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Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary. (English) Zbl 1139.35057

Summary: This paper considers the properties of positive solutions for a nonlocal equation

u t (x,t)=Δu+ Ω u q (y,t)dy-ku p (y,t),inΩ×(0,T),

with nonlocal boundary condition

u(x,t)= Ω f(x,y)u(y,t)dy,onΩ×(0,T),

and initial condition

u(x,0)=u 0 (x),forxΩ

where p,q1,k>0, and Ω n is a bounded domain with smooth boundary Ω. Conditions for the existence and nonexistence of global positive solutions are given. Moreover, we establish uniform blow-up estimates for the blow-up solution.

MSC:
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
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