The Existence Of Exponential Attractor For Semigroup In Banach Space And Its Application

In this doctoral dissertation, we consider the existence of exponential attractor in Banach space when there isn't a compact absorbing set for equations. As an application, we prove the existence of exponential attractor in L^2p(Ω) for the following Eq., (2).Let {Sⁿ}_n=1^∞be the discrete semigroup of Banach space X, and (?) be the global attractor for {Sⁿ}_n=1^∞. We assume that S is of class C¹ on 5_∈0((?)), and on every point of B_∈0((?)), the linearized operator can be decomposed as L=K+C with K compact and ||C|| <λ< 1. Then we can prove the existence of exponential attractor for discrete semigroup {Sⁿ}_n=1^∞, see Theorem 3.1.1.In the proof of the existence of exponential attractor for discrete semigroup {Sⁿ}_n=1^∞,we note that if B_∈0((?)) is a positively invariant set for S, then for any integer m≥1, we haveand then, construct an exponential attractor on some proper subset (?)For the continuous semigroup {S(t)}_t≥0 on Banach space X, and its global attractor (?). We note that for any∈₀> 0, there exist T^*, such that S(T^*) : B_∈0((?))→B_∈0((?)). Letting S=S(T^*), by proving S is of class C¹, we obtain the existence of exponential attractor M_d for discrete semigroup {Sⁿ}_n=1^∞. and then applying the standard manner (e.g., see [47] Chap. 3), we can obtain the existence of exponential attractor for the continuous case, see Theorem 3.2.1.Meanwhile, we apply it to the following reaction-diffusion equation.whereΩis a bounded smooth domain, p≥2, the initial value u(0)∈L²(Ω), the external force g∈L²(Ω), the nonlinear term f∈C² and satisfying the following assumption: We prove the existence of exponential attractor in L^2p(Ω) by our method, which is difficult to obtain it by the general method.In particular, it is difficult to prove the Frechet differential for the semigroup {S(t)}_t≥0 in L^2p(Ω) directly. However we give a translation S₁(t)(u₀-u^*)(?) S(t)u₀-u^*, where u* is the global minimal solution of the following elliptic equation:Obviously, the translation semigroup (?)(t)(?)S₁(t)(u₀-u^*) satisfies the following equation:Using the same argument as in [143] given by Marion. We can show that S₁(t)(u₀-u^*)∈L^∞(Ω), for t>0. It implies that with the simple translation, we obtain an more regular system. If we prove that the {S₁(t)}_t≥0 is Fr(?)chet differential in L^2p(Ω), from the equality "S₁(t)(u₀+Ï…₀-u^*)-S₁(t)(u₀ - u^*) = (S(t)(u₀+Ï…₀)-u^*)-(S(t)u₀-u^*) = S(t)(u₀+Ï…₀) - S(t)u₀", it is easy to prove that semigroup {S(t)}_t≥0 is also Frechet differential in L^2p(Ω).This thesis consists of four chapters:In Chapter one, the background and major results on the theory of dynamical systems, in particular, about exponential attractor, and the main idea and results of this doctoral dissertation are introduced.In Chapter two, some preliminary results and definitions that we will used in this thesis are presented.In Chapter three, we prove the existence of exponential attractor in Banach space for the semigroup when there isn't a compact absorbing set.In Chapter four, As an application, we prove the existence of exponential attractor in L^2p(Ω) for Eq., (2) by our method presented in Chapter three.