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Journal of the London Mathematical Society Advance Access published online on June 5, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp018
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© 2009 London Mathematical Society

Boundary blow-up solutions for logistic-type porous media equations with nonregular source

Huiling Li

Department of Mathematics
Southeast University
Nanjing 210018
People's Republic of China
lihuiling_seu@yahoo.com.cn

Peter Y. H. Pang

Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543, Singapore

Mingxin Wang

Department of Mathematics
Southeast University
Nanjing 210018
People's Republic of China
Science Research Center
Harbin Institute of Technology
Harbin 150080
People's Republic of China
mxwang@seu.edu.cn

In this paper, we establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to the porous media equations of logistic type –{Delta} u = a(x)u1/m – b(x)f(u) with m > 1. We first consider the existence of such solutions for the general function f(u), and then study the uniqueness and the blow-up rate for the function f(u) whose variation at infinity is not regular. We also note the difference in the treatment of the blow-up rate for the cases where f varies regularly or not regularly at infinity.

2000 Mathematics Subject Classification 35J25, 35J65, 35K57.

Part of the paper was done during the first and third authors’ visit to the National University of Singapore. The first author was partially supported by NSFC Grants 10701024 and 10601011, the second author was supported by the NUS ARF Grant R-146-000-088-112 and the third author was supported by NSFC Grant 10771032 and the Natural Science Foundation of Jiangsu Province Grant BK2006088.

Received August 5, 2008; revised February 18, 2009;
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