The Asymptotic Behaviors Of Solutions For Some Evolutionary Equations With Delays

In this doctoral dissertation, we consider the asymptotic behaviors of solutions for the following evolutionary equations with dalays:(1) nonclassical di?usion equations with delays ?tu- ?t?u- ?u = f(u) + g(t, ut) + k(t) in ? × [τ, ∞),u(x, t) = 0 on ?? × [τ, ∞),u(x, τ + θ) = ?(x, θ), x ∈ ?, θ ∈ [-h, 0],(2) p-Laplacian equations with delays ?tu- div(|?u|p-2?u) = f(u) + g(t, ut) + k(t) in ? ×(τ, ∞),u(x, t) = 0 on ?? ×(τ, ∞),u(x, τ + θ) = ?(x, θ) x ∈ ?, θ ∈ [-h, 0],(3) weakly damped wave equations with delays ?ttu + η?tu- ?u = f(u) + g(t, ut) + k(t) in ? × [τ, ∞),u(x, t) = 0 on ?? × [τ, ∞),u(x, t) = ?(x, t- τ), x ∈ ?, t ∈ [τ- h, τ ],?tu(x, t) = ?t?(x, t- τ), x ∈ ?, t ∈ [τ- h, τ ],where τ ∈ R, the constant η > 0, g is a operator acting on the solutions containing some hereditary characteristics, k(·) is the time-dependent external force term, ?is the initial datum on the interval [τ- h, τ ], h(> 0) is the length of the delay e?ects, and for each t ≥ τ, we denote by utthe function de?ned in [-h, 0] with ut(θ) = u(t + θ), θ ∈ [-h, 0].The main works of this paper are to establish the existence of the pullback attractors for the above evolutionary equations with delays when the nonlinearity f satis?es critical growth or polynomial growth of arbitrary order p- 1(p ≥ 2).This paper is organized as follows.In Chapter 3, we prove the existence of the pullback attractors for the nonclassical di?usion equations with delays. The nonclassical di?usion equation is atype of special equation between parabolic equations and wave equations. If we add a disturbance coe?cient in the damped term-?t?u, we will establish some relation between parabolic equations and wave equations, and our work will be signi?cant. In this section, we consider two types of nonlinearity f : one is the case of critical growth, and the other one is the polynomial growth of arbitrary order p-1(p ≥ 2). For the case of critical nonlinearity, we use the decomposed technique to overcome the di?culty brought by the critical exponent and delays term, then obtain the existence of pullback attractor in CH10(?). For the case of polynomial growth of arbitrary order p- 1(p ≥ 2), ?rstly, we establish the well-posedness of solutions; then, we verify the existence of pullback attractor in CH10(?)for the process {U(t, τ)}t≥τby devising the energy inequality.In Chapter 4, we consider the p-Laplacian equations with delays, and the nonlinearity f satis?es the polynomial growth of arbitrary order p- 1(p ≥ 2).p-Laplacian equation is a type of strongly damped quasilinear parabolic equation.Our problem contains the delay term g(t, ut) and the time-dependent external forcing k(·), which lead to some methods and techniques of obtaining compactness for semilinear parabolic equations be invalid, so we will confront some di?culties when we verify the compactness for the process {U(t, τ)}t≥τ. In this section, we?rstly establish the well-posedness of the solutions; then, we prove the existence of the pullback attractors in CL2(?)for the process {U(t, τ)}t≥τby devising special energy functional.In Chapter 5, we prove the existence of the pullback attractors for the weakly damped wave equations with delays, and the nonlinearity f satis?es critical growth.The weakly damped wave equation with critical growth is a typical weakly damped equation. In the development process of in?nite dimensional dynamical systems,many researchers often make it as the studying object to introduce some methods and theories. In this section, we will obtain the existence of the pullback attractors in CH10(?)×CL2(?)for the process {U(t, τ)}t≥τby constructing the energy functional and combining with the idea of contractive function method.Finally, in Chapter 6, we end this paper by some problems that we will con-sider in the future.