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

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.