Global Sobolev solutions of quasilinear parabolic equations
type: Item
instance of: scholarly article
title: Global Sobolev solutions of quasilinear parabolic equations
zbMATH Open document ID: 1004.35068
author: Kevin McLeod
author: Albert Milani
published in: Differential and Integral Equations
publication date: 2000-05-29T00:00:00Z
review text: Global existence, uniqueness and a priori estimates of solutions to the initial and homogeneous Dirichlet boundary value problem for the equation \[ u_t - \sum _{i,j=1}^{n} a_{i,j}(\nabla u) \partial _i \partial _j u = f(x,t)\quad\text{on} \Omega \times (0,T) \] is proved in Sobolev spaces \(X_{s+2}(T)\) for sufficiently large \(s.\) Here \[ X_m(T) = \{ u \in L^2(0,T;H^{m}(\Omega)),\;\partial ^{m'}_t u \in L^2(0,T;H^{m - 2m'}(\Omega))\} \] with \(m' = [\frac {m+1}{2}]\). The data of the problem satisfy natural smoothness assumptions. The proof uses as a starting point classical existence results of O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva combined with higher-order estimates obtained with the help of Gagliardo-Nirenberg interpolation inequalities.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 1453647
zbMATH Keywords: homogeneous Dirichlet boundary value problem
zbMATH Keywords: Gagliardo-Nirenberg interpolation inequalities
reviewed by: J. Stará
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1453647