A Class Of Stochastic Partial Differential Equation

In the past years, the standard K-S equationhas been payed widely attention. Many results, such as existence, uniqueness, regular-ity and the property of attractor, have been established. In contrast with the standardK-S equation, the stochastic model describes more real world. With development ofmathematics, it appears more considerate model which re?ect the behavior's details:Moreover, stochastic Kuramoto-Sivashinsky equation with a nonlocal termwas studied by many authors, whereσ> 0 is a constant and the noise term W(t) is aninfinite dimensional Wiener process.Recently, Y. Wang et al. studied a generalized stochastic Kuramoto-Sivashinsky equation with a nonlocal termwhereαand L are positive constants.In this paper, according to the stochastic K-S equation, we introduce many recentresults and methods.According to the conservation law and the studying phase separation of binaryalloys and some other media, we haveFor example, binary alloys consisting of two atomic species, u denotes the local con-centration of one of the species. The inter-di?usion current j is given bywhere M denotes the mobility and m is the local chemical potential, defined as theFrechet derivative of the free-energy functionalThe usual form ofψiswhereβ> 0 for temperatures lower than the critical one, and consequentlyψhas the double well structure below the critical point ([1]). The noise termθsatisfiesAssuming M = 1,from deterministic Cahn-Hilliard equation we obtain the equationwhere ?(u) =ψ(u).In the past 30 years, the deterministic Cahn-Hilliard equation was widely studied.Elliott and Zheng Songmu established existence, uniqueness and regularity[5]. Andthe asymptotic behavior was also studied[6].We suppose that ? is a bounded smooth domain and we supplement equation bythe zero-mass ?ux boundary conditionand by the natural boundary conditionWe introduce the results and methods according to the mentioned stochastic Cahn-Hilliard equation together with initial and boundary conditions and the related mutativestochastic Cahn-Hilliard equation.Fractional equations, both partial and ordinary ones, have received more attentionin recent years. Various phenomena in physics, such as di?usion in a disordered, fractalmedium, in image analysis, or in risk management have been modeled by means of fractional equations. Some of them are of high order. However, comparatively fewpublications treat stochastic fractional partial di?erential equations. These operatorsgenerate symmetric stable semigroups when the order of derivation is less than 2. Theaim of this work is to generalize, in the framework of the multi-parameter processes,the results of Walsh [27] to nonlinear stochastic fractional partial di?erential equations(SFPDE) of high order containing also derivatives of entire order.In the paper, we simple introduce some fractional equations and the property ofits solution.