Blow-Up Of P-Laplacian Evolution Equations With Variable Source Power

Keywords

blow-up; p-Laplacian; variable source power

Abstract

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|p−2∇u)+uq(x)inΩ×(0,T), where Ω is either a bounded domain or the whole space ℝN, q(x) is a positive and continuous function defined in Ω 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 Ω, compared with the case of constant source power. For the case that Ω 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 Ω 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.

Publication Date

3-1-2017

Publication Title

Science China Mathematics

Volume

60

Issue

3

Number of Pages

469-490

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s11425-016-0091-0

Socpus ID

85002291915 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85002291915

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