Asymptotic Behavior Of A Class Of Fourth-order Parabolic Equation

In this paper,we mainly consider the large time behavior of global solutions to the Cauchy problem of fourth order parabolic equation in one dimension space:with f(u) satisfyingHere are our main results:(i) Suppose the initial data satisfies .and its norms are sufficiently small, we establish the time-decay rate of global solution in L^p-norm :Also , we can prove that for q > 2,the large time behavior of global solution,inL^P(R)(1 < p ≤∞), is described by U = e^t(△-△)u₀:Here U = e^t(△-△)u₀ is the solution of the following homogeneous equation with initial data u₀which satisfies the following estimates(ii) Moreover,for the case p = 1,we suppose :Here,I_β is the Riesz Potential,which satisfies I_βu₀=C(?)_R|x-y|^β-1u₀(y)dy. Under these assumption ,We have the L¹-decay rate of the solution to equation (0.9):thenthenThis is the first result about L¹-decay to fourth order parabolic equation.Remark 1: We emphasis q > 2 and the smallness of initial data in the above results .For the case 1 < q ≤ 2 and the general initial data ,it remains to be considered.Remark 2: The method developed in this paper can also be used to consider the high dimensional case ,and to consider the large time behavior of global solution to fourth order parabolic equation with second order homogeneous parts, such as...