup $( \mathbf{S}_{0}(\tau) )_{\tau \ge 0}$ to locate the spectrum of the free operator $\mathbf L_0$. Indeed, by \cite{EngNag00}, p.~55, Theorem 1.10, we immediately infer
\begin{equation}
\label{spectrumLo}
 \sigma(\mathbf L_0)\subseteq \{\lambda\in \mathbb C: \text{Re}\lambda\leq -1\}.
 \end{equation}

\subsection{The spectrum of the full linear operator} Next, we need to derive a suitable growth estimate for the semigroup $\mathbf{S}(\tau)$ and therefore turn our attention to the spectrum of the operator $\mathbf{L}$. To begin with, we consider the point spectrum.
\begin{prop} \label{OhaioProp}
We have
\begin{align*}
\sigma_p (\mathbf{L}) \subseteq \{ \lambda \in \mathbb{C}:~~\mathrm{Re} \lambda <0 \} \cup \{1\}.
\end{align*}
\end{prop}
\begin{proof}
We argue by contradiction and assume there exists a $\lambda \in \sigma_p(\mathbf{L}) \setminus \{ 1\}$ with $\mathrm{Re}\lambda \geq 0$. The latter means that there exists an element $\mathbf{u}=(u_{1},u_{2}) \in \mathcal{D} (\mathbf{L}) \setminus \{ 0 \}$ such that $\mathbf{u} \in$ $\ker(\lambda-\mathbf{L})$. A straightforward calculation shows that the spectral equation $(\lambda-\mathbf{L}) \mathbf{u} =0$ implies
\begin{small}
\begin{align*}
\big (1-\rho ^2 \big) u_{1}^{\prime \prime} (\rho) + \Bigg( \frac{d+1}{\rho} - 2(\lambda +2) \rho \Bigg)u_{1}^{\prime} (\rho)- \Bigg( (\lambda + 1) (\lambda +2)+ \frac{d-1}{2} V(\rho) \Bigg )  u_{1} (\rho) =0,
\end{align*}
\end{small}
for $\rho \in (0,1)$, where
\begin{align*}
V(\rho):=\frac{\eta'(f_0(\rho))}{\rho ^2} = \frac{-16(d-2)}{(\rho^2+d-2 )^2}.
\end{align*}
Since $\mathbf u\in \mathcal H$, we see that $u_{1}$ must lie in $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})$. To proceed, we set $v_{1}(\rho):=\rho u_{1}(\rho)$. A straightforward computation implies that $v_{1}$ solves the second order ordinary differential equation
\begin{align}
\label{eq:specode}
\big (1-\rho ^2 \big) v_{1}^{\prime \prime} (\rho) + \Bigg( \frac{d-1}{\rho} - 2(\lambda +1) \rho \Bigg)v_{1}^{\prime} (\rho)- \Bigg( \lambda  (\lambda +1)+ \frac{d-1}{2}\hat{V}(\rho) \Bigg )  v_{1} (\rho) =0,
\end{align}
for $\rho \in (0,1)$, where
\begin{align*}
\hat{V}(\rho):= 2\frac{ \rho ^4 -6(d-2)\rho ^2 +(d-2)^2 }{\rho ^2 ( \rho^2 +d-2 )^2   }.
\end{align*}
We remark that this is the spectral equation studied in \cite{CosDonXia16, CosDonGlo16}.
Since all coefficients in \eqref{eq:specode} are smooth functions in $(0,1)$, we immediately get the a priori regularity $v_{1} \in C^{\infty}(0,1)$.
We claim that $v_1\in C^\infty[0,1]$. To prove this, we employ Frobenius' method. The point $\rho =0$ is a regular singularity with Frobenius indices $s_{1}=1$ and $s_{2}=-(d-1)$. Therefore, by Frobenius theory, there exists a solution of the form
\begin{align*}
v^{1}_{1} (\rho)=\rho \sum _{i=0}^{\infty} x_{i} \rho ^{i} = \sum _{i=0}^{\infty} x_{i} \rho ^{i+1},
\end{align*}
which is analytic locally around $\rho=0$. Moreover, since $s_{1}-s_{2}=d \in \mathbb{N}_{\text{odd}}$, there exists a second linearly independent solution of the form
\begin{align*}
v^{2}_{1} (\rho) &=C \log(\rho) v^{1}_{1} (\rho)  + \rho ^{-(d-1)} \sum _{i=0}^{\infty} y_{i} \rho ^{i}
\end{align*}
for some constant $C\in \mathbb C$ and $y_0=1$.
However, $v^{2}_{1}(\rho)/\rho$ does not lie in the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ due to the strong singularity in the second term, no matter the value of the constant $C$. Consequently, $v_1$ must be a multiple of $v_1^1$ and we infer $v_1\in C^\infty[0,1)$. Similarly, the point $\rho =1$ is a regular singularity with Frobenius indices
$s_{1}=0$ and $s_{2}=\frac{d-1}{2}-\lambda$.
Now we need to distinguish different cases. If $\frac{d-1}{2} -\lambda \notin \mathbb{Z}$, we have two linearly independent solutions of the form
\begin{align*}
& v_{1} ^{1}(\rho) =\sum_{i=0}^\infty x_i (1-\rho)^i, \\
& v_{1} ^{2}(\rho)=(1-\rho)^{\frac{d-1}{2}-\lambda}\sum_{i=0}^\infty y_i (1-\rho)^i
\end{align*}
with $x_0=y_0=1$. The solution $v_{1} ^{2}(\rho)/\rho$ does not belong to the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ and thus, $v_1\in C^\infty[0,1]$. In the case
 $\frac{d-1}{2} - \lambda:=k \in \mathbb{N}_{0}$, we have two fundamental solutions of the form
\begin{align*}
 v_{1} ^{1}(\rho)&=(1-\rho)^{k}\sum_{i=0}^\infty x_i(1-\rho)^i,\qquad x_0=1 \\
 v_{1} ^{2}(\rho)&= \sum_{i=0}^\infty y_i (1-\rho)^i+C\log (1-\rho)v_1^1(\rho),\qquad y_0=1
\end{align*}
near $\rho= 1$. By assumption, $\mathrm{Re}\lambda\geq 0$ and thus, $k \leq \frac{d-1}{2}$. Hence, $v_1^2(\rho)/\rho$ does not lie in the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ unless $C=0$ and we conclude $v_1\in C^\infty[0,1]$.
Finally, if $\frac{d-1}{2}-\lambda=:-k$ is a negative integer, the fundamental system around $\rho=1$ has the form
\begin{align*}
v_1^1(\rho)&=\sum_{i=0}^\infty x_i(1-\rho)^i \\
v_1^2(\rho)&=C\log(1-\rho)v_1^1(\rho)+(1-\rho)^{-k}\sum_{i=0}^\infty y_i (1-\rho)^i
\end{align*}
with $x_0=y_0=1$.
Again, $v_1^2(\rho)/\rho$ does not belong to $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$ and we infer $v_1\in C^\infty[0,1]$ also in this case.
In summary, we have found a nontrivial solution $v_1\in C^\infty[0,1]$ to Eq.~\eqref{eq:specode} with $\mathrm{Re}\lambda \geq 0$, $\lambda\not=1$, but this contradicts \cite{CosDonXia16, CosDonGlo16}.
\end{proof}

The fact that $\mathbf L'$ is compact implies that the result on the point spectrum from Proposition \ref{OhaioProp} is already sufficient to obtain the same information on the full spectrum.

