Asymptotic Behavior Of The Solution To A Non-local Nonlinear Diffusion Equation

In this paper, we study the Dirichlet problem to a non-local diffusion equation with local and non-local sources By using the comparison principle, the super and sub-solution method and constructing auxiliary functions method, we discuss the blow-up behavior of the solution to above equation. Namely, we give the blow-up rate at the maximum point. In particular, when the localized source dominates the equation, the uniform blow-up profiles is obtained. Furthermore, we get the result on the blow-up set. In the other words, under some con-ditions on initial data, we prove that the solution of the equation occurs total blow-up if the localized source dominates this equation, i.e. p≤ q+1; If the local source dominates this equation, i.e., p> q+1, the one dimension radially symmetric solution blows up only at the origin.