Asymptotic behavior of solutions to a degenerate parabolic equation with a gradient source term

Article type: Research Article

Authors: Zhou, Shuangshuang^; | Zheng, Sining^;

Affiliations: School of Mathematics and Computational Science, Hunan City University, Yiyang, China | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Note: [] Corresponding author. E-mail: snzheng@dlut.edu.cn

Abstract: In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+uq|∇u|2−ur, subject to null Dirichlet boundary condition. We study the existence of global solutions and the large time behavior for them. The main effort is paid to obtain uniform asymptotic profiles for decay solutions, under various dominations of the nonlinear diffusion or absorption. It is shown that the large time property of the solution u behaves just like (1+(r−1)t)(−1/r−1) if the decay is governed by the nonlinear absorption with 1<r<p+1. Otherwise, the asymptotic profiles would possess the form of (1+pt)−1/pW, with W solving various homogeneous Dirichlet elliptic equations: (i) −ΔW=W1−p−W if r=p+1<q+2; (ii) −ΔW=W1−p−W+W−1|∇W|2 if r>p+1=q+2; and (iii) −ΔW=W1−p if r,q+2>p+1.

Keywords: degenerate parabolic equation, gradient term, global solution, large time behavior

DOI: 10.3233/ASY-141254

Journal: Asymptotic Analysis, vol. 91, no. 2, pp. 91-102, 2015