International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 11, Pages 1809-1818

A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system

Alan V. Lair

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.


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(F1(G(s)))= (or, equivalently, ds/g(G1(F(s)))=), where F(s)=0sf(r)dr and G(s)=0sg(r)dr. The domain ΩN(N1) 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 pq1.