Electron. J. Diff. Equ., Vol. 2016 (2016), No. 143, pp. 1-13.

Evolutionary p(x)-Laplacian equation free from the limitation of the boundary value

Huashui Zhan, Jie Wen

Abstract:
In this article we consider the evolutionary $p(x)$-Laplacian equation
$$ {u_t}= \hbox{div} ({\rho^\alpha}{| {\nabla u} |^{p(x) - 2}}\nabla u),\quad (x,t) \in \Omega \times (0,T), $$
where $\rho(x) = \hbox{dist} (x,\partial \Omega )$. If the diffusion coefficient degenerates on the boundary, then and the solution may be free from any limitations of the boundary condition.

Submitted June 2, 2015. Published June 14, 2016.
Math Subject Classifications: 35L65, 35K85, 35R35.
Key Words: p(x)-Laplacian equation; diffusion coefficient; Fichera-Oleinik theory; boundary condition; Fichera function.

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Huashui Zhan
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: 2012111007@xmut.edu.cn
Jie Wen
School of Sciences
Jimei University
Xiamen, Fujian 361021, China
email: 1195103523@qq.com

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