Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Construction and stability of blowup solutions for a non-variational
  semilinear parabolic system   We consider the following parabolic system whose nonlinearity has no gradient
structure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + |v|^{p-1}v,
\quad & \partial_t v = \mu \Delta v + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad
& v(\cdot, 0) = v_0, \end{array}\right. $$ in the whole space $\mathbb{R}^N$,
where $p, q > 1$ and $\mu > 0$. We show the existence of initial data such that
the corresponding solution to this system blows up in finite time $T(u_0, v_0)$
simultaneously in $u$ and $v$ only at one blowup point $a$, according to the
following asymptotic dynamics: $$\left\{\begin{array}{c} u(x,t)\sim
\Gamma\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log
(T-t)|}\right)\right]^{-\frac{(p + 1)}{pq - 1}},\\ v(x,t)\sim \gamma\left[(T-t)
\left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(q + 1)}{pq
- 1}}, \end{array}\right.$$ with $b = b(p,q,\mu) > 0$ and $(\Gamma, \gamma) =
(\Gamma(p,q), \gamma(p,q))$. The construction relies on the reduction of the
problem to a finite dimensional one and a topological argument based on the
index theory to conclude. Two major difficulties arise in the proof: the
linearized operator around the profile is not self-adjoint even in the case
$\mu = 1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the
problem. In the last section, through a geometrical interpretation of
quantities of blowup parameters whose dimension is equal to the dimension of
the finite dimensional problem, we are able to show the stability of these
blowup behaviors with respect to perturbations in initial data.

33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn