Energy Decay For Two Classes Of Variable-coefficient Wave Equations With Acoustic Boundary Conditions

In the sciences of physics,neuroscience and chemistry,the state of the system depends not only on time but also on space.It often accompanies partial differential equation when it is described in precise mathematical language.Partial differential equations have a long history and a huge background.The view is strong,and as you can see from its physical background,such as the elastic mechanical systems which corresponds to the elastic mechanical equations,the fluid mechanics which corresponds to the hydrodynamic equations,quantum mechanics which corresponds to the Schr¨dinger equation.We mainly discuss two elastic mechanical systems — the wave equations,one of which is the variable-coefficient wave equation with acoustic boundary conditions and time delay,and the second is the variable-coefficient wave equation with nonlinear acoustic boundary conditions and source term.The thesis is divided into three chapters.In chapter 1,firstly,we provide a simple physical significance and research summary of the variable-coefficient wave equation with acoustic boundary conditions,delay and source term.Then we present some notations on Riemann geometry which are applied in this paper.In chapter 2,we study the energy decay of variable-coefficient wave equation with acoustic boundary conditions and delay.Firstly,we use a transformation to convert the original system into another system,then using the Galerkin method to prove the existence of the solution.After that,we define the corresponding energy functional of the system,and using the multiplier method and the Riemann geometry method to estimate each term of energy.Finally,the exponential stability result of the system is obtained.In chapter 3,we discuss the energy decay of variable-coefficients wave equation with nonlinear acoustic boundary conditions and source term.Firstly,we use the Galerkin method to prove the existence of the solution,then defining the system of energy functional,we deal with the source term by the principle of potential well,and using the multiplier method and the Riemann geometry method to estimate each term of energy.Finally,we prove that decay rate of system consistents with an ODE.