Blow-Up Of Solutions To Wave Equations And Coupled System With Memory Terms

Wave equation is a kind of important hyperbolic equation,which can explain and simulate many physical phenomenon in the applied disciplines.Because damping term could affect the energy decay rate of solution to wave equation and nonlinear external force could cause the wave to become steep in the process of propagation until it blows up.Therefore,it is of great scientific value to investigate the influence of damping term and nonlinear term on blow-up behaviors of solution to wave equation.This paper is mainly concerned with blow-up results of solutions to the Cauchy problem or initial boundary value problem for single wave equation with memory term and the corresponding coupled system in flat Minkowski spacetime and Schwarzschild black hole spacetime.Upper bound lifespan estimates of solutions to the problem are established by employing test function method and iteration approach.It is illustrated that the effects of damping terms,nonlinear memory terms and spatial dimensions on blow-up phenomena and lifespan estimates of solutions are discussed in different spacetimes.In flat Minkowski spacetime,the scattering damping terms dependent on the time variable are handled by constructing suitable multipliers.Upper bound lifespan estimates of solutions to single wave equation and the corresponding coupled system are studied by making use of iteration approach.The nonlinear terms are presented in the form of power nonlinearity,derivative nonlinearity,combined nonlinearities as well as the corresponding memory terms.The initial boundary value problem is considered on exterior domain when the spatial dimensions are n=1,n=2 and n≥ 3 by choosing different test functions,respectively.We observe that lifespan estimate of solution to wave equation with memory term when α→0+ coincides with the lifespan estimate of solution to nonlinear wave equation with |u|~p,|u_t|~p,|u_t|~p+|u|~q.In Schwarzschild black hole spacetime,non-existence of global solutions to the initial value problems(ε=1)for single wave equation with power type memory term and the corresponding coupled system are demonstrated.The main tools in proofs are based on the Regge-Wheeler coordinate transformation and test function method,where test function is related to the Riemann-Liouville fractional derivative.Furthermore,formation of singularities of solutions and upper bound lifespan estimates of solutions to the small initial value problems(0<ε<1)are derived under the assumptions that initial values have compact supports.