Abstract:
We study equations of the form
$$
\begin{gathered}
u_{tt}+Lu+b(x,t)u_t=a(x,t)|u|^{\sigma-1}u,
\\-u_t+Lu=a(x,t)|u|^{\sigma-1}u,
\end{gathered}
$$
where $L$ is a uniformly elliptic operator and $0<\sigma<1$. In the half-cylinder $\Pi_{0,\infty}=\{(x,t):x=(x_1,\dots,x_n)\in \Omega,\ t>0\}$, where $\Omega\subset\mathbb R^n$ is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition for $x\in\partial\Omega $ and $t>0$. We find conditions under which these solutions have compact support and prove statements of the following type: $u(x,t)=o(t^\gamma)$ as $t\to\infty$, then there exists a $T$ such that $u(x,t)\equiv0$ for $t>T$. In this case $\gamma$ depends on the coefficients of the equation and on the exponent $\sigma$.
Fulltext:
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