\begin{cor}
\label{cor:spec}
We have
 \begin{align*}
\sigma (\mathbf{L}) \subseteq \{ \lambda \in \mathbb{C}:~~\mathrm{Re} \lambda <0 \} \cup \{1\}.
\end{align*}
\end{cor}

\begin{proof}
Suppose there exists a $\lambda\in \sigma(\mathbf L)\setminus \{1\}$ with $\mathrm{Re}\lambda\geq 0$.
Then $\lambda\notin\sigma(\mathbf L_0)$ and thus, $\mathbf R_{\mathbf L_0}(\lambda)$ exists.
From the identity $\lambda-\mathbf L=[1-\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)](\lambda-\mathbf L_0)$ we see that $1\in \sigma(\mathbf L'\mathbf R_{\mathbf L_0}(\lambda))$. Since $\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)$ is compact, it follows that $1\in \sigma_p(\mathbf L'\mathbf R_{\mathbf L_0}(\lambda))$ and thus, there exists a nontrivial $\mathbf f \in \mathcal H$ such that $[1-\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)]\mathbf f=0$.
Consequently, $\mathbf u:=\mathbf R_{\mathbf L_0}(\lambda)\mathbf f\not= 0$
satisfies $(\lambda-\mathbf L)\mathbf u=0$ and thus, $\lambda\in \sigma_p(\mathbf L)$.
This contradicts Proposition \ref{OhaioProp}.
\end{proof}

Next, we provide a uniform bound on the resolvent.
To this end, we define
\begin{align*}
\Omega _{\epsilon,R} :=  \{  \lambda \in \mathbb{C}:~~~\text{Re}\lambda \geq -1+\epsilon, |\lambda|\geq R \}
\end{align*}
for $\epsilon, R >0$.

\begin{prop} \label{O2}
Let $\epsilon >0$. Then there exist constants $R_{\epsilon}, C_{\epsilon}>0$ such that the resolvent $\mathbf{R}_{\mathbf{L}}$ exists on $\Omega _{\epsilon,R_\epsilon}$ and satisfies
\begin{align*}
\| \mathbf{R} _{\mathbf{L} } (\lambda)\| \leq C_{\epsilon}
\end{align*}
for all $\lambda\in \Omega_{\epsilon,R_\epsilon}$.
\end{prop}

\begin{proof}
Fix $\epsilon>0$ and take $\lambda\in \Omega_{\epsilon,R}$ for an arbitrary $R>0$.
Then $\lambda\in \rho(\mathbf L_0)$ and the identity $(\lambda-\mathbf L)=[1-\mathbf L' \mathbf R_{\mathbf L_0}(\lambda)](\lambda-\mathbf L_0)$ shows that
$\mathbf R_{\mathbf L}(\lambda)$ exists if and only if
$1-\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)$ is invertible.
By a Neumann series argument this is the case if $\|\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)\|<1$.

 To prove smallness of $\mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L}_{0}} (\lambda)$, we recall the definition of $\mathbf L'$, Eq.~\eqref{3},
\begin{align*}
\mathbf{  L ^{\prime} } \mathbf{u} (\rho)=
\begin{pmatrix}
   0 \\
   - \frac{d-1}{2} V(\rho )u _{1} (\rho)
 \end{pmatrix},
\quad V(\rho)=\frac{\eta'(f_0(\rho))}{\rho ^2} = \frac{-16(d-2)}{(\rho^2+d-2 )^2}.
\end{align*}
Let $\mathbf u=\mathbf R_{\mathbf L_0}(\lambda)\mathbf f$ or, equivalently,
$(\lambda - \mathbf{L}_{0}) \mathbf{u} =\mathbf{f}$. The latter equation implies
\begin{align*}
(\lambda +1) u_{1} (\rho )  =u_{2} (\rho)- \rho u_{1}^{\prime} (\rho) + f_{1} (\rho).
\end{align*}
Now we use Lemma 4.1 from \cite{DonSch16} and $\|V^{(k)}\|_{L^\infty(0,1)}\lesssim 1$ for all $k\in \{0,1,\dots,m-1\}$ to obtain
\begin{align*}
 |\lambda +1| \| \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) \mathbf{f}  \| & = |\lambda +1|  \| \mathbf{L}^{\prime} \mathbf{u} \|  \simeq \big \| V \big( u_{2} - (\cdot) u_{1}^{\prime} +f_{1} \big) \big \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } \\
&  \lesssim \| u_{2} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) }  + \| (\cdot) u_{1}^{\prime} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) }  + \| f_{1} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } \\
&  \lesssim \| u_{2} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) }  + \|  u_{1} \|_{ H_{\text{rad}}^{m} (\mathbb{B}^{d+2} ) }  + \| f_{1} \|_{ H_{\text{rad}}^{m-1} (\mathbb{B}^{d+2} ) } \\
& \simeq  \| \mathbf{u} \| + \| \mathbf{f} \|  \lesssim \Big( \frac{1}{\text{Re}\lambda +1}  + 1 \Big) \| \mathbf{f} \| \\
&\lesssim \|\mathbf f\|,
\end{align*}
where we have used the bound
\[ \|\mathbf u\|=\|\mathbf R_{\mathbf L_0}(\lambda)\mathbf f\|\leq \frac{1}{\mathrm {Re}\lambda+1}\|\mathbf f\| \]
which follows from semigroup theory, see \cite{EngNag00}, p.~55, Theorem 1.10.
In other words,
\begin{align*}
\| \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda)   \| \lesssim \frac{1}{ |\lambda +1| } \leq \frac{1}{|\lambda| -1} \leq \frac{1}{R-1}
\end{align*}
and by choosing $R$ sufficiently large, we can achieve the desired $\|\mathbf L'\mathbf R_{\mathbf L_0}(\lambda)\|<1$.
As a consequence, $[1-  \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) ]^{-1}$
exists and we obtain the bound
\begin{align*}
\| \mathbf{R}_{\mathbf{L}} (\lambda) \| & = \| \mathbf{R}_{\mathbf{L}_{0}} (\lambda) [1-  \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) ]^{-1} \| \\
& \leq  \| \mathbf{R}_{\mathbf{L}_{0}} (\lambda) \| \| [1-  \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) ]^{-1} \| \\
& \leq  \| \mathbf{R}_{\mathbf{L}_{0}} (\lambda) \| \sum _{i=0}^{\infty} \| \mathbf{L}^{\prime} \mathbf{R}_{\mathbf{L} _{0}} (\lambda) \|^{i} \\
& \leq C_\epsilon.
\end{align*}
\end{proof}

\subsection{The eigenspace of the isolated eigenvalue} In this section, we convince ourselves that the eigenspace of the isolated eigenvalue $\lambda=1$ for the full linear operator $\mathbf{L}$ is spanned by
\begin{align} \label{g}
\mathbf{g} (\rho):=
\begin{pmatrix}
   g_{1} (\rho) \\
   g_{2} (\rho)
 \end{pmatrix}
 =
 \begin{pmatrix}
   \frac{1}{\rho ^2 +d-2} \\
    \frac{2(d-2)}{(\rho ^2 + d-2)^2}
 \end{pmatrix},
~~\rho \in [0,1].
 \end{align}
Consequently, we are looking for all $\mathbf{u}=(u_{1},u_{2}) \in \mathcal{D} (\mathbf{L}) \setminus \{ 0 \}$ such that $\mathbf{u} \in \ker(1-\mathbf{L})$. A straightforward calculation shows that the spectral equation $(1-\mathbf{L}) \mathbf{u} =0$ is equivalent to the following system of ordinary differential equations,
\begin{align*}
 \begin{cases}
      u_{2} (\rho)= \rho u_{1}^{\prime} (\rho)+ 2 u_{1} (\rho), & \\
      \big (1-\rho ^2 \big ) u_{1}^{\prime \prime} (\rho) + \Big( \frac{d+1}{\rho} - 6\rho \Big)u_{1}^{\prime} (\rho)- \Big( 6+ \frac{d-1}{2} \frac{\eta'(f_0(\rho))}{\rho ^2} \Big )  u_{1} (\rho) =0,  \
     \end{cases}
\end{align*}
for $\rho \in (0,1)$. One can verify that a fundamental system of the second equation is given by
\begin{align*}
\Big  \{ \frac{1}{\rho ^2 + d-2},~~\frac{Q_{d-1}(\rho)}{\rho ^d (\rho ^2 + d-2)} \Big \}
\end{align*}
where $Q_{d-1}$ is a polynomial of degree $d-1$ with non-vanishing constant term. We can write the general solution for the second equation as
\begin{align*}
u_{1} (\rho) = C_{1}  \frac{1}{\rho ^2 + d-2} + C_{2} \frac{Q_{d-1}(\rho)}{\rho ^d (\rho ^2 + d-2)}.
\end{align*}
We must ensure that $\mathbf{u} \in \mathcal{D}( \mathbf{L})$ which in particular implies that $u_{1}$ must lie in the Sobolev space $H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2} )$. This requirement yields $C_{2}=0$ which in turn gives $\mathbf u \in \langle \mathbf{g} \rangle$. In conclusion,
\begin{align} \label{ker}
\text{ker}(1-\mathbf{L}) = \langle \mathbf{g} \rangle,
\end{align}
as initially claimed.
\subsection{Time evolution for the linearized problem}
We now focus on the time evolution for the linearized problem \eqref{Evolution}. Due to the presence of the eigenvalue $\lambda =1$, there exists a one dimensional subspace $\langle \mathbf{g} \rangle$ of initial data for which the solution grows exponentially in time. We call this subspace the unstable space. On the other hand, initial data from the stable subspace lead to solutions that decay exponentially in time. As we will show now, this time evolution estimates can be established using semigroup theory together with the previous results on the spectrum of the linear operators $\mathbf{L}_{0}$ and $\mathbf{L}$. To make this rigorous, we follow \cite{DonSch16} and use the fact that the unstable eigenvalue $\lambda=1$ is isolated to introduce a (non-orthogonal) projection $\mathbf{P}$. This projection decomposes the Hilbert space of initial data $\mathcal{H}$ into the stable and the unstable space.
Most importantly, we must ensure that $\langle \mathbf{g}\rangle$ is the only unstable direction in $\mathcal{H}$. This is the key statement of the following proposition and it is equivalent to the fact that
the algebraic multiplicity of the isolated eigenvalue $\lambda =1$,
\begin{align*}
m_{a} (\lambda =1):=\mathrm{rank}\, \mathbf{P} =\dim\mathrm{rg}\, \mathbf{P},
\end{align*}
is equal to one. We denote by $\mathcal{B}(\mathcal{H})$ the set of bounded operators from $\mathcal{H}$ to itself and prove the following result.
\begin{prop} \label{projection}
There exists a projection
\begin{align*}
\mathbf{P} \in \mathcal{B} (\mathcal{H}),\quad \mathbf{P}: \mathcal{H} \longrightarrow \langle \mathbf{g} \rangle,
\end{align*}
which commutes with the semigroup $\big( \mathbf{S}(\tau) \big) _{\tau \ge 0}$. In addition, we have
\begin{equation}
\mathbf{S}(\tau) \mathbf{P} \mathbf{f} = e ^{\tau} \mathbf{P} \mathbf {f}, \label{semigroup1}
\end{equation}
and there exist constants $C, \epsilon>0$ such that
\begin{equation}
 \|  (1-\mathbf{P}) \mathbf{S}(\tau) \mathbf{f} \| \leq C e^{-\epsilon \tau } \| (1- \mathbf{P}) \mathbf{f} \|,\label{semigroup2}
\end{equation}
for all $\mathbf {f} \in \mathcal{H}$ and $\tau \geq 0$.
\end{prop}

\begin{proof}
We argue along the lines of \cite{DonSch16}. Since the eigenvalue $\lambda =1$ is isolated, we can define the spectral projection
\begin{align*}
\mathbf{P}: \mathcal{H} \longrightarrow  \mathcal{H}, \quad \mathbf{P} := \frac{1}{2 \pi i} \int _{\gamma} \mathbf{R}_{\mathbf{L}} (\mu) d \mu,
\end{align*}
where $\gamma : [0,2\pi] \longrightarrow \mathbb{C}$ is a positively orientated circle around $\lambda =1$ with radius so small that $\gamma \big( [0,2 \pi] \big) \subseteq \rho (\mathbf{L})$, see e.g.~\cite{Kat95}. The projection $\mathbf P$ commutes with the operator $\mathbf{L}$ and thus with the semigroup $\mathbf S(\tau)$. Moreover, $\mathbf{P}$ decomposes the Hilbert space as $\mathcal{H} =\mathcal M \oplus \mathcal N$, where $\mathcal M:=\rg \mathbf P$ and $\mathcal N:=\rg(1-\mathbf P)=\ker \mathbf{P}$. Most importantly, the operator $\mathbf{L}$ is decomposed accordingly into the parts $\mathbf{L}_{\mathcal M}$ and $\mathbf{L}_{\mathcal N}$ on $\mathcal M$ and $\mathcal N$, respectively. The spectra of these operators are given by
\begin{align} \label{spectrum}
\sigma \left( \mathbf L_{\mathcal N} \right ) = \sigma (\mathbf{L}) \setminus \{1\},\qquad \sigma \left( \mathbf{L}_{\mathcal M} \right ) = \{1\}.
\end{align}
We refer the reader to \cite{Kat95} for these standard results.

To proceed, we break down the proof into the following steps: \\ \\
Step 1: We prove that $\rank\mathbf{P}:=\dim\rg\mathbf{P}<+\infty$. We argue by contradiction and assume that $\rank\mathbf{P}=+\infty$. Using \cite{Kat95}, p.~239, Theorem 5.28, the fact that $\mathbf{L}^{\prime}$ is compact (see Proposition $\ref{growthestimatelinear}$), and the fact that the essential spectrum is stable under compact perturbations (\cite{Kat95}, p.~244, Theorem 5.35), we obtain
\begin{align*}
\mathrm{rank}\,\mathbf{P} = +\infty  \Longrightarrow 1 \in \sigma _{e} (\mathbf{L}) = \sigma _{e} (\mathbf{L} -\mathbf{L}^{\prime})=\sigma _{e}(\mathbf{L}_{0}) \subseteq \sigma (\mathbf{L}_{0}).
\end{align*}
This contradicts \eqref{spectrumLo}.
\\ \\
Step 2: We prove that $\langle \mathbf{g} \rangle=\mathrm{rg}\,\mathbf{P}$. It suffices to show $\mathrm{rg}\,\mathbf{P} \subseteq \langle \mathbf{g} \rangle$ since the reverse inclusion follows from the abstract theory. From Step 1, the operator $1-\mathbf{L}_{\mathcal M}$ acts on the finite-dimensional Hilbert space $\mathcal M=\rg \mathbf P$ and, from $\eqref{spectrum}$, $\lambda =0$ is its only spectral point. Hence, $1-\mathbf{L}_{\mathcal M}$ is nilpotent, i.e., there exists a $k\in \mathbb N$ such that
\begin{align*}
\big(  1-\mathbf{L}_{\mathcal M}  \big)^{k}  \mathbf{u}= 0
\end{align*}
for all $\mathbf{u} \in \mathrm{rg}\, \mathbf{P}$
and we assume $k$ to be minimal.
Recall $\eqref{ker}$ to see that the claim follows immediately if $k=1$. We proceed by contradiction and assume that $k\geq 2$. Then, there exists a nontrivial function $\mathbf{u} \in \rg \mathbf{P} \subseteq \mathcal{D}( \mathbf{L})$ such that
$(1-\mathbf L_{\mathcal M})\mathbf u$ is nonzero and belongs to $\ker(1-\mathbf L_{\mathcal M})\subseteq \ker(1-\mathbf L)=\langle\mathbf g\rangle$.
This means that $\mathbf{ u } \in \rg\mathbf{P} \subseteq \mathcal{D} (\mathbf{L})$ satisfies $(1- \mathbf{ L }) \mathbf{ u } = \alpha \mathbf{ g }$, for some $\alpha \in \mathbb{ C }\setminus \{0\}$. Without loss of generality we set $\alpha=-1$ and a straightforward computation shows that the first component of $\mathbf u$ solves the second order differential equation
\begin{align*}
\left(1-\rho ^2\right) u_{1}^{\prime \prime} (\rho) + \left( \frac{d+1}{\rho} -6 \rho \right ) u_{1} ^{\prime} (\rho) - \left ( 6 + \frac{d-1}{2} \frac{\eta'(f_0(\rho) ) }{\rho ^2} \right ) u_{1} (\rho) =  G(\rho),
\end{align*}
for $\rho \in (0,1)$, where
\begin{align*}
G(\rho):= \frac{\rho ^2 +5(d-2)}{ (\rho ^2 + d-2)^2},~~\rho \in [0,1].
\end{align*}
In order to find the general solution to this equation, recall $\eqref{g}$ to see that
\begin{align*}
\hat{u}_{1} (\rho):= g_{1} (\rho) = \frac{1}{\rho ^2 +d-2},~~~\rho \in (0,1)
\end{align*}
is a particular solution to the homogeneous equation
\begin{align*}
\left(1-\rho ^2 \right) u_{1}^{\prime \prime} (\rho) + \left( \frac{d+1}{\rho} -6 \rho \right ) u_{1} ^{\prime} (\rho) - \left ( 6 + \frac{d-1}{2} \frac{\eta'(f_0(\rho) ) }{\rho ^2} \right ) u_{1} (\rho) = 0.
\end{align*}
To find another linearly independent solution, we use the Wronskian
\begin{align*}
\mathcal{W} (\rho) := (1-\rho ^2)^{\frac{d-5}{2} } \rho ^{-d-1}
\end{align*}
to obtain
\begin{align*}
\hat{u}_{2} (\rho) := \hat{u}_{1} (\rho) \int _{\rho _1}^{\rho}  (1- x ^2)^{\frac{d-5}{2}} x^{-d-1} (x^2 + d-2)^2 dx,
\end{align*}
for some constant $\rho_{1} \in (0,1)$ and for all $\rho \in (0,1)$.
Note that we have the expansion
\[ \hat u_2(\rho)=\rho^{-d}\sum_{j=0}^\infty a_j\rho^j,\quad a_0\not= 0 \]
near $\rho=0$. Furthermore, if $d\geq 5$, $\hat u_2\in C^\infty(0,1]$ and we choose $\rho_1=1$ which yields the expansion
\[ \hat u_2(\rho)=(1-\rho)^{\frac{d-3}{2}}\sum_{j=0}^\infty b_j (1-\rho)^j,\qquad b_0\not= 0 \]
near $\rho=1$.
For $d=3$, we set $\rho_1=\frac12$ and the expansion of $\hat u_2$ near $\rho=1$ contains a term $\log(1-\rho)$.
We invoke the variation of constants formula to see that $u_1$ can be expressed as
\begin{align*}
u_{1} (\rho) & = c_{1}   \hat{u}_{1} (\rho) + c_{2} \hat{u}_{2} (\rho) \\
& + \hat{u}_{2} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2}  } } dy  - \hat{u}_{1} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{2}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2}  } } dy,
\end{align*}
for some constants $c_{1}, c_{2} \in \mathbb{C}$ and for all $\rho \in (0,1)$.
The fact that $u_1\in H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})$ implies
$c_2=0$ and we are left with
\begin{equation}
\label{eq:expru1}
u_{1} (\rho)  = c_{1}   \hat{u}_{1} (\rho)
 + \hat{u}_{2} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2}  } } dy  - \hat{u}_{1} (\rho) \int _{0}^{\rho} \frac{ \hat{u}_{2}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2}  } } dy.
\end{equation}
If $d=3$, $\hat u_2(\rho)\simeq \log(1-\rho)$ near $\rho=1$ and thus, the last term in Eq.~\eqref{eq:expru1} stays bounded as $\rho\to 1-$ whereas the second term diverges unless
\[ \int_0^1  \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1-y^2)^{ \frac{d-3}{2}  } } dy=0, \]
which, however, is impossible since the integrand is strictly positive on $(0,1)$.
This contradicts $u_1\in H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})$ and we arrive at the desired $k=1$.

Next, we focus on $d\geq 5$, where the last term in Eq.~\eqref{eq:expru1} is smooth on $[0,1]$. To analyze the second term, we set
\begin{align}
\label{def:Id}
\mathcal{I} _{d}(\rho) :=  \hat{u}_{2} (\rho) \int _{0}^{\rho} \frac{ F_{d}(y) }{(1-y)^{ \frac{d-3}{2}  } } dy, \quad \text{~~~~~}   ~ F_{d}(y):= \frac{ \hat{u}_{1}(y)G(y)y^{d+1} }{(1+y)^{\frac{d-3}{2}}}=\frac{y^{d+1}(y^2+5(d-2))}{(1+y)^{\frac{d-3}{2}}(y^2+d-2)^3}.
\end{align}
Note that $F_{5}(1)\not= 0$ and thus, the expansion of $\mathcal I_5(\rho)$ near $\rho=1$ contains a term of the form $(1-\rho)\log(1-\rho)$.
Consequently, $\mathcal I_5''\notin L^2(\frac12,1)$ and this is a contradiction to $u_1\in H^4_{\mathrm{rad}}(\mathbb B^7)$.
The general case is postponed to the appendix (Proposition \ref{prop:Id}) where it is shown that the function $\mathcal I_d$ is not analytic at $\rho=1$. This implies that the expansion of $\mathcal{I}_d(\rho)$ near $\rho =1$ contains a term $(1-\rho)^{\frac{d-3}{2}}\log(1-\rho)$ which again contradicts $u_1\in H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})$.\\ \\
Step 3: Finally, we prove the estimates $\eqref{semigroup1}$ and $\eqref{semigroup2}$ for the semigroup. First, note that $\eqref{semigroup1}$ follows immediately from the facts that $\lambda =1$ is an eigenvalue of $\mathbf{L}$ with eigenfunction $\mathbf{g}$ and $\rg\mathbf{P}=\langle \mathbf{g} \rangle$.
Furthermore, from Corollary \ref{cor:spec} and Proposition \ref{O2} we infer the existence of $C,\epsilon>0$ such that
\[ \|\mathbf R_{\mathbf L}(\lambda)(1-\mathbf P)\|\leq C \]
for all $\lambda \in \mathbb C$ with $\mathrm{Re}\lambda\geq -2\epsilon$.
Consequently, the Gearhart-Pr\"uss Theorem, see \cite{EngNag00}, p.~302, Theorem 1.11, yields the bound \eqref{semigroup2}.
\end{proof}

