Rate of decay to 0 of the solutions to a nonlinear parabolic equation

Abstract

We study the decay rate to $0$ as $t\to +\infty$ of the solution of the equation ${\psi }_t-\Delta \psi +|\psi {|}^{p-1}\psi =0$ with Neumann boundary conditions in a bounded smooth open connected domain of ${\mathbb{R}}^n$ where $p>1$. We show that either $\psi (t,\cdot )$ converges to $0$ exponentially fast or $\psi (t,\cdot )$ decreases like $t^{-\frac{1}{(p-1)}}$.