Communications in Mathematical Sciences

Volume 22 (2024)

Number 2

Lifespan estimates of solutions to the weakly coupled system of semilinear wave equations with space dependent dampings

Pages: 375 – 393

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n2.a4

Authors

Sen Ming (Department of Mathematics, North University of China, Taiyuan, China)

Han Yang (Department of Mathematics, Southwest Jiaotong University, Chengdu, China)

Xiongmei Fan (Data Science And Technology, North University of China, Taiyuan, China)

Abstract

$\def \lv{\lvert}\def\rv{\rvert}$ This paper is devoted to investigating the weakly coupled system of semilinear wave equations with space dependent dampings and power nonlinearities ${\lv v \rv}^p, {\lv u \rv}^q$, derivative nonlinearities ${\lv v_t \rv}^p, {\lv u_t \rv}^q$, mixed nonlinearities ${\lv v \rv}^q, {\lv u_t \rv}^p$, combined nonlinearities ${\lv v_t \rv}^{p_1} + {\lv v \rv}^{q_1}, {\lv u_t \rv}^{p_2} + {\lv u \rv}^{q_2}$, combined and power nonlinearities ${\lv v_t \rv}^{p_1} + {\lv v \rv}^{q_1}, {\lv u \rv}^{q_2}$, combined and derivative nonlinearities ${\lv v_t \rv}^{p_1} + {\lv v \rv}^{q_1}, {\lv u_t \rv}^{p_2}$, respectively. Formation of singularities and lifespan estimates of solutions to the problem in the sub-critical and critical cases are illustrated by making use of test function technique. The main innovation is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent.

Keywords

weakly coupled system, semilinear wave equations, test function technique, formation of singularities, lifespan estimates

2010 Mathematics Subject Classification

35L70, 58J45

Received 19 March 2023

Received revised 28 June 2023

Accepted 8 July 2023

Published 1 February 2024