\subsection{Estimates for the nonlinearity}
The aim of this section is to establish a Lipschitz-type estimate for the nonlinearity. Recall that the nonlinear term in $\eqref{Evolution}$ is given by
\begin{align*}
\mathbf{  N } ( \mathbf{u}  ) (\rho) =
  \begin{pmatrix}
   0  \\
   \hat N (\rho, u_{1} (\rho))
 \end{pmatrix}
 :=
 \begin{pmatrix}
   0 \\
   - \frac{d-1}{2} \frac{N( \rho u _{1} (\rho) )}{\rho ^3}
 \end{pmatrix}.
\end{align*}
To begin with, we claim that
\begin{align*}
&\hat N(\rho,u_1(\rho)) \\
&=4 (d-1) u_{1}^{2} (\rho) \int _{0}^{1}  \int _{0}^{1}\int _{0}^{1} \cos \left( 2z \left( f_0(\rho) +xy \rho u_{1} (\rho) \right) \right) \left( \frac{ f_0 (\rho)}{\rho} + xy u_{1} (\rho)\right) x dz dy dx. \\
\end{align*}
To see this, we use the fundamental theorem of calculus and the fact that $\eta ^{\prime \prime} (0)=0$ to write
\begin{align*}
N( \rho u _{1} (\rho) ) & = \eta( f_0 (\rho) + \rho u _{1} (\rho) ) - \eta ( f_0 (\rho)) - \eta ^{\prime} ( f_0 (\rho)) \rho u _{1} (\rho) \\
&= \int _{f_0 (\rho)}^{f_0(\rho) + \rho u_{1} (\rho)} \eta ^{\prime} (s) ds - \eta ^{\prime} ( f_0 (\rho)) \rho u _{1} (\rho) \\
&=\rho u_{1} (\rho) \int _{0}^{1} \eta ^{\prime} (f_0 (\rho) + x \rho u_{1} (\rho)) dx - \eta ^{\prime} ( f_0 (\rho)) \rho u _{1} (\rho) \\
&=\rho u_{1} (\rho)  \int _{0}^{1}\left( \eta ^{\prime} (f_0 (\rho) + x \rho u_{1} (\rho)) - \eta ^{\prime} ( f_0(\rho)) \right) dx \\
&=\rho u_{1} (\rho)  \int _{0}^{1}\left( \int _{f_0(\rho)}^{f_0 (\rho) + x\rho u_{1} (\rho)} \eta ^{\prime \prime} (s) ds \right) dx \\
&=\rho^{2} u_{1}^{2} (\rho)  \int _{0}^{1}x  \int _{0}^{1} \eta ^{\prime \prime} (f_0 (\rho) + xy \rho u_{1} (\rho)) dy dx \\
&=\rho^{2} u_{1}^{2} (\rho)  \int _{0}^{1}x  \int _{0}^{1} \int _{0}^{f_0(\rho) + xy \rho u_{1}(\rho)} \eta ^{\prime \prime \prime } (s) ds dy dx \\
&=\rho^{2} u_{1}^{2} (\rho)  \int _{0}^{1}x  \int _{0}^{1} \int _{0}^{1} \eta ^{\prime \prime \prime } \left(  (f_0(\rho) +xy \rho u_{1} (\rho))z \right) \left( f_0 (\rho) + xy \rho u_{1} (\rho)\right)  dz dy dx \\
&=\rho^{3} u_{1}^{2} (\rho)  \int _{0}^{1} x \int _{0}^{1} \int _{0}^{1} \eta ^{\prime \prime \prime } \left(  (f_0(\rho) +xy \rho u_{1} (\rho))z \right) \left(\frac{ f_0 (\rho)}{\rho} + xy u_{1} (\rho)\right)  dz dy dx. \\
\end{align*}
For later purposes, we note that the function
\begin{align*}
\hat N (\rho, \zeta)=4 (d-1)  \zeta ^{2}   \int _{0}^{1} \int _{0}^{1} \int _{0}^{1} \cos \left( 2z \left( f_0(\rho) +xy \rho  \zeta  \right) \right) \left( \frac{ f_0 (\rho)}{\rho} + xy  \zeta \right) x dz dy dx,
\end{align*}
defined for all $(\rho, \zeta) \in [0,1] \times \mathbb{R},$ is perfectly smooth in both variables since
\begin{align*}
\frac{f_0 (\rho)}{\rho} =  \frac{2}{\rho} \arctan \left ( \frac{\rho}{\sqrt{d-2}}  \right)
\end{align*}
is smooth at $\rho=0$.
Moreover, we define
\begin{align} \label{DefM}
M (\rho, \zeta) := \partial _{\zeta} \hat N (\rho, \zeta) = 4 (d-1) \left (  A(\rho, \zeta) + B(\rho, \zeta) +C(\rho, \zeta) +D(\rho, \zeta)  \right),
\end{align}
where
\begin{align*}
& A(\rho,\zeta):= 2 \frac{f_0(\rho)}{\rho} \zeta  \int _{0}^{1} \int_{0}^{1}  \int _{0}^{1}  \cos  \left( 2z \left( f_0 (\rho) +xy \rho  \zeta \right) \right) x  dz dy dx, \\
& B(\rho, \zeta):= -2 f_0(\rho) \zeta ^{2}  \int_{0}^{1} \int_{0}^{1} \int_{0}^{1}  \sin \left( 2z \left(f_0 (\rho) + xy \rho \zeta \right) \right)  x^{2} y z dz dy dx, \\
& C(\rho, \zeta):=  3  \zeta ^{2}  \int_{0}^{1} \int_{0}^{1} \int_{0}^{1}  \cos \left( 2z \left( f_0(\rho) + xy \rho \zeta \right) \right)  x^{2} y  dz dy dx, \\
& D(\rho, \zeta):=  -2 \rho \zeta ^{3}  \int_{0}^{1} \int_{0}^{1} \int_{0}^{1}  \sin \left( 2z \left( f_0 (\rho) + xy \rho \zeta \right) \right)  x^{3} y^{2}  zdz dy dx.
\end{align*}
We denote by $\mathcal{B}_{\delta} \subseteq \mathcal{H}$ the ball of radius $\delta$ in $\mathcal{H}$ centered at zero, i.e.,
\begin{align*}
\mathcal{B}_{\delta}:= \left \{\mathbf{u} \in \mathcal{H}:~\left \| \mathbf{u} \right \|= \left \| (u_{1},u_{2}) \right \|_{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) \times H_{\text{rad}}^{\frac{d+1}{2}} (\mathbb{B}^{d+2}) }
\leq \delta  \right \}.
\end{align*}
The main result of this section is the following Lipschitz-type estimate.

