Study Of Blow-up Of Solutions To A Class Of Fourth-order Nonlinear Parabolic Equations

This thesis mainly deals with the behavior of blow-up solutions to a class of fourth-order nonlinear parabolic equation.This thesis is divided into three chapters.In the first chapter,we introduce some background of our problem studied and the related works.At the same time,our studied problem are given as follows:(?)where Ω is a bounded smooth domain in RN.Chapter two will be devoted to studying the properties of blow-up solutions as well as the estimate of upper bound of blow-up time.Our main tools are energy estimate method and the concavity method.To be more precisely,our main result is as follows:Theorem 0.1.Assume the weight function k(t),the initial energy Ψ(0),and p,m satisfy the following conditions(?)Then the solution of Problem(0.2)blows up in finite time.That is,there is a positive number T]such thatIn the last chapter,we apply basic knowledge of Real Analysis to prove a new Sobolev inequality.Furthermore,we construct a control function and apply the new Sobolev inequal-ity and energy inequality to establish a differential inequality which implies the estimate of lower bound of blow-up time.For simplicity,we only consider space dimension N = 2,3.That is Ω(?)RN,N = 2,3.Our main result is as follows:Theorem 0.2.Assume that p>m/2 and that k(t)satisfies k’/k ≤αfor some constant α≥0.Then the blow-up time T1 satisfies the following estimate:(?)where β,γ,θ(0)are defined as follows(?)...