Electron. J. Diff. Equ., Vol. 2016 (2016), No. 209, pp. 1-15.

Weak solutions for parabolic equations with p(x)-growth

Ning Pan, Binlin Zhang, Jun Cao

In this article we study nonlinear parabolic equations with p(x)-growth in the space $W^{1,x}L^{p(x)}(Q)\cap L^\infty(0,T; L^2(\Omega))$. By using the method of parabolic regularization, we prove the existence and uniqueness of weak solutions for the equation
 \frac{\partial u}{\partial t}=\hbox{div}(a(u)
 |\nabla u|^{p(x)-2}\nabla u)+f(x,t).
Also, we study the localization property of weak solutions for the above equation.

Submitted April 2, 2016. Published August 2, 2016.
Math Subject Classifications: 35K15, 35K20, 35K55.
Key Words: Parabolic equation; p(x)-growth condition.

Show me the PDF file (282 KB), TEX file for this article.

Ning Pan
Department of Mathematics
Northeast Forestry University
Harbin 150040, China
email: hljpning@163.com
Binlin Zhang
Department of Mathematics
Heilongjiang Institute of Technology
Harbin 150050, China
email: zhangbinlin2012@163.com
Jun Cao
College of Mechanical and Electrical Engineering
Northeast Forestry University
Harbin 150040, China
email: zdhcaojun@163.com

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