Global Exponential κ-dissipative Semigroups And Global Attractors For Inhomogeneous Reaction-diffusion Equations In Unbounded Domains

In this doctoral dissertation, we mainly consider the long-time behaviors of the solutions of infinite dimensional dissipative dynamical systems. This paper is divided into two parts. In the first part(Chapter 3), by the concept of noncom-pact measure κ, we firstly introduce a new concept for semigroups, called global exponential κ-dissipativeness. Then, we prove that if a continuous semigroup {S(t)}t＞o has a bounded absorbing set and is global exponentially κ-dissipative , then there exist a compact set A*, which is positive invariant and exponentially attracts any bounded subset, see Theorem 3.2. Further more, the Hausdorff di-mension of the compact set.A* is finite if the global attractor of{S(t)}t≥o is also finite dimensional, see Theorem 3.6. In order to apply this concept, several suffi-cient conditions to verify that a semigroup is global exponentially κ-dissipative are given, see Theorems 3.8,3.9,3.12,3.13 and 3.14. At the end of this chapter, the results are illustrated with two applications:reaction-diffusion equations and damped semilinear wave equations, see Theorem 3.17 and Theorem 3.20.In the second part(Chapter 4 and Chapter 5), we consider the existence of global attractors for some inhomogeneous reaction-diffusion equations in the whole space Rn. In Chapter 4, we obtain the global attractors for a class of equations with distribution derivatives terms in L2(Rn) by the ω-limit compact-ness method and the cutting function technique, see Theorem 4.7. In Chapter 5, We discuss two types of the real Ginzburg-Landau equations with inhomoge-neous terms, in the first type, the primary operator -Δ is not positive.In the second type, the primary operator is strongly indefinite near the origin and the nonlinear term is coercive. Since the primary operator of first type is not pos-itive definite in H1(Rn), so the Gronwall inequality can not be derived and the corresponding semigroup does not possess any bounded absorbing set in L2(Rn). Thus, by a special monotonicity method, we first prove that the equation has a bounded absorbing set. Then using the cutting function technique, we obtain the existence of global attractor A in Lp(Rn), which attracts any bounded subset in L2(Rn) ∩Lp(Rn), see Theorem 5.5. In the second type of equations, we first prove that the equation has a absorbing set in Hk1(Rn). Then we obtain the existence of global attractor (?)in L2(Rn) by Sobolev embeddings, see Theorem 5.12. Fur-thermore by using the Z2-index theory, we verify that (?) is infinite dimensional, see Corollary 5.14.