Abstract
This paper considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the blow-up rate estimate for the blow-up solution.
Highlights
This paper studies the singularity properties of the following reaction-diffusion system with nonlocal boundary condition: ut = △u + uαVp, Vt = △V + uqVβ, x ∈ Ω, t > 0, u (x, t) = ∫ f (x, y) u (x, y) dy, Ω (1)V (x, t) = ∫ g (x, y) V (x, y) dy, x ∈ ∂Ω, t > 0,u (x, 0) = u0 (x), V (x, 0) = V0 (x), x ∈ Ω, where Ω is a bounded domain of RN, N ≥ 1, with smooth boundary ∂Ω and Ω is the closure of Ω. α, β, p, and q are positive numbers which ensure that the equations in (1) are completely coupled with the nonlinear reaction terms
This paper considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition
U (x, 0) = u0 (x), V (x, 0) = V0 (x), x ∈ Ω, where Ω is a bounded domain of RN, N ≥ 1, with smooth boundary ∂Ω and Ω is the closure of Ω. α, β, p, and q are positive numbers which ensure that the equations in (1) are completely coupled with the nonlinear reaction terms
Summary
There are some important phenomena formulated as parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [24,25,26]) In this case, the solution could be used to describe the entropy per volume of the material. Motivated by the above cited works, in this paper, we deal with singularity analysis of the parabolic system (1) with nonlocal boundary condition and it is seems that there is no work dealing with this type of systems except the single equations case, this is a very classical model.
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