Decay Properties And Well-Posedness Of Coupled Evolution Equations With Damping

The present dissertation is concerned with some coupled evolution systems with damping effect given by memory terms. These systems are mainly from the viscoelastic science. By applying Lyapunov auxiliary function method, the multiplier method and other theoretical methods, we get well-posedness and decay estimates for the coupled systems. Moreover, applications to some concrete systems are provided as well.First of all, in Part 1 (Chapters 2-5), we consider an abstract system of two coupled evolution equations with memory term only in one equation.· If the memory kernel function is positive, using the properties of the positive k-ernel and some semigroup theories, we obtain existence of global solutions (with small initial data for semi-linear case). The details can be found in Chapter 2;· In Chapter 3, let the memory kernel function be monotonous and integrable and the system, linear or semi-linear, have the same wave speed. By constructing new auxiliary functions to control the memory energy term, we obtain uniform energy decay at the rate of t-1, which is an optimal energy decay rate;· In Chapter 4, if the memory kernel function is monotonous and integrable and the system has different wave speeds and is linear, we still obtain the energy decays to zero at the rate of t-1 and the estimate is also an optimal energy decay rate;· In Chapter 5, if the memory kernel function is only oscillating (not necessarily decreasing), with some other assumptions we show that the energy of the system decays to zero.In Part 2, we study an abstract system of two coupled linear equations with two memory terms. Through using the strongly positive kernel theory and multiplier method, we obtain a control for the kinetic energy. Noting the system itself, we also find that the potential energy can be controlled by the kinetic energy. Then, we obtain the energy decays at the polynomial rate, when the kernel is monotonous and integrable. Moreover, this result is interesting and new even for the corresponding case of single equation with damping term. The details can be found in Chapter 6.