An antimaximum principle for a degenerate parabolic problem

Abstract

We obtain an anti\-maximum principle for the following quasilinear parabolic problem: \begin{equation*} \tag*{\rm (P)} \left\{ \begin{alignedat}{2} \frac{\partial u}{\partial t} - \Delta_p u & = \lambda\, |u|^{p-2} u + f(x,t), & & \quad (x,t)\in \Omega\times (0,T); \\ u(x,t)& = 0, & & \quad (x,t)\in \partial\Omega\times (0,T); \\ u(x,0)& = u_0(x), & & \quad x\in \Omega, \end{alignedat} \right. \end{equation*} which involves the $p$-Laplace operator $\Delta_p u\equiv \mathrm{div}(|\nabla u|^{p-2}\nabla u)$ (with Dirichlet boundary conditions, $1 < p < \infty$) and a spectral parameter $\lambda\in \mathbb{R}^N$ taking values near the first eigenvalue $\lambda_1$ of $-\Delta_p$. We show that {\it any\/} weak solution $u\colon \Omega\times [0,T)\to \mathbb{R}$ of problem (P) (suitably defined in a standard way) eventually becomes positive for all $x\in \Omega$ and all times $t\geq T_{+}$, provided, for instance, $f(x,t)\geq \underline{f}(x) >0$ for some function $\underline{f}\in L^\infty(\Omega)$, $u_0\in W_0^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, and $\lambda_1 < \lambda < \lambda_1 + \delta$. Here, the ``key'' constants $\delta\equiv \delta(\underline{f}, u_0) >0$ and $T_{+}\equiv T_{+}(f,u_0)\in (0,T)$ depend on $f$ (or $\underline{f}$ only) and $u_0$. In particular, a solution $u$ eventually becomes positive even if the initial data $u_0$ are "arbitrarily" negative as long as they are smooth enough.