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

Abstract

We study the decay rate to $0$ as $t\rightarrow + \infty$ 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 $t^{-\frac{1}{(p-1)}}$.