The Cauchy--Dirichlet Problem for the Porous Media Equation in Cone-Like Domains
Daniele Andreucci, Anatoli Fedorovich Tedeev · SIAM Journal on Mathematical Analysis · 2014
We investigate the behavior for large times of nonnegative solutions to the Dirichlet problem in cone-like domains for the porous media equation. We obtain optimal estimates for the sup norm of the solution and for the size of its support. We also consider the case where a damping term depending on the space gradient of the solution appears. In this case we also identify the critical behavior of the damping term discriminating between decay to zero of a suitable moment of the solution as $t\to+\infty$, and stabilization of the moment to a positive constant.