Document Type

Article

Publication Date

2005

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.

Comments

Sourced from the publisher version of record at Hindawi:
Lair, A. V. (2005). A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system. International Journal of Mathematics and Mathematical Sciences, 2005(11), 1809–1818. https://doi.org/10.1155/IJMMS.2005.1809

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. CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/)

MathSciNet: MR2178254

DOI

10.1155/IJMMS.2005.1809

Source Publication

International Journal of Mathematics and Mathematical Sciences

Share

COinS