Advances in Differential Equations

An antimaximum principle for a degenerate parabolic problem

Juan Francisco Padial, Peter Takáč, and Lourdes Tello

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 15, Number 7/8 (2010), 601-648.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854621

Mathematical Reviews number (MathSciNet)
MR2650583

Zentralblatt MATH identifier
1195.35237

Subjects
Primary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 35J20: Variational methods for second-order elliptic equations 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05] 47J30: Variational methods [See also 58Exx]

Citation

Padial, Juan Francisco; Takáč, Peter; Tello, Lourdes. An antimaximum principle for a degenerate parabolic problem. Adv. Differential Equations 15 (2010), no. 7/8, 601--648. https://projecteuclid.org/euclid.ade/1355854621.


Export citation