Existence Of Global Attractors For Semilinear Parabolic Quation

This thesis is mainly a contribution to the study of longtime behaviors of semilinear parabolic equation on unbounded domains. Next we consider the asymptotic behavior of global strong solution for a class of nonlinear hyperbolic equations on bounded domains.As for our problem, we shall confront two difficulties when considering the existence of (L²(Rⁿ),H¹(Rⁿ))-global attractor. One difficulty is to verify the necessary asymptotic compactness of the semigroup associated with the solution. As we known, the key is to get the asymptotic compactness of the semigroup when we consider the existence of global attractor, The problem is more challenging whenΩis unbounded since the embeddings are no longer compact, so that the existence of compact attracting set may not be directly sufficient. Lately some authors have developed methods for verifying the necessary asymptotic compactness, e.g. asymptotic smoothness, which is no longer available for us since the external term Dⁱfⁱ belongs to H^-1. The other one is that we can't directly use the methods of [19] proving the existence of (L²(Rⁿ),L^p(Rⁿ))-global attractor to obtain our result, because the nonlinear function f has no any restriction and there are no nested relation between H¹(Rⁿ) and L^p(Rⁿ) as p large enough. So we find a new method named contractive function, it comes from the reference [17]. Here we firstly use the result of [19] to obtain the asymptotic compactness of bounded absorbing set in L^p(Rⁿ) and L²(Rⁿ), then we construct the contractive function of H¹(Rⁿ) and get the asymptotic compactness of bounded absorbing set in H¹(Rⁿ), the existence of the global attractor follows from it immediately.In Chapter 2 we firstly recall some basic concepts and theories about the global attractor for the infinite dimensional dynamical systems, and present a criterion for the asymptotic compactness using the concept of the contractive function. As an application, in Chapter 3, we consider the longtime behavior of the semilinear parabolic equation on unbounded domains and get the existence of global attractor.In Chapter 4 we study the asymptotic behaviors of global strong solution for nonlinear damping hyperbolic equation and improve the result of reference [34].