Regularity in Sobolev spaces for doubly nonlinear parabolic equations
type: Item
instance of: scholarly article
title: Regularity in Sobolev spaces for doubly nonlinear parabolic equations
zbMATH Open document ID: 1029.35142
DOI: 10.1016/S0022-0396(02)00093-1
author: José Miguel Urbano
author: C. Ebmeyer
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: Let \(m>1\) and \(p>(m+1)/m\) be constants, and define \(A(\xi)=|\xi|^{p-2}\xi\) and \(a(z)=|z|^{m-1}z\). The authors examine the regularity of solutions of the Cauchy problem \[ u_{t} =\operatorname {div} A(D[a(u)])\quad \text{in } \mathbb R^{n}\times(0,T), \qquad u(x,0)=u_0(x) \quad\text{in } \mathbb R^{n} \] for \(n \geq 2\) and \(T>0\). If \(q<\max\{p/(p+1),2/3\}\) and \(r<1/q\), it is shown that \[ a(u) \in L^{q}(0,T;W^{r,q}(\mathbb R^{n})). \] Similar estimates are proved for powers of \(|Da(u)|\). The proof is based on difference quotient arguments developed by the first author [Z. Anal. Angew. 15, 637-650 (1996; Zbl 0858.35069)] for elliptic \(p\)-Laplace type equations.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1884227
zbMATH Keywords: porous medium equation
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: finite speed of propagation
zbMATH Keywords: difference quotient
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1884227