Critical exponent for nonlocal diffusion equations with Dirichlet boundary condition

Abstract

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