Electron. J. Differential Equations, Vol. 2017 (2017), No. 311, pp. 1-15.

Existence of solutions to an evolution p-Laplacian equation with a nonlinear gradient term

Huashui Zhan, Zhaosheng Feng

Abstract:
We study the evolution p-Laplacian equation with the nonlinear gradient term
$$ {u_t} = \hbox{div} (a(x){| {\nabla u} |^{p - 2}}\nabla u) -B(x)|\nabla u|^q, $$
where $a(x), B(x)\in C^1(\overline{\Omega})$, p>1 and p>q>0. When a(x)>0 and B(x)>0, the uniqueness of weak solution to this equation may not be true. In this study, under the assumptions that the diffusion coefficient a(x) and the damping coefficient B(x) are degenerate on the boundary, we explore not only the existence of weak solution, but also the uniqueness of weak solutions without any boundary value condition.

Submitted April 9, 2017. Published December 31, 2017.
Math Subject Classifications: 35L65, 35K85, 35R35.
Key Words: Evolution p-Laplacian equation; weak solution; uniqueness; boundary value condition.

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Huashui Zhan
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: huashuizhan@163.com
Zhaosheng Feng
Department of Mathematics
University of Texas-Rio Grande Valley
Edinburg, TX 78539, USA
email: zhaosheng.feng@utrgv.edu

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