Research On Well-Posedness For Fourth-Order Boussinesq Equation And Cahn-Hilliard Equation

This thesis aims to reveal the relationship between the initial data and the well-posedness of solutions to the initial boundary value problem for Cahn-Hilliard equations with two different growth type source terms(such as the exponential source terms,the polynomial source terms)and the Cauchy problem for the generalized Boussinesq equation.In the framework of potential well 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 arbitrarily positive initial energy)in order to reveal the effect of the initial data on the well-posedness of solutions in detail.We focus on a comprehensive investigation of the qualitative behavior for the initial boundary value problem for 2D hyperbolic Cahn-Hilliard equations with polynomial nonlinearity in Chapter 2.This equation can be used to describe the phenomenon that when a molten alloy such as iron-nickel is rapidly quenched to a low temperature,the alloy is decomposed into particles.Based on the potential well method,multiplyer method and concave method,the global existence,exponential decay and finite time blow up for the subcritical and critical initial energy are established.Further,it is demonstrated that there exists some initial data assuring the finite time blow up for arbitrarily positive initial energy with the aid of an adapted concave method.We deal with the qualitative behavior for the initial boundary value problem for 2D hyperbolic Cahn-Hilliard equations with exponential nonlinearity in Chapter 3.In the framework of potential well,the global existence,exponential decay and finite time blow up for subcritical and critical initial energy are obtained through the multiplyer method,concave method and Trudinger-Moser inequality.Further,an arbitrarily positive initial energy blow up result is displayed by adapted concave method.The result of the Cauchy problem for a class of generalized Boussinesq equations finite time blowup is considered in Chapter 4.This equation not only describes the longitudinal displacement of the rod but also characterizes the propagation of nonlinear waves in the waveguide.A finite time blow up result for the Cauchy problem of this equation with exponential type source term is considered in this Chapter.Based on an adapted concave method,we derive a sufficient condition on initial data leading to the arbitrarily positive initial energy finite time blow up.