Asymptotic Behavior For A Class Of Random Nonclassical Reaction-diffusion Equations

We consider the asymptotic behavior for the following class of nonclassical reaction-diffusion equationswhere u(x,t) is a unknown function, f(u)∈C1(R,R) is the nonlinear external forces which satisfies some conditions.∑jm=1gjdwj(t) is the random term.Eq(0.2), which appears as a class of nonclassical reaction-diffusion equations, like the fluid mechanics, solid mechanics and heat conduction theory. Since there is a term-Δut in system (0.2), the regularity of the solution semigroup is not as high as reaction-diffusion equation. The compactness of the system cannot be obtained by Sobolev compact embedding theorem. We must use new methods to overcome this difficulty.In Chapter 3, we give a conception of exponent absorbing property. It is proved that the global attractor (?)1 in H01(Ω) is bounded in D(A) by resolution of operator and asymptotic priori estimate method. Furthermore, it is proved that (?)1 is the same to the global attractor in D(A).In Chapter 4, we consider the non-autonomous nonclassical diffusion equation in unbounded domain. We will make use of cut-off function and contractive function to get the existence of the uniform attractors, where the external force f is not translation compact.We discuss the random system in Chapter 5. By using asymptotic priori es-timate to obtain the compactness of term f. The main method of dealing with the random term is to change the system into a system with random parameter. Furthermore, we establish the general existence theorem of random attractor.