Asymptotic analysis for a localized nonlinear diffusion equation

In this paper we study a localized nonlinear diffusion equation ut=Δum+λ1up+λ2uq(0,t)ut=Δum+λ1up+λ2uq(0,t) subject to null Dirichlet boundary condition with p,q≥0p,q≥0, max{p,q}>m>1max{p,q}>m>1, and λ1,λ2>0λ1,λ2>0. We investigate interactions among the localized and local sources, nonlinear diffusion with the zero boundary value condition to establish blow-up rates and uniform blow-up profiles of solutions under different dominations. In addition, as results of the interactions of multiple nonlinearities, the blow-up sets of solutions, namely, total versus single point blow-up of solutions are also determined.