We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power where Ω is either a bounded domain or the whole space RN and q（x） is a positive and continuous...We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power where Ω is either a bounded domain or the whole space RN and q（x） is a positive and continuous function defined in with 0 〈 q- infq（x） = q（x） 〈 ∞supq（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 Ω, 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 Ω = RN the Fujita phenomenon occurs if 1 q＋ q＋ ≤p--1＋p/N, while if q_ 〉 p -- 1 ＋p/N there exist global solutions.展开更多
基金supported by Shanxi Bairen Plan of China and Ng-Jhit-Cheong Foundation
文摘We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power where Ω is either a bounded domain or the whole space RN and q（x） is a positive and continuous function defined in with 0 〈 q- infq（x） = q（x） 〈 ∞supq（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 Ω, 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 Ω = RN the Fujita phenomenon occurs if 1 q＋ q＋ ≤p--1＋p/N, while if q_ 〉 p -- 1 ＋p/N there exist global solutions.
