Critical exponent for nonlocal diffusion equations with Dirichlet boundary condition

This paper is concerned with a nonlocal diffusion equation with Dirichlet boundary conditions and a reaction term {∂∂tu(x,t)=∫RNJ(x−y)(u(y,t)−u(x,t))dy+eβtup(x,t),x∈Ω,t>0,u(x,t)=0,x∉Ω,t≥0,u(x,0)=u0(x)≥0,x∈Ω. Our main result shows that this nonlocal diffusion problem has a critical exponent pβ∗ which coincides with that of the corresponding local diffusion problem. The blow-up rate of solutions is also discussed.