The Research On Well-posedness For Some High Order Damped Wave Equations

This thesis aims to reveal the relationship between the initial data and the well-posedness of solutions to the initial boundary value problem for a class of nonlinear high-order dissipative wave equations with three various source terms of different growth type(such as the exponential source terms,the polynomial source terms and the logarithmic source terms)in the framework of potential well along with the functional analysis.Moreover,this thesis makes a deep investigation on the qualitative behavior of solutions to classify three different initial energy(subcritical initial energy,critical initial energy and supercritical initial energy),which implies the effect of the initial data on the well-posedness of solutions in detail.Chapter 2 is concerned with the arbitrarily positive initial energy finite time blow up of the solution to the initial boundary value problem for a class of fourth-order damped wave equation with exponential growth nonlinearity.By introducing a new auxiliary function and utilizing the adapted concave method,the finite time blow up of the solution with arbitrarily positive initial energy is given.Chapter 3 investigates the local existence and uniqueness as well as the global well-posedness of solutions for a class of nonlinear high-order damped wave equation involving the initial boundary value problem with nonlinear polynomial source terms at different initial energy levels.In the spirit of Galerkin method and Contraction Mapping Principle,both the existence and uniqueness of local solution are established.It is also shown that the global existence,exponential decay and finite time blow up of the solution for subcritical and critical initial energy are established in the framework of potential well.Especially for arbitrarily positive initial energy a finite time blow up result is given.Further,a lower bound of blow up time is estimated.Chapter 4 is devoted to the qualitative behavior to the initial boundary value problem for a class of fourth-order strongly damped evolution equation with nonlinear logarithmic nonlinearity source terms at different initial energy levels.In the spirt of Galerkin method and Contraction Mapping Principle along with logarithmic Sobolev inequality,the existence and uniqueness of local solution are proved.In the framework of potential well,the global existence,exponential decay and infinite time blow up are given for subcritical and critical initial energy.Further,an infinite time blow up result for arbitrarily positive initial energy is derived.