Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder

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 $\sigma$.