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

In this paper we study the blow-up phenomenon for the non-local p-laplacian equation with a reaction term, u(t)(x, t) = integral(J)(A)(x -y)vertical bar u(y, t) - u(x, t)vertical bar(p) (2)(u(y, t) - u(x, t))dy + u(q)(x, t), x is an element of Omega, t is an element of [ 0, T], with Dirichlet conditions (A = R-N, u = 0 in R-N \ Omega) or Neumann conditions (A = Omega). Those problems are the non-local analogous to the equation v(t) = Lambda(p)v + v(q) 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 blowup rate is the same as the one that holds for the ODE u(t) = u(q), that is, lim(t NE arrow T)(T - t)(1/q-1)parallel to u(., t)parallel to(infinity) = (1/q-1)(1/q-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. (C) 2012 Elsevier Ltd. All rights reserved.