\begin{lemma}
Let $\delta>0$.
Then we have
\begin{align} \label{Lipschitz}
\big \|  \mathbf{N (u)} -  \mathbf{N(v)}\big \|  \lesssim  (\|  \mathbf{u}  \|  +\|  \mathbf{v} \| ) \|  \mathbf{u}-\mathbf{v} \|
\end{align}
for all $\mathbf{u}, \mathbf{v} \in \mathcal{B}_{\delta}$.
\end{lemma}

\begin{proof}
We start by fixing a $\delta>0$, we pick two elements $\mathbf{u}, \mathbf{v} \in \mathcal{B}_{\delta}$ and define the auxiliary function
\begin{align*}
\zeta (\sigma)(\rho)= \sigma u_{1}(\rho) + (1-\sigma) v_{1} (\rho),
\end{align*}
for $\rho \in (0,1)$ and $\sigma \in [0,1]$. The triangle inequality implies
\begin{align*}
\mathbf{u},\mathbf{v} \in \mathcal{B}_{\delta} \Longrightarrow \left \| u_{1} \right \|_{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})}  \leq \delta ,~\left \| v_{1} \right \|_{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})} \leq \delta
\Longrightarrow  \left  \|  \zeta (\sigma) \right  \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})  }  \leq  \delta,
\end{align*}
for all $\sigma \in [0,1]$. In other words,
\begin{align*}
\zeta (\sigma) \in \mathscr{B}_{\delta}:= \left \{f \in H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})  :~\left \| f \right \|_{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})  } \leq \delta  \right \},
\end{align*}
for all $\sigma \in [0,1]$. Now, we claim that to show $\eqref{Lipschitz}$, it suffices to establish the estimate
\begin{align} \label{M}
\left \| M(\cdot, f(\cdot) ) \right \|_{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})   } \lesssim \left \| f \right \|_{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})  }
\end{align}
for all $f \in \mathscr{B}_{\delta}$,
where $M$ is given by $\eqref{DefM}$. To see this, we use the algebra property
\begin{align*}
\| fg \|_{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \lesssim \| f \|_{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } \| g \|_{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) } ,
\end{align*}
which holds since $\frac{d+3}{2}>\frac{d+2}{2}$,
to estimate
\begin{align*}
\big \| \mathbf{N(u)} -\mathbf{N(v)} \big\|  &= \big \|  \hat N (\cdot, u_{1}(\cdot) ) - \hat N (\cdot , v_{1}(\cdot)) \big \| _{H_{\text{rad}}^{\frac{d+1}{2}} (\mathbb{B}^{d+2}) } \\
&\leq
\big \|  \hat N (\cdot, u_{1}(\cdot) ) - \hat N (\cdot , v_{1}(\cdot)) \big \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \\
&=\left \|  \int _{v_{1}(\cdot)}^{u_{1}(\cdot)}  \partial _{2} \hat N (\cdot, \zeta )  d \zeta  \right \|  _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2})}   \\
&=\left \|   \left( u_{1}(\cdot)  -  v_{1}(\cdot) \right) \int _{0}^{1} \partial _{2} \hat N (\cdot , \underbrace{  \sigma u_{1}(\cdot)+(1- \sigma) v_{1}(\cdot)}_{\zeta(\sigma) })  d \sigma \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }   \\
 \nonumber
&\lesssim \left \|   u_{1} - v_{1} \right \| _{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \left \|  \int _{0}^{1} \partial _{2} \hat N (\cdot , \zeta (\sigma) ) d \sigma \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \\
& \lesssim \left \|   u_{1} - v_{1} \right \| _{ H ^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }   \int _{0}^{1}  \left \|  M (\cdot , \zeta (\sigma)(\cdot) ) \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  d \sigma   \\
&  \lesssim  \left \| u_{1} - v_{1}  \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \int _{0}^{1}  \left \|  \zeta (\sigma)  \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }   d \sigma  \\
& \lesssim \left \| u_{1} - v_{1}  \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \int _{0}^{1} \left( \sigma  \left \|  u_{1} \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  +(1-\sigma) \left \|  v_{1}    \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \right)  d \sigma   \\
& \lesssim \left \| u_{1} - v_{1}  \right \| _{ H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \left( \left \|   u_{1} \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  + \left \|  v_{1}    \right \| _{H_{\text{rad}}^{\frac{d+3}{2}} (\mathbb{B}^{d+2}) }  \right) \\
&\lesssim   \left \| \mathbf{u} - \mathbf{v}  \right \| \left( \left \|   \mathbf{u} \right \|  + \left \|  \mathbf{v}   \right \| \right).
\end{align*}
It remains to prove $\eqref{M}$. To this end we use a simple extension argument (see e.g.~Lemmas B.1 and B.2 in \cite{DonSch16}) and Moser's inequality (\cite{Rau12}, p.~224, Theorem 6.4.1) to infer the existence of a smooth function $h: [0,\infty)\to [0,\infty)$ such that
\[ \|M(\cdot,f(\cdot))\|_{H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})}\leq h\left (\|f\|_{L^\infty(\mathbb B^{d+2})}\right )\|f\|_{H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})} \]
for all $f\in \mathscr B_\delta$.
By Sobolev embedding we have $\|f\|_{L^\infty(\mathbb B^{d+2})}\lesssim \|f\|_{H^{\frac{d+3}{2}}_\mathrm{rad}(\mathbb B^{d+2})}\leq \delta$
for all $f\in \mathscr B_\delta$ and \eqref{M} follows.
This concludes the proof.
\end{proof}

\subsection{The abstract nonlinear Cauchy problem} In this section, we focus on the existence and uniqueness of solutions to the Cauchy problem $\eqref{Evolution}$. In fact, by appealing to Definition $\ref{def}$, we consider the integral equation
\begin{align} \label{integralequation}
\Phi(\tau)= \mathbf{S}(\tau) \mathbf{u} + \int _{0}^{\tau} \mathbf{S}(\tau - s) \mathbf{N} \big( \Phi (s) \big)  ds,
\end{align}
for all $\tau \ge 0$ and $\mathbf u\in \mathcal H$. We introduce the Banach space
\begin{align*}
\mathcal{X}:= \{ \Phi \in C( [0,\infty);\mathcal{H}) : ~~\| \Phi \|_{\mathcal{X}} := \sup _{\tau >0} e^{\epsilon \tau} \| \Phi(\tau) \| < + \infty  \}
\end{align*}
with $\epsilon>0$ from Proposition \ref{projection}. Moreover, we denote by $\mathcal{X}_{\delta}$ the closed ball
\begin{align*}
\mathcal{X}_{\delta} :=\left \{ \Phi \in \mathcal{X}: \| \Phi \|_{\mathcal{X}} \leq \delta \right \} = \left \{ \Phi \in C( [0,\infty);\mathcal{H}): \| \Phi \| \leq \delta e^{-\epsilon \tau},~~\forall \tau >0 \right \}.
\end{align*}

In the following, we will only sketch the rest of the proof and discuss the main arguments since they are analogous to \cite{Don11, DonSch12, DonSch14, Don14, DonSch16}. To prove the main theorem, we would like to apply a fixed point argument to the integral equation $\eqref{integralequation}$. However, the exponential growth of the solution operator on the unstable subspace prevents from doing this directly. We overcome this obstruction by subtracting the correction term\footnote{All integrals here exist as Riemann integrals over continuous functions.}
\begin{align} \label{Correction}
\mathbf{C}(\Phi,\mathbf{u}) := \mathbf{P} \left(  \mathbf{u} + \int_{0}^{\infty} e^{-s} \mathbf{N} \big( \Phi (s) \big) ds \right)
\end{align}
from the initial data. Consequently, we consider the fixed point problem
\begin{align} \label{modified}
\Phi (\tau)= \mathbf{K} ( \Phi, \mathbf{u})(\tau)
\end{align}
where
\begin{align} \label{K}
\mathbf{K} (\Phi, \mathbf{u}) (\tau):=\mathbf{S} (\tau) [\mathbf{u} - \mathbf{C}(\Phi,\mathbf{u})] + \int_{0}^{\tau} \mathbf{S} (\tau - s) \mathbf{N} \big( \Phi(s) \big) ds.
\end{align}
This modification stabilizes the evolution as the following result shows.

