Existence and uniqueness of solutions for a quasilinear evolution equation in an Orlicz space
Volume 117 / 2016
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.
