Global existence and blowup of solutions to semilinear fractional reaction-diffusion equation with singular potential

Abstract

We consider the following semilinear fractional reaction-diffusion equation with singular potential{Asu=−ut|x|2s+|u|p−2u,(x,t)∈Ω×(0,∞),u=0,(x,t)∈(RN∖Ω)×[0,∞),u(x,0)=u0,x∈Ω. Here As(0<s<1) represents spectral fractional Laplacian operator, 2<p<2s⁎, 2s⁎=2NN−2s is critical exponent of fractional Sobolev trace embedding inequality, N>2s, and Ω is bounded domain in RN with smooth boundary. We first prove that the solution exists globally under appropriate hypotheses. And due to the non-locality of fractional Laplacian operator, we use the Caffarelli-Silvestre extension method to convert the non-local problem into a variable local problem. Being based upon this, by means of potential well, we obtain decay estimate and long time asymptotic behavior of global solution, as well as blow-up behavior of local solution.