Study On The Properties Of Solutions For Two Classes Of Coupled Nonlinear Evolution Equations

Partial differential equation is an important branch of modern mathematics,which has a variety of important applications in many fields,such as physics,differential geometry,computational mathematics,image processing and so on.Nonlinear evolution equations is very significant partial differential equations,and the study for the decay rate and blow-up of the solutions to nonlinear evolution equations have been the one of the vital parts of partial differential equations.In this thesis,we mainly study initial boundary value problems of two classes of coupled nonlinear evolution equations,and show that the global solutions are decay and the local solutions blow up.First of all,we discuss the properties of the solutions for coupled nonlinear viscoelastic plate equations with initial-boundary value conditions.We prove that,for certain initial data in the stable set,the decay rate estimate of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s inequality and modified potential well method.Conversely,for certain initial data in the unstable set,we use the perturbed energy method to show that the solution blows up in finite time when the initial energy is not larger than some positive number.Secondly,we discuss coupled higher-order nonlinear wave equations with initial-boundary value conditions.For certain class of relaxation functions and certain initial data,we show that the decay rate of the energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type.Also,we prove that nonlinear source of polynomial type is able to force solution to blow up in finite time even in presence of stronger damping.