The blow-up of solutions to a coupled system of semilinear wave equations in the sub-critical and critical cases is investigated. The system contains scattering dampings and combined nonlinearities. The upper bound lifespan estimates of solutions which are related to the indexes of nonlinear terms are established by applying the functional method and iteration technique.

The local well-posedness for the Cauchy problem of a weakly dissipative shallow water equation is established using the Littlewood–Paley theory and a priori estimates of solutions to the transport equation. The norm estimates, the blow-up mechanisms, and the global existence of solutions to the problem are investigated. The novelty is that the effects of the weakly dissipative coefficients and nonlinear index to guarantee the global existence of solutions are presented.

This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity $ |u|^{p} $, derivative type nonlinearity $ |u_{t}|^{p} $ and combined type nonlinearities $ |u_{t}|^{p}+|u|^{q} $ in the case of single equation, combined type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u_{t}|^{p_{2}}+|u|^{q_{2}} $, combined and power type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u|^{q_{2}} $, combined and derivative type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u_{t}|^{p_{2}} $ in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.