\begin{theorem} \label{th2}
There exist constants $\delta,C>0$ such that for every $\mathbf{u} \in \mathcal{H}$ with $\| \mathbf{u} \| \leq \frac{\delta}{C}$, there exists a unique $\mathbf{\Phi} (\mathbf{u}) \in \mathcal{X}_{\delta}$ that satisfies
\begin{align*}
\mathbf{\Phi} (\mathbf{u}) = \mathbf{K} (\mathbf{\Phi} (\mathbf{u}),\mathbf{u}).
\end{align*}
In addition, $\mathbf{\Phi}(\mathbf u)$ is unique in the whole space $\mathcal X$ and the solution map $\mathbf{u} \mapsto \mathbf{\Phi }(\mathbf{u})$ is Lipschitz continuous.
\end{theorem}
\begin{proof}
The proof is based on a fixed point argument and the essential ingredient is the Lipschitz estimate \eqref{Lipschitz} for the nonlinearity. Although the proof coincides with the one of Theorem 4.13 in \cite{DonSch16}, we sketch the main points for the sake of completeness. We pick $\delta>0$ sufficiently small and fix $\mathbf{u} \in \mathcal{H}$ with $\| \mathbf{u} \| \leq \frac{\delta}{C}$, where $C>0$ is sufficiently large. First, note that the continuity of the map
\begin{align*}
\mathbf{K} (\Phi, \mathbf{u}): [0,\infty) \longrightarrow \mathcal{H},\quad \tau \longmapsto \mathbf{K} (\Phi, \mathbf{u})(\tau)
\end{align*}
follows immediately from the strong continuity of the semigroup $\left( \mathbf{S}(\tau) \right)_{\tau >0}$. Next, to show that $\mathbf K(\cdot,\mathbf u)$ maps $\mathcal X_\delta$ to itself,
we pick an arbitrary $\Phi \in \mathcal{X}_{\delta} $ and decompose the operator according to
\begin{align*}
\mathbf{K} ( \Phi, \mathbf{u})(\tau) = \mathbf{P} \mathbf{K} ( \Phi, \mathbf{u})(\tau) + (1-\mathbf{P} ) \mathbf{K} ( \Phi, \mathbf{u})(\tau).
\end{align*}
The Lipschitz bound \eqref{Lipschitz} implies
\begin{align*}
\left \|  \mathbf{N} \left(  \Phi (\tau) \right) \right \| \lesssim \delta^2 e^{-2 \epsilon \tau}
\end{align*}
and together with the time evolution estimates for the semigroup on the unstable and stable subspaces (see Proposition \ref{projection}), we get
\begin{align*}
 \left \|  \mathbf{P} \mathbf{K} \left( \Phi, \mathbf{u} \right) (\tau) \right \| \lesssim \delta ^{2} e^{-2 \epsilon  \tau}, \quad \left \| \left(1- \mathbf{P} \right) \mathbf{K} \left( \Phi, \mathbf{u} \right) (\tau) \right \| \lesssim (\tfrac{\delta}{C}+\delta^2) e^{-\epsilon  \tau} .
\end{align*}
Clearly, these estimates imply that $ \mathbf{K} ( \Phi, \mathbf{u}) \in \mathcal{X}_{\delta}$ for sufficiently small $\delta$ and sufficiently large $C>0$. Finally, we need to show the contraction property. To this end, we pick two elements $\Phi,\widetilde{\Phi} \in \mathcal{X}_{\delta}$.
As before, the Lipschitz estimate \eqref{Lipschitz} together with Proposition \ref{projection} imply
\begin{align*}
 \left \| \mathbf{P} \left( \mathbf{K} ( \Phi, \mathbf{u})(\tau) - \mathbf{K} ( \widetilde{ \Phi }, \mathbf{u})(\tau)  \right) \right \| &\lesssim  \delta e^{-\epsilon \tau} \left \| \Phi - \widetilde{\Phi} \right \|_{\mathcal X}, \\
 \left \| \left(1- \mathbf{P}\right) \left( \mathbf{K} ( \Phi, \mathbf{u})(\tau) - \mathbf{K} ( \widetilde{ \Phi }, \mathbf{u})(\tau)  \right) \right \| &\lesssim  \delta e^{-\epsilon\tau} \left \| \Phi - \widetilde{\Phi} \right \|_{\mathcal X}
\end{align*}
and by choosing $\delta$ sufficiently small we conclude
\begin{align*}
\left \| \mathbf{K} ( \Phi, \mathbf{u}) - \mathbf{K} ( \widetilde{ \Phi }, \mathbf{u})  \right \|_{\mathcal{X}} \leq \frac{1}{2}  \left \| \Phi - \widetilde{\Phi} \right \|_{\mathcal{X}}.
\end{align*}
Consequently, the claim follows by the contraction mapping principle. Uniqueness in the whole space $\mathcal X$ and the Lipschitz continuity of the solution map are routine and we omit the details.
\end{proof}

Now we turn to the particular initial data we prescribe. To this end, we define the space
\begin{align*}
\mathcal{H}^{R} := H_{\text{rad}}^{m} (\mathbb{B}_{R}^{d+2} ) \times H_{\text{rad}}^{m-1} (\mathbb{B}_{R}^{d+2}), \quad m\equiv m_{d} =\frac{d+3}{2}
\end{align*}
for $R>0$, endowed with the induced norm
\begin{align*}
\left \| \mathbf{w} \right \|_{\mathcal{H}^{R}}^{2} = \left \| (w_{1},w_{2}) \right \|_{\mathcal{H}^{R}}^{2}  =\left  \| w_{1}\right  \|_{ H_{\text{rad}}^{m} \left( \mathbb{B}^{d+2}_R \right) } +
\left   \| w_{2} \right \|_{ H_{\text{rad}}^{m-1} \left( \mathbb{B}^{d+2}_R \right) }.
\end{align*}
Recall the definition of the initial data operator $\mathbf U(\mathbf v, T)$ from Eq.~\eqref{5}.

\begin{lemma} \label{th1}
Fix $T_0>0$.
Let $\delta>0$ be sufficiently small and $\mathbf{v}$ with $|\cdot|^{-1} \mathbf{v} \in \mathcal{H}^{T_{0} + \delta}$. Then, the map
\begin{align*}
\mathbf{U} (\mathbf{v},\cdot):[T_{0}-\delta,T_{0}+\delta] \longrightarrow \mathcal{H},\quad T \longmapsto \mathbf{U} (\mathbf{v},T)
\end{align*}
is continuous. Furthermore, for all $T \in [T_{0} -\delta,T_{0}+\delta]$,
\begin{align*}
\big \| |\cdot|^{-1} \mathbf{v}\big \|_{\mathcal{H}^{T_{0}+\delta} } \leq \delta \Longrightarrow  \big \| \mathbf{U} (\mathbf{v},T)\big  \|  \lesssim \delta.
\end{align*}
\end{lemma}

\begin{proof}
The statements are straightforward consequences of the very definition of $\mathbf U(\mathbf v,T)$, the smoothness of $\frac{f_0(\rho)}{\rho}$, and the continuity of rescaling in Sobolev spaces. We omit the details.
\end{proof}

