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Existence to nonlinear parabolic problems with unbounded weights

  • Iwona ChlebickaEmail author
  • Anna Zatorska-Goldstein
Article
  • 51 Downloads

Abstract

We consider the weighted parabolic problem of the type
$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} u_t-\mathrm{div}(\omega _2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega _1(x) |u|^{p-2}u,&{} x\in \Omega ,\\ u(x,0)=f(x),&{} x\in \Omega ,\\ u(x,t)=0,&{} x\in \partial \Omega ,\ t>0,\\ \end{array}\right. \end{aligned} \end{aligned}$$
for quite a general class of possibly unbounded weights \( \omega _1,\omega _2\) satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that \(\lambda >0\) is smaller than the optimal constant in the inequality. The domain is assumed to be bounded or quasibounded. The obtained solution is proven to belong to
$$\begin{aligned} L^p({{\mathbb {R}}}_+; W_{(\omega _1,\omega _2),0}^{1,p}(\Omega ))\cap L^\infty ({{\mathbb {R}}}_+; L^2(\Omega )). \end{aligned}$$

Keywords

Existence of solutions Hardy inequalities Parabolic problems Weighted p-Laplacian Weighted Sobolev spaces 

Mathematics Subject Classification

35K55 35A01 47J35 

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland

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