The Finite-time Blow-up Of The Solutions Of Several Types Of Evolution Equations Under High Initial Energy

Once the local existence result for the solution of the evolution equation has been established,a natural problem follows:whether or not does a solution exist globally?If the maximal existence T~*of the solution is finite,some norm of this solution usually turns out to be unbounded on the life span[0,T~*),that is to say,this solution blows up in finite time.In this thesis,by applying the potential well method,Levine's concavity method,differential inequality and critical point theory,we study some kinds of evo-lution equations and give the sufficient conditions for finite time blow-up of solutions,life-span estimation and some other related conclusions,with the initial data at high energy level.In Chapter 1,we introduce the history and current research states for the global existence and finite time blow-up of solutions for evolution equations,and review the history and development of some common approaches,in particular,the potential well method and Levine's concavity method.We also introduce the structure of this thesis and give some conventions and symbols at the end of Chapter 1.In Chapter 2,we study the initial boundary value problem for a class of pseudo-parabolic(or parabolic)equations with time varing coefficients.By using the potential well method,Levine's concavity method and some differential inequality techniques,we obtain the finite time blow-up results under some assumptions involving the initial energy and initial data and establish the lower and upper bounds for the blow-up time.In particular,we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.In Chapter 3,we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term.Under suitable assumptions on the initial energy,initial data,the exponent p of the source term and the relaxation function g,we obtain the finite time blow-up results of solutions,by using the concavity method,an improved method of potential well fam-ilies involving time t and some differential inequality techniques.We also derive the upper bounds for the blow-up time.Finally,we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.In Chapter 4,we consider the initial boundary value problem for a class of thin-film equations with a p-Laplace term and a nonlocal source term.By using the fountain theorem,the concavity method and some differential inequality techniques,we prove that there exist weak solutions for the problem with arbitrarily initial energy that blow up in finite time.We also obtain the upper bounds for the blow-up time.In Chapter 5,we study a three-dimensional(3D)viscoelastic wave equation with nonlinear weak damping,supercritical sources and prescribed past history.By estab-lishing a special Liapunov's function and using the concavity method,we obtain some finite time blow-up results with high initial energy.In particular,for the case of linear weak damping,we obtain the finite time blow-up result with the estimation for the up-per bound of the blow-up time.We also obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.In Chapter 6,we study the initial boundary value problem for a Petrovsky type equation with a memory term,nonlinear weak damping and a superlinear source.By establishing a special Liapunov's function,we obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.For the case of linear weak damping,we obtain the finite time blow-up result with the estimation for the upper bound of the blow-up time,under some relative strong assumptions.