Abstract
Throughout this paper, we mainly consider the parabolic p-Laplacian equation with a weighted absorption u_{t}-operatorname{div} (|nabla u|^{p-2}nabla u )=-lambda |x|^{alpha} {chi}_{{u>0}}u^{-beta} in a bounded domain Omegasubseteqmathbb{R}^{n} (ngeq1) with Lipschitz continuous boundary subject to homogeneous Dirichlet boundary condition. Here lambda>0 and alpha>-n are parameters, and betain(0,1) is a given constant. Under the assumptions u_{0}in W_{0}^{1,p}(Omega)cap L^{infty}(Omega), u_{0}geq0 a.e. in Ω, we can establish conditions of local and global in time existence of nonnegative solutions, and show that every global solution completely quenches in finite time a.e. in Ω. Moreover, we give some numerical experiments to illustrate the theoretical results.
Highlights
1 Introduction In this paper, we mainly study the following initial-boundary value problem for the pLaplacian equation
We suppose that u0 satisfies the following assumptions: u0 ≥ 0 a.e. in and u0 ∈ W01,p( ) ∩ L∞( )
Galaktionov and Vazquez [20] systematically studied the properties of several equations, such as complete or incomplete blowup and extinction
Summary
We mainly study the following initial-boundary value problem for the pLaplacian equation. Galaktionov and Vazquez [20] systematically studied the properties of several equations, such as complete or incomplete blowup and extinction. They studied the problem ut = um + uq, with m > 1, q > 1. Assumed that the initial function u0 = u0(r) is strictly positive, bounded away from zero and has an inverse bell-shaped form They studied another kind of singularity of the equation ut = um – u–q, with m > 1, q > 0, and proved that extinction is complete if and only if q + m ≤ 0. Giacomoni et al [22] showed that problem (1.1) has a global in time bounded weak solution.
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