Blow-up of p-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power u(t)(x, t) = div(vertical bar del u vertical bar(p-2)del u) + uq((x)) in Omega x (0, T), where Omega is either a bounded domain or the whole space R-N , q(x) is a positive and continuous function defined in Omega with 0 < q(-) = inf q(x) <= q(x) <= sup q(x) = q(+) < infinity. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Omega, compared with the case of constant source power. For the case that Omega is a bounded domain, the exponent p - 1 plays a crucial role. If q(+) > p - 1, there exist blow-up solutions, while if q(+) < p - 1, all the solutions are global. If q(-) > p - 1, there exist global solutions, while for given q(-) < p - 1 < q(+), there exist some function q(x) and Omega such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Omega = R-N , the Fujita phenomenon occurs if 1 < q(-) <= q(+) <= p - 1 + p/N, while if q(-) > p - 1 + p/N, there exist global solutions.