Geodesic
Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder
Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 700-711 
G. V. Grishina. Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder. Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 700-711. http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a7/
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     author = {G. V. Grishina},
     title = {Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder},
     journal = {Matemati\v{c}eskie zametki},
     pages = {700--711},
     year = {1997},
     volume = {62},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\sigma$.

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