International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 11, Pages 1809-1818
doi:10.1155/IJMMS.2005.1809
A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system
Department of Mathematics and Statistics, Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB 45433-7765, OH, USA
Received 21 July 2004
Copyright © 2005 Alan V. Lair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We show that the reaction-diffusion system ut=Δφ(u)+f(v), vt=Δψ(v)+g(u), with homogeneous Neumann boundary conditions, has a positive global solution on Ω×[0,∞) if and only if ∫∞ds/f(F−1(G(s)))=∞
(or, equivalently, ∫∞ds/g(G−1(F(s)))=∞), where F(s)=∫0sf(r)dr and G(s)=∫0sg(r)dr. The domain Ω⊆ℝN(N≥1) is bounded with smooth boundary. The functions φ, ψ, f, and g are nondecreasing, nonnegative C([0,∞)) functions satisfying φ(s)ψ(s)f(s)g(s)>0 for s>0 and φ(0)=ψ(0)=0. Applied to the special case f(s)=sp and g(s)=sq, p>0, q>0, our result proves that the system has a global solution if and only if pq≤1.