Stabilizaiton Of Wave Equations With Several Dynamical Boundary Conditions

This dissertation gives a treatment of stabilization of wave equations with several dynamical boundary conditions.In chapter 1,we give a brief introduction to the research object,and list some inequalities and theorems.In chapter 2,we consider the wave equation with variable coefficient subject to general dynamic boundary conditions in a Riemannian manifold.This equation has an acceleration term ?_tt(x,t)and a delayed velocity term ?_t(x,t-?)on the boundary?₁.Under suitable geometric conditions(G)and proper assumptions on the functions of the system,we obtain the exponential decay for the solutions.Our proof relies on the multiplier method and Lyapunov approach.In chapter 3,we deal with the semilinear wave equation in a Riemannian manifold with a local internal damping f(x,?_t)and dynamic Wentzell boundary conditions with a memory term₀^?tg(t-s)(?)_Tu(s)ds.The stabilization estimate is more difficult to obtain since the physical energy of the system not only contains the H¹Sobolev norm of the solution but also depends on the memory term on the boundary.Under suitable geometric conditions(G)and proper assumptions(A)on the functions of the system,the exponential stabilization is attained by constructing new Lyapunov functionals and using multiplier methods.To illustrate the results,numerical simulations are given in the last part,which implies the necessity of interior and boundary dissipations to reduce the energy.In chapter 4,we deal with the wave equation in a Riemannian manifold with acoustic boundary conditions.Under suitable geometric conditions(G)and proper assumptions(A)on the functions of the system,the exponential stabilization is obtained by Lyapunov ap-proach and the multiplier method.With another assumption(R),we can rewrite the wave equations with memory type acoustic boundary conditions containing the memory term of(?)_?? and(?)_??ton?₁,which is not available in the literature,as the first one,so we can get the the exponential stabilization.Finally an example is given in the end to show that the assumptions are reasonable.In(4.1.1)we have considered the case M=0 in(1.1.2),but in remark 4.2 we also point out that letting M=l(x)and modifying the energy functional,exponential stabilization can also be obtained.In the last chapter,I make a brief summary of the whole paper and explain my next thoughts and plans.