Finite-time Blowup And Existence Of Global Solutions Of Stochastic Cahn-Hilliard Equations

In the thesis, we consider the stochastic Cahn-Hilliard equations formally given by(?)where D ? R is a bounded domain with smooth boundary ?D,κandγare constants withκ> 0, and {Wt, t≥0} is a standard one-dimensional Brownian motion on a stochasticbasis (Ω, F, (Ft)t≥0, P). The stochastic Cahn-Hilliard equations [8, 15, 21] describes phaseseparation of a binary alloy in the presence of thermal ?uctuations. In the model, u representsthe separate ratio between the two species and the noise term W˙t accounts for the thermal?uctuations.There is a large amount of literature on the mathematical analysis of determine Cahn-Hilliard equations as a well model [5, 11, 13, 22]. The model describes the process of phaseseparation of a binary ?uid when is quenched blow a critical temperature, spontaneouslyseparates and forms domains pure in each component. In determine case, the properties ofthe solution for Cahn-Hilliard equations are given as follow [13]: i.e.κ= 0, ifγ< 0,the solution of Cahn-Hilliard equation must blow up in a finite time for large initial data;and ifγ> 0, there exist global solutions. Da Prato and Debussche [9] first proposed thestochastic Cahn-Hilliard equation, and studied the existence, uniqueness and regularity ofglobal mild solutions. Cardon-Webber [6, 7] used the Green function to prove existence anduniqueness of an almost surely continuous solution and used the Malliavin calculus to studythe existence of the density, as well as strict positivity of the density.Recently, many authors studied the properties of the solutions of SPDE. C. Bandle etal. [1] studied a stochastic reaction-di?usion equation with a polynomial nonlinearity, andgave a su?cient condition for the blow up of the solution in finite time. M. Dozzi and J. A.Lo′pez-Mimbela [12] studied a stochastic semi-linear equation, estimated the probability of finite-time blowup of positive solutions, and obtained the probability of existence of positiveglobal solutions.Our purpose of this paper is to study the long time behavior of stochastic Cahn-Hilliardequations by using of a related random Cahn-Hilliard equation .(?)The property of the solution in (1) depends the coe?cientγin the nonlinear term f. InSection 3, we study the blowup behavior of the solution v for (2) whenγ< 0. We considerthe weak solution of (2), and solve a Bernoulli equation. The solution of the Bernoulliequation can be written as integrals of exponential Brownian motion with drift. Using theresults of Marc Yor [24] (or M. Dozzi and J. A. Lo′pez-Mimbela [12]), we can estimate theprobability in finite time of u and v. In another words, the solution v may blow up in a senseof a probability, and we still obtain lower bounded of the probability. In Section 4, since wecan easily obtain local existence and uniqueness by applying the Picard iteration scheme, weuse a priori estimates on v and obtain global solutions u and v whenγ> 0.The main results are given as follow:Theorem 0.1 Ifγ< 0, the lower bound to the probability for finite-time blowup ofsolutions of (1) is given by (?)Theorem 0.2 Ifγ> 0, then for any initial data g(x)∈H2(D) and T > 0 there exists aunique global solution H4,1(D×[0.T]).