Finally, given $T_{0}>0$ and $\mathbf{v} \in \mathcal{H}^{T_{0} + \delta}$ with $\| |\cdot|^{-1} \mathbf{v} \|_{\mathcal{H}^{T_{0}+\delta} } \leq \frac{\delta}{M}$ for $\delta>0$ sufficiently small and $M>0$ sufficiently large, we apply Lemma \ref{th1} to see that $\mathbf{u}:=\mathbf{U} (\mathbf{v},T)$ satisfies the assumptions of Theorem \ref{th2} for all $T\in [T_0-\delta,T_0+\delta]$. Hence, for all $T \in [T_{0}-\delta,T_{0}+\delta]$, the map $\mathbf K(\cdot,\mathbf U(\mathbf v,T))$ has a fixed point $\Phi_{T}:= \mathbf{\Phi} (\mathbf U(\mathbf v,T)) \in \mathcal{X}_{\delta}$. In the last step we now argue that for each $\mathbf v$, there exists a particular $T_{\mathbf{v}} \in [T_{0}-\delta,T_{0}+\delta]$ that makes the correction term vanish, i.e., $\mathbf C(\Phi_{T_{\mathbf v}},\mathbf U(\mathbf v,T_{\mathbf v}))=0$. Since $\mathbf C$ has values in $\rg \mathbf P=\langle\mathbf g\rangle$, the latter is equivalent to
\begin{align}
\label{eq:Tv}
\exists T_{\mathbf{v}} \in [T_{0}-\delta,T_{0}+\delta]: \quad \Big< \mathbf{C} \left( \Phi _{T_{\mathbf{v}}},\mathbf{U} \left( \mathbf{v},T_{\mathbf{v}} \right) \right),\mathbf{g} \Big>_{\mathcal H} = 0.
\end{align}
The key observation now is that
\[ \partial_T \left . \left ( \begin{array}{c}
\frac{1}{\rho}f_0(\frac{T}{T_0}\rho) \\
\frac{T^2}{T_0^2}f_0'(\frac{T}{T_0}\rho) \end{array} \right )\right |_{T=T_0}=\frac{2\sqrt{d-2}}{T_0}\,\mathbf g(\rho) \]
and thus, we have the expansion
\[ \Big< \mathbf{C} \left( \Phi_T,\mathbf U (\mathbf{v},T ) \right),\mathbf{g} \Big>_{\mathcal H}=\frac{2\sqrt{d-2}}{T_0}\|\mathbf g\|^2(T-T_0)
+O((T-T_0)^2)+O(\tfrac{\delta}{M}T^0)+O(\delta^2T^0).
\]
Consequently, a simple fixed point argument proves \eqref{eq:Tv}, see \cite{DonSch16}, Theorem 4.15 for full details.
In summary, we arrive at the following result.

\begin{theorem} \label{correction}
Fix $T_0>0$. Then there exist $\delta,M >0$ such that for any $\mathbf{v}$ with
\[ \| |\cdot|^{-1} \mathbf{v} \|_{\mathcal{H}^{T_{0}+\delta} } \leq \frac{\delta}{M} \] there exists a $T \in [T_{0}-\delta,T_{0}+\delta]$ and a function $\Phi \in \mathcal{X}_{\delta}$ which satisfies
\begin{align}
 \label{eq:int}
\Phi (\tau) = \mathbf{S} (\tau) \mathbf{U} (\mathbf{v},T) + \int_{0}^{\tau} \mathbf{S} (\tau-s) \mathbf{N} \big( \Phi (s) \big) ds
\end{align}
for all $\tau \geq 0$. Furthermore, $\Phi$ is unique in $C \big( [0,\infty);\mathcal{H} \big)$.
\end{theorem}

\subsection{Proof of the main theorem} With the results of the previous section at hand, we can now prove the main theorem. Fix $T_0>0$ and suppose the radial initial data
$\psi [0]$ satisfy
\begin{align*}
\left  \| |\cdot|^{-1} \Big( \psi [0] -\psi^{T_{0}}[0]  \Big) \right \|_{ H^{\frac{d+3}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2}) \times H^{\frac{d+1}{2}} (\mathbb{B}_{T_{0}+\delta}^{d+2}  )} \leq \frac{\delta}{M}
\end{align*}
with $\delta,M>0$ from Theorem \ref{correction}.
We set $\mathbf v:=\psi[0]-\psi^{T_0}[0]$, cf.~Section \ref{sec:sim}.
Then we have
\begin{align*}
\left \| |\cdot|^{-1} \mathbf{v} \right \|_{\mathcal{H}^{T_{0}+\delta}}  = \left \| |\cdot|^{-1} \Big( \psi [0] -\psi^{T_{0}}[0] \Big)
\right \|_{\mathcal{H}^{T_{0}+\delta}} \leq \frac{\delta}{M}
\end{align*}
and Theorem $\ref{correction}$ yields the existence of
$T \in [T_{0}-\delta,T_{0}+\delta]$ such that Eq.~\eqref{eq:int} has a unique solution $\Phi \in \mathcal{X}$ that satisfies $\| \Phi (\tau) \| \leq \delta e^{-\epsilon \tau}$ for all $\tau\geq 0$.
By construction,
\[ \psi(t,r)=\psi^T(t,r)+\frac{r}{T-t}\phi_1\left (\log\frac{T}{T-t},\frac{r}{T-t}\right ) \]
is a solution to the original wave maps problem \eqref{cauchy}. Furthermore,
\[ \partial_t \psi(t,r)=\partial_t \psi^T(t,r)+\frac{r}{(T-t)^2}\phi_2 \left (\log\frac{T}{T-t},\frac{r}{T-t}\right ). \]
Consequently,
\begin{align*}
(T-t)^{k-\frac{d}{2}}&\left \||\cdot|^{-1}\left (\psi(t,\cdot)-\psi^T(t,\cdot)\right )
\right \|_{\dot H^k(\mathbb B^{d+2}_{T-t})}  \\
&=(T-t)^{k-\frac{d}{2}-1}\left \|\phi_1\left (\log\frac{T}{T-t},\frac{|\cdot|}{T-t}\right ) \right \|_{\dot H^k(\mathbb B^{d+2}_{T-t})} \\
&=\left \|\phi_1\left (\log\frac{T}{T-t},\cdot\right ) \right \|_{\dot H^k(\mathbb B^{d+2})}
\leq \left \|\Phi\left (\log\frac{T}{T-t}\right )\right \| \\
&\leq \delta (T-t)^\epsilon
\end{align*}
for all $t\in [0,T)$ and $k=0,1,2,\dots,\frac{d+3}{2}$.
Analogously,
\begin{align*}
(T-t)^{\ell-\frac{d}{2}+1}&\left \||\cdot|^{-1}\left (\partial_t \psi(t,\cdot)-\partial_t \psi^T(t,\cdot)\right )\right \|_{\dot H^\ell(\mathbb B^{d+2}_{T-t})} \\
&=(T-t)^{\ell-\frac{d}{2}-1}\left \|\phi_2\left (\log\frac{T}{T-t},\frac{|\cdot|}{T-t}\right ) \right \|_{\dot H^\ell(\mathbb B^{d+2}_{T-t})} \\
&=\left \|\phi_2\left (\log\frac{T}{T-t},\cdot\right ) \right \|_{\dot H^\ell(\mathbb B^{d+2})}
\leq \left \|\Phi\left (\log\frac{T}{T-t}\right )\right \| \\
&\leq \delta (T-t)^\epsilon
\end{align*}
for all $\ell=0,1,2,\dots,\frac{d+1}{2}$.