In this paper we investigates the blow-up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition, we obtain the blow-up condition and its blow-up rate estimate.
This paper deals withp-Laplacian systemsut−div(|∇u|p−2∇u)=∫Ωvα(x,t)dx,x∈Ω,t>0,vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx,x∈Ω, t>0,with null Dirichlet boundary conditions in a smooth bounded domainΩ⊂ℝN, wherep,q≥2,α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈ℝN:|x|<R} (R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.
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