Abstract
Abstract In this paper, a two-species cooperating model with free diffusion and self-diffusion is investigated. The existence of the global solution is first proved by using lower and upper solution method. Then the sufficient conditions are given for the solution to blow up in a finite time. Our results show that the solution is global if the intra-specific competition is strong, while if the intra-specific competition is weak and the self-diffusion rate is small, blow-up occurs provided that the initial value is large enough or the free diffusion rate is small. Numerical simulations are also given to illustrate the blow-up results. MSC:35K57, 92D25.
Highlights
The global existence or blow-up problem for parabolic equations describing the ecological models have been considered by many authors, e.g
In this paper, we are concerned with the following nonlinear reaction-diffusion system: ⎧⎪⎪⎪⎨uu tt [(d + α u )u ] = u (a – b u + c u ) [(d + α u )u ] = u (a + b u – c u )⎪⎪⎪⎩u (x, t) = u (x, t) =in × (, T), in × (, T), on ∂ × (, T), ( . )u (x, ) = η (x), u (x, ) = η (x) in, where = N i= ∂ /∂
The global existence or blow-up problem for parabolic equations describing the ecological models have been considered by many authors, e.g. [ – ]
Summary
The global existence or blow-up problem for parabolic equations describing the ecological models have been considered by many authors, e.g. ) exists and is uniformly bounded in × [ , +∞) if b c > b c , while if b c > b c the solution blows up in a finite time for big ai with any nontrivial nonnegative initial data or for any ai with big initial data. They gave a sufficient condition on the initial data for the solution to blow up in a finite time.
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