Dynamical Properties Of Two Kinds Of Nonlinear Wave Equations With Dispersed Terms

The partial differential equation(PDE) is developed rapidly in the 19 th century, it is used widely in many ?elds, which has been an signi?cant part of the contemporary mathematics especially in recent years. On the other hand, from the point of the mathematic,the study of the PDE promotes the development of the ordinary differential equation,calculus of variations, differential geometry, theory of functions, algebra, series development and so on, and it becomes an important bridge between the branches of the pure mathematics and the ?elds of the natural science, the engineering technology.The partial differential equation includes three parts: the hyperbolic equation, the parabolic equation and the elliptic equation. In this paper, we will prove the uniform decay of energy for a viscoelastic equation(s) with damping by drawing into the energy function.The thesis is divided into three chapters according to contents.In Chapter 1, we introduce the history of the partial differential equation and the viscoelastic equation.In Chapter 2, by drawing into a new function and creating the energy method, we prove the exponential and polynomial decay for a quasilinear viscoelastic problem with damping:where ? is a bounded domain in Rnwith smooth boundary, ρ > 0 is a number and g(t)is a positive function that represents the kernel of the memory term which satis?es some conditions to be speci?ed below.In Chapter 3, we investigate the uniform decay rate of the energy for solutions for thenonlinear viscoelastic problem with damping without drawing into the auxiliary function:where ? is a bounded domain in Rn(n ≥ 1) with smooth boundary ??, g(t) is a positive function that represents the kernel of the memory term which satis?es some conditions to be speci?ed below, a, b, h and f are real valued functions which satisfy appropriate conditions. We will establish a general uniform stability result for kernels of generaltype decay, from which the usual exponential decay and the polynomial decay results are...