Blow-up phenomena of semilinear wave equations and their weakly coupled systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Ikeda, Masahiro^1,2  Sobajima, Motohiro³  Wakasa, Kyouhei⁴ 

[1] Keio Univ, Fac Sci & Technol, Dept Math, Kohoku Ku, 3-14-1 Hiyoshi, Yokohama, Kanagawa 2238522, Japan

[2] RIKEN, Ctr Adv Intelligence Project, Wako, Saitama, Japan

[3] Tokyo Univ Sci, Fac Sci & Technol, Dept Math, 2641 Yamazaki, Noda, Chiba 2788510, Japan

[4] Kushiro Coll, Natl Inst Technol, Dept Creat Engn, 2-32-1 Otanoshike Nishi, Kushiro, Hokkaido 0840916, Japan

关键词: Semilinear wave equations;    Weakly coupled system;    Blowup;    Upper bounds of lifespan;    Test function methods;   

DOI  :  10.1016/j.jde.2019.05.029

来源: Elsevier

【 摘 要 】

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function methods and give a simple proof of the derivation of sharp upper bounds for lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical cases, we use a family of self-similar solutions to the linear wave equation including Gauss's hypergeometric functions, which are originally introduced by Zhou [59]. We emphasize that our framework does not require the pointwise positivity of the initial data even in the high dimensional case N >= 4. Moreover, we find a new (p, q)-curve for the system partial derivative(2)(t)u - Delta u = vertical bar v vertical bar(q), partial derivative(2)(t)v - Delta v = vertical bar partial derivative(t)u vertical bar(P) with lifespan estimates for small solutions in a new region. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free