Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blow-up for wave equation with the scale-invariant damping and combined
  nonlinearities   In this article, we study the blow-up of the damped wave equation in the
\textit{scale-invariant case} and in the presence of two nonlinearities. More
precisely, we consider the following equation: $$u_{tt}-\Delta
u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), $$
with small initial data.\\ For $\mu < \frac{N(q-1)}{2}$ and $\mu \in (0,
\mu_*)$, where $\mu_*>0$ is depending on the nonlinearties' powers and the
space dimension ($\mu_*$ satisfies $(q-1)\left((N+2\mu_*-1)p-2\right) = 4$), we
prove that the wave equation, in this case, behaves like the one without
dissipation ($\mu =0$). Our result completes the previous studies in the case
where the dissipation is given by $\frac{\mu}{(1+t)^\beta}u_t; \ \beta >1$
(\cite{LT3}), where, contrary to what we obtain in the present work, the effect
of the damping is not significant in the dynamics. Interestingly, in our case,
the influence of the damping term $\frac{\mu}{1+t}u_t$ is important.

15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga