Global existence for semilinear wave equations with the critical blow-up term in high dimensions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Takamura, Hiroyuki¹  Wakasa, Kyouhei^2,3 

[1] Future Univ Hakodate, Fac Syst Informat Sci, Dept Complex & Intelligent Syst, 116-2 Kamedanakano Cho, Hakodate, Hokkaido 0418655, Japan

[2] Hokkaido Univ, Dept Math, Sapporo, Hokkaido 0600810, Japan

[3] Muroran Inst Technol, Coll Liberal Arts, Math Sci Res Unit, 27-1 Mizumoto Cho, Muroran, Hokkaido 0508585, Japan

关键词: Semilinear wave equation;    High dimensions;    Critical exponent;    Lifespan;   

DOI  :  10.1016/j.jde.2016.03.036

来源: Elsevier

【 摘 要 】

We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in two space dimensions, quadratic terms in three space dimensions and quadratic terms including a square of unknown functions itself in four space dimensions. Except for the last case, criteria to classify nonlinear terms into the almost global, or global existence case, are well-studied and known to be so-called null condition and non-positive condition. Our motivation of this work is to find such a kind of the criterion in four space dimensions. In our previous paper, an example of the non-single term for the almost global existence case is introduced. In this paper, we show an example of the global existence case. These two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions. But it may help us to describe the final form of the criterion. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free