Nonlinear Evolution Equations The Overall Solution Of The Asymptotic Behavior Of Steady State Solutions

Nonlinear evolution equations, i.e., partial differential equations with time t as one of the independent variables, arise not only from many fields of mathematics, but also from other branches of science such as physics, biological science and ma-terial science. The asymptotic behavior of global solutions of nonlinear evolution equations, in particular, convergence to a certain stationary solution as time tends to infinity, is one of the main concern in the field of nonlinear evolution equations.The main concern of this thesis is these nonlinear evolution equations which are autonomous and (i) they have unique bounded global solution for any given initial value; (ii) their nonlinearity are analytic; (iii) they can generate a gradient system. For these equations, to study whether the global solution will converge to a certain stationary solution as t→+∞, we will use different methods in the high-dimensional (n≥1)case and one-dimensional (n=1) case, respectively.In the high-dimensional (n≥1) case, we will get the convergence by proving the Lojasiewicz-Simon inequality. This method has been developed from 1983, when L. Simon studied a system whereΩ(?)Rn is a bounded domain with smooth boundaryΓ. He proved that if there exists a bounded global solution and the nonlinearity f is analytic in the dependent variable, then the convergence holds. His idea relies on the extension of a gradient inequality by S. Lojasiewicz for analytic function defined in Rn to the infinite-dimensional space. In the one-dimensional (n=1) case, the corresponding stationary problem becomes a two-points boundary value problem for a nonlinear ordinary equation. Thus the set of stationary solutions can be studied more clearly. Especially, we can usually use the phase plane analysis method to prove that the set of stationary solutions is discrete. Then we can apply the well-known results in the gradient system to conclude the convergence. The main idea of the phase plane analysis is reduced problems to finding intersection points of two analytic functions of one variable.We have to notice that the idea of getting the convergence result by proving the discreteness of the set of stationary solutions usually can not be used in the high-dimension case. In 2002, P. Polacik and F. Simondon [72] has shown that even if the nonlinearity f in (0.0.2) belongs to C∞, the w-limit set of its bounded global solution is diffeomorphic to the unit circle S1 (see [71] for another counterexample).The present thesis will use the above two methods to study two high-dimensional problems and two one-dimensional problems, respectively. More precisely, the thesis is organized as follows:Chapter 1 is a preliminary chapter in which we not only recall the history as well as current state of related research in the literature, but also illustrate the main idea of the proof of our convergence result. We discuss the new features and corresponding mathematical difficulties of the problem under consideration. Some basic materials and frequently used inequalities are also presented.Chapter 2 will be divided into two sections. A high-dimensional semilinear parabolic chemotaxis model and a high-dimensional quasilinear nonuniformly parabolic chemotaxis model will be considered, respectively. We will establish the correspond-ing Lojasiewicz-Simon inequality, and use it to get the convergence results for the above two problems. Furthermore, we will get the convergence rate. We would like to mention that first, the energy functional related to our problem is not analytic on the underlying space and only belongs to C1. Consequently, the Lojasiewicz-Simon approach must be considerably modified. Observing that the gradient of the func- tional is of type "monotone operator+linear compact perturbation", we establish a non-smooth version of the Lojasiewicz-Simon inequality based on [26]. Second, since the problem in Section 2 is not uniformly parabolic, the traditional methods are not applicable here. In fact, to overcome the difficulty resulted by the degener-acy, in Section 2 we introduce an auxiliary system which is uniformly parabolic and get the uniform boundedness of its solution by Moser technique. Then by dedicate comparison of the original system and the auxiliary system, we finally obtain the uniform boundedness of the solution to the original system.Chapter 3 is concerned with a one-dimensional thin film equation and a one-dimensional semilinear parabolic chemotaxis model. We use the phase plane analysis method to prove that the set of stationary solutions is discrete. Then we apply the well-known results in the gradient system to conclude the convergence. The phase plane analysis method has been used in several equations in the literature. However, due to the complexity of the nonlinearity in our problem, some methods used in the related papers do not work here. Therefore, we use some new methods which are the extension of some results in Schaff[77]. Moreover, the methods here are applicable to problems with more general nonlinearity.