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

Alan V. Lair

International Journal of Mathematics and Mathematical Sciences, 2005, vol. 2005, 1-10

Abstract:

We show that the reaction-diffusion system u t = Δ φ ( u ) + f ( v ) , v t = Δ ψ ( v ) + g ( u ) , with homogeneous Neumann boundary conditions, has a positive global solution on Ω × [ 0 , ∞ ) if and only if ∫ ∞ d s / f ( F − 1 ( G ( s ) ) ) = ∞ (or, equivalently, ∫ ∞ d s / g ( G − 1 ( F ( s ) ) ) = ∞ ), where F ( s ) = ∫ 0 s f ( r ) d r and G ( s ) = ∫ 0 s g ( r ) d r . The domain Ω ⊆ ℝ N ( 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 ) = s p and g ( s ) = s q , p > 0 , q > 0 , our result proves that the system has a global solution if and only if p q ≤ 1 .

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:428957

DOI: 10.1155/IJMMS.2005.1809

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