Blow-up At Infinity Of Solutions To A Fourth-order Parabolic Equation With Logarithmic Nonlinearity

In this paper,we consider blow-up at infinity of solutions to a fourth-order parabolic equation with logarithmic nonlinearity(?)where Ω is a bounded domain of Rn(n≥ 1)with smooth boundary(?)Ω,v is the unit Outward normal on(?)Ω,u0(x)∈H02(Ω).This paper is divided into three parts.In the first part,we summarize the background of the problem studied in this paper and the related work at home and abroad,and then we introduce the discussed problems and the methods used to prove the conclusion.In the second part,we introduce some preliminaries,including sets and functions related to potential well and lemmas needed to prove the main conclusions of this paper.In the third part,we combine Galerkin approximation with a priori estimate to prove the local existence of weak solutions of the problem(0.1),and then we prove that the weak solutions blow up at infinity under weaker conditions.The main results of this paper are the following:Theorem 1.Let u0 ∈ H02(Ω),then there is a constant T0>0 such that the problem(0.1)has a weak solution u(x,t)in Ω x[0,T0).In addition,the weak solution u(x,t)satisfies the following energy identity(?)Here(?).Theorem 2.Suppose u0∈H02(Ω)satisfies I(u0)<0,then the weak solution u=u(x,t)blows up at infinity,i.e(?)Here(?).