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Self-similar solutions and asymptotic behaviour for a class of degenerate and singular diffusion equations

Published online by Cambridge University Press:  21 May 2007

Chunpeng Wang
Affiliation:
Department of Mathematics, Jilin University, Changchun, Jilin 130012, People's Republic of China (wangcp@jlu.edu.cn), (yjx@jlu.edu.cn)
Tong Yang
Affiliation:
Department of Mathematics, City University of Hong Kong, Hong Kong (matyang@cityu.edu.hk)
Jingxue Yin
Affiliation:
Department of Mathematics, Jilin University, Changchun, Jilin 130012, People's Republic of China (wangcp@jlu.edu.cn), (yjx@jlu.edu.cn)
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Abstract

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In this paper, we study the self-similar solutions and the time-asymptotic behaviour of solutions for a class of degenerate and singular diffusion equations in the form

$$ u_t=(|(p(u))_x|^{\lambda-2}(p(u))_x)_x,\quad -\infty<x<+\infty,\quad t>0, $$

where $\lambda>2$ is a constant. The existence, uniqueness and regularity for the self-similar solutions are obtained. In particular, the behaviour at two end points is discussed. Based on the monotonicity property of the self-similar solutions and the comparison principle, we also investigate the time convergence of the solution for the Cauchy problem to the corresponding self-similar solution when the initial data have some decay in space variable.

Type
Research Article
Copyright
2007 Royal Society of Edinburgh