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 ut(x,t)=div(|?u|?u)+u~(q(x)) in?×(0,T),where ? is either a bounded domain or the whole space R~N,and q(x) is a positive and continuous function defined in ? with 0<q=inf q(x)<=q(x)<=sup q(x)=q<∞.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 ?,compared with the case of constant source power.For the case that ? 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 ? such that all nontrivial solutions will blow up,which is called the Fujita phenomenon.For the case ?=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.