Research On Blow-Up Of Solutions For A Class Of Forth-Order Parabolic Equations

In this paper,we deal with the initial-boundary value problem for a class of fourth order parabolic equation involving both the Hessian and the heat source function,where ?(?)R2 is an open,bounded domain with smooth boundary(?)?,0??,?<+?and ?,? are not zero at the same time,1<p<+?.Under some appropriate conditions on the initial value and relevant parameters,the concavity method is used to prove that the solution of the above problem blows up in finite time with different norms.Furthermore,using the energy integral inequalities,we get estimates for upper and lower bounds of blow-up time,with L2-norm and H02-norm,respectively.This paper is divided into five sections.Section 1,we introduce the development of the fourth order parabolic problems with Hessian and the main work of this paper.Section 2,we discuss the existence space of the solution to the above problem and give some preliminary lemmas.Section 3,Under some appropriate conditions on the initial value and relevant pa-rameters,we prove that the solution of the above problem blows up in finite time with different norms.Section 4,we estimate the upper and lower bounds of the blow-up time of the solution to the above problem.Section 5,we list several questions which need to study in the future.