The Global Existence,uniqueness And Decay Estimates For Solutions To Semilinear Structural Damped Wave Equations

Wave equations is a kind of second order partial differential equations,which are mainly used to describe the diffusion and propagation of waves.Their solutions can help us to understand the characteristics of waves more intuitively and we can use these characteristics to obtain some effective information in practical project application.In the process of propagation of waves,the existence of media causes dissipation.Therefore,we have to take damping effect into consideration when analyzing some properties of wave equations.In this thesis,we consider a kind of semilinear structural damped wave equation as follows:where,??1,a?[0,?),??0.we prove the global existence and uniqueness of small data solutions,at the same time,decay rates of solutions are also estimated.The main contribution of this paper is that we further expand the range of p when nonlinear term has the form of ?D|au|p.In Chapter 1,according to the form of damped term,we present background of fractional wave equations.The decay estimates and asymptotic profiles of semilinear and linear structural damped wave equations are introduced.In Chapter 2,we consider the linear part of SSDW equations.First of all,partial differential equation(0.1)is transformed into ordinary differential equation through Fourier transform.Then,the solutions to ordinary differential equations are divided into three categories according to the value of parameter ? and we solve them in turn.Based on Riesz-Thorin interpolation theorem and generalized Holder inequality,we can obtain that the decay rates of solutions to ordinary differential equations satisfy exponential form.Finally,we give the decay rate of original partial differential equations by Fourier inverse transformation.In Chapter 3,the global existence,uniqueness and decay estimates of small data solutions to SSDW equations are derived based on Duhamel's principle,Gagliardo-Nirenberg inequality and decay estimates of linear part.In the last chapter,the main content of this paper is summarized.Meanwhile,we provide the main contribution and introduce the research directions in the future.