Blow-up for the non-local pp-Laplacian equation with a reaction term

In this paper we study the blow-up phenomenon for the non-local pp-laplacian equation with a reaction term, ut(x,t)=∫AJ(x−y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy+uq(x,t),x∈Ω,t∈[0,T], with Dirichlet conditions (A=RN,u≡0 in RN∖ΩRN∖Ω) or Neumann conditions (A=ΩA=Ω). Those problems are the non-local analogous to the equation vt=Δpv+vqvt=Δpv+vq with the corresponding conditions. We determine in both cases which are the global existence exponents, that coincide with the exponents of the corresponding local problem for Neumann boundary conditions. However, we observe differences with respect to the global existence exponents of the local Dirichlet problem. Moreover, we show that the blow-up rate is the same as the one that holds for the ODE ut=uqut=uq, that is, limt↗T(T−t)1q−1‖u(⋅,t)‖∞=(1q−1)1q−1. We also find some differences between the local and the non-local models concerning the blow-up sets. Precisely we show that regional blow-up is not possible for non-local problems. Finally, we include some numerical experiments which illustrate our results.