Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}\)-data
type: Item
instance of: scholarly article
title: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}\)-data
zbMATH Open document ID: 1231.35106
author: Mostafa Bendahmane
author: Petra Wittbold
author: Aleksandra Zimmermann
published in: Journal of Differential Equations
publication date: 2010-09-17T00:00:00Z
review text: The authors prove well-posedness, existence and uniqueness of a renormalized solution of a nonlinear parabolic equation with variable exponents and \(L^1\)-data, namely \[ \begin{cases} \partial_t u-\text{div}(|\nabla u|^{p(x)-2}\nabla u)=f &\text{in}\;Q_T,\\ u=0 &\text{on}\;\Sigma_T,\\ u(0,\cdot)=u_0(\cdot) &\text{in}\;\Omega, \end{cases} \] where \(\Omega\subset \mathbb R^ N\) is a~bounded open Lipschitz domain, \(N\geq2\), \(Q_Y=(0,T)\times\Omega\) and \(\Sigma_T=(0,T)\times\partial\Omega\). The authors work in the functional setting of the Lebesgue and Sobolev variable-exponent spaces. The main result states that when the variable-exponent function \(p:\overline{\Omega}\to[1,\infty)\) is continuous and moreover satisfies \[ 1<\min_{x\in\overline{\Omega}} p(x)\leq\max_{x\in\overline{\Omega}} p(x)<N, \] then there exists at least one renormalized solution \(u\) to the above-stated problem. Moreover, if \(\min_{x\in\overline{\Omega}} p(x)>2-\frac1{N+1}\), then \(u\in L^{q^-}(0,T;W_0^{1,q(\cdot)}(\Omega))\) for every continuous variable-exponent function \(q\) satisfying \[ 1\leq q(x)<\frac{N(p(x)-1)+p(x)}{N+1} \text{ for all } x\in\overline{\Omega}, \text{ where }q^-=\min_{x\in\overline{\Omega}} q(x). \] The proof is deep and involved and several side results of independent interest are established in the course of it. The techniques used in the proof involve the semigroup approach, a rather comprehensive study of various properties of renormalized solutions, a priori estimates, various results on several types of convergence of functions approximating the solution with a special emphasis on strong convergence. The uniqueness of the solution is proved separately. Some connections to entropy solutions are observed.
reviewed by: Luboš Pick
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5787090
zbMATH Keywords: nonlinear parabolic equation
zbMATH Keywords: renormalized solution
zbMATH Keywords: well-posedness
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: variable-exponent spaces
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Lebesgue spaces
MaRDI profile type: Publication
full work available at URL: j.jde.2010.05.011
OpenAlex ID: W1971062478
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DOI: 10.1016/J.JDE.2010.05.011
description: scientific article; zbMATH DE number 5787090