Extinction in finite time of solutions to the nonlinear diffusion equations involving p ( x , t ) -Laplacian operator

Peng Sun,Chunling Cao,Mingji Liu

doi:10.1186/1687-2770-2013-259

Abstract

The aim of this paper is to study the extinction of solutions of the initial boundary value problem for . The authors discuss how the relations of and dimension N affect the properties of extinction in finite time. MSC:35K35, 35K65, 35B40.

Highlights

In this paper, we consider the following nonlinear degenerate parabolic equation:⎧ ⎪⎪⎨ut = div(|∇u|p(x,t)– ∇u) + b(x, t)|u|q – a u, (x, t) ∈ × (, T) = QT,⎪⎪⎩uu((xx, t) =, ) = u (x),(x, t) ∈ ∂ × (, T) = T, ( . )x∈, where QT = ×
The aim of this paper is to study the extinction of solutions of the initial boundary value problem for ut = div(|∇u|p(x,t)–2∇u) + b(x, t)|u|q – a0u
For more complete physical background, the readers may refer to [ – ]. These models include parabolic or elliptic equations which are nonlinear with respect to gradient of the thought solution and with variable exponents of nonlinearity; see [ – ] and references therein

Summary

Introduction

We consider the following nonlinear degenerate parabolic equation:. x∈ , where QT = × We consider the following nonlinear degenerate parabolic equation:. It will be assumed throughout the paper that the exponent p(x, t) is continuous in Q = QT with logarithmic module of continuity. ) may describe some properties of electro-rheological fluids which change their mechanical properties dramatically when an external electric field is applied. ) is a function of the external electric field |→–E | , which satisfies the quasi-static The variable exponent p in model ( . ) is a function of the external electric field |→–E | , which satisfies the quasi-static

Maxwell equations

Vn is dense in

Qτ τ

Multiplying the above identity by h

Assume that

Similarly as α