Blow-Up Of Solutions For Several Classes Of Nonlinear Evolution Equations

The problem on the solution of partial differential equation came from geometry,physic-s,chemistry,biology,etc.Therefore,to research the problem is extremely significance.In mathematics,the problem covers many topics such as existence,uniqueness,decay,asymp-totic behavior and so on.This paper will study the initial boundary value problems on several classes of partial differential evolution equations.The main considerations are existence and blow-up of solutions.According to the content,this paper is divided into the following four chapters.Chapter 1 Preference,we introduce the development of nonlinear wave equation and parabolic equation,and the main contents of this paper.Chapter 2 We consider the nonlinear wave equation utt-μ1△ut-div(σ(▽u)▽u)-μ2div(β(▽ut)▽ut)+f(ut)=g(u),with Dirichlet initial boundary conditions.Under certain assumptions on functions σ,β,f and g,we show that any weak solution with negative initial energy blows up at finite time.Chapter 3 We discuss the nonlinear wave equation with variable exponents:utt-△ut-div(|▽u|α(·)-2▽u)-div(|▽ut|β(·)-2▽ut)+a|ut|m(·)-2ut＝b|u|p(·)-2u,where a,b＞0 are constants and the exponents α(·),β(·),m(·)and p(·)are given measurable functions.Under certain assumptions,we establish the existence of a local weak solution and show that the weak solution with negative initial energy blows up in finite time.Chapter 4 We establish some sufficient conditions on the heat source function and the heat conduction function of the parabolic equation to guarantee that u(x,t)blows up at finite time,and give upper and lower bounds of the blow-up time in multi-dimensional space.