Existence Of Weak Solution And Attractors For Semilinear Nonlocal Reaction Diffusion Equation

As early as 300 years ago, mathematicians had proposed the fractional derivative. But until recent decades, mathematicians and natural scientists focus their eyesight on it for which successfully describes many nonlocal and time-delayed phenomena.In mathematics, the diffusion of the classical reaction diffusion equation is reflected by the Laplace operator, but Laplace operator only reflects local effects of the space. When we replace the diffusion into nonlocal operator, it will produce anomalous diffusion. In this paper, we consider a nonlocal reaction diffusion equation, we replace the Laplace operator of classical equation into the nonlocal operator A?, where A?is a fractional Laplace operator,defined by singular integrals.For classical reaction diffusion equation, the theory of existence of weak solutions and attractor are more complete than the nonlocal equation. In this paper, we discuss the reaction diffusion equation with such a nonlocal operator exists weak solutions in the certain Sobolev space and global attractors in the space L2(D).In this paper, we review and introduce some notions about fractal Sobolev space,nonlocal vector calculus and some imbedding theorem about Sobolev spaces, then we prove that the weak solution of the nonlocal reaction diffusion equations exists in the certain Sobolev spaces by the Galerkin approximation method. Furthermore, on the basis of the existence of weak solutions, we consider long time behavior of the dynamical system. Since the global attractor is a proper concept which characterizes long time dynamic behavior of the dynamical system, it indicates that the system will eventually tend to some compact invariant set. In this article, we obtain that a specific model of the nonlocal equation exists global attractors in the space L~2(D).