Energy Decay And Blow-up Of Higher-order Nonlinear Wave Equations

Partial differential equation is a very important content in the field of mathematics.For the partial differential equations,we mainly study the regularity,well-posedness,stability,controllability and decay of solutions.While wave equation is an important part of partial differential equation,which mainly describes various wave phenomena in nature,such as sound waves,light waves and water waves.It appears in different fields,such as acoustics,electromagnetism and fluid mechanics.Therefore,studying this kind of equation has great practical significance and application value.Most of the literatures have studied blow up of solutions for the wave equations with different damping terms and the viscoelastic wave equations with variable coefficients and exponentials,but there are few researches on the higher order wave equations.Therefore,this paper extends the wave equations with nonlinear damping terms to higher-order cases,and discusses the energy decay and blow-up of solutions for higher-order viscoelastic wave equations with nonlinear damping terms and the decay of higher-order wave equations with time-dependent nonlinear damping terms.The first chapter is the introduction,which introduces some research status of wave equation blow-up and energy estimation.In Chapter 2,we consider the energy decay and blow-up results of solutions for a class of higher-order viscoelastic wave equations with nonlinear damping terms and source terms.Under different initial conditions and , satisfying suitable assumptions,it proved that the blow-up and asymptotic properties of the solutions for the nonlinear high-order viscoelastic wave equations by using the energy perturbation method and constructing Lyapunov functional method.In Chapter 3,we study the energy decay of solutions for a class of high-order wave equations with nonlinear time-dependent damping.In this chapter,we construct the corresponding energy functional,then use multiplier method to deal with the terms in the energy functional and use some general weighted integral inequalities,finally obtain the decay rate of the system.