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
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 , , with homogeneous Neumann boundary conditions, has a positive global solution on if and only if
(or, equivalently, ), where and . The domain is bounded with smooth boundary. The functions , , , and are nondecreasing, nonnegative functions satisfying for and . Applied to the special case and , , , our result proves that the system has a global solution if and only if .