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Existence and uniqueness of solutions for a quasilinear evolution equation in an Orlicz space

Volume 117 / 2016

Zheng Zhou, Fei Fang Annales Polonici Mathematici 117 (2016), 269-289 MSC: Primary 35K55; Secondary 35K65. DOI: 10.4064/ap3861-4-2016 Published online: 4 October 2016

Abstract

We consider the following quasilinear evolution equation in an Orlicz space: $$ u_t=\mathrm {div}(a(|\nabla u|)\nabla u)+f(x,t,u), $$ where $a\in C^1(\mathbb {R})$ and $f\in C^1(\overline {\varOmega }\times [0,T]\times \mathbb {R})$. We use the difference method to transform the evolution problem to a sequence of elliptic problems. Then by making some uniform estimates for these elliptic problems, we obtain the existence of global solutions for the evolution problem. Uniqueness is also proved.

Authors

  • Zheng ZhouSchool of Applied Mathematical Sciences
    Xiamen University of Technology
    Xiamen 361024, China
    e-mail
  • Fei FangDepartment of Mathematics
    Beijing Technology and Business University
    Beijing 100048, China
    e-mail

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