Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - 2010 - Volume 140, Issue 01  

Research Article

Boundary blow-up of a logistic-type porous media equation in a multiply connected domain

Huiling Lia1, Peter Y. H. Panga2 and Mingxin Wanga3

a1 Department of Mathematics, Southeast University, Nanjing 210018, People's Republic of China

a2 Department of Mathematics, National University of Singapore, 2 Science Drive 2, 117543 Republic of Singapore, (matpyh@nus.edu.sg)

a3 Department of Mathematics, Southeast University, Nanjing 210018, People's Republic of China and Science Research Center, Harbin Institute of Technology, Harbin 150080, People's Republic of China

Abstract

We study positive solutions to the porous media equation of degenerate logistic type −Δu = a(x)u1/m – b(x)f(u), m > 1, which blow up at the inner boundary and satisfy a homogeneous boundary condition at the outer boundary of a multiply connected domain. In particular, we investigate uniqueness and blow-up rate near the inner boundary for such solutions. We also look at the limiting behaviour as m ↘ 1 for the special case where b(x) > 0 and f(u) = up/m with p > m.

(Received August 26 2008)

(Accepted March 18 2009)