Nonlinear Evolution Equations And Asymptotic Behavior Of Global Solutions Of Equations

The complexity and challenges in the theoretical study of nonlinear evolution equations which arise not only from many fields of mathematics but also from other branches of science such as physics, material science, mechanics etc. have attracted a lot of interests from many mathematicians for a long time. The asymptotic behavior of global solutions of nonlinear evolution equations, in particular convergence to a certain equilibrium as time goes to infinity has become one of the main concern in the field of nonlinear evolution equations since 1980s.For one space dimension case, significant progress has been made. For instance, H. Matano [95] in 1978 proved that, if the global solution of a nonlinear parabolic equation is bounded with respect to time, then it will converge to an equilibrium (see also [34,134,136] and references therein). However, the situation in higher space dimension case is much more complicated. It is well-known that for a gradient system or a gradient-like system, theω-limit set of its global solution is a connected subset of the system's equilibria. If we have known the set of equilibria is discrete, then the convergence easily follows (e.g., [29] ). However, unlike 1-d case, the set of equilibria is in general not discrete and can form a continuum (see e.g., [65,140] ). On the other hand, a counterexample in P. Pola(?)ik＆F. Simondon [105] has shown that even if the nonlinearity of a semilinear parabolic equation belongs to C~∞, theω-limit set of its bounded global solution could be diffeomorphic to the unit circle S~1 (see also [104] for another example). In the literature, a lot of assumptions have been made to ensure the convergence for bounded global solutions. In 1983, L. Simon [117] proved that if the nonlinearity of a nonlinear parabolic equation is analytic in the dependent variable, then the convergence holds. His idea relies on the extension of a gradient inequality by S. (?)ojasiewicz for analytic function defined in R~n to the infinite-dimensional space. After his breakthrough, a lot of work has been done in this direction. However, most of the previous results are concerned with evolution equations subject to homogeneous (Dirichlet) boundary condition. For many nonlinear evolution equations with other boundary conditions (for instance, the dynamical boundary condition) which are very important from the physical point of view, the current framework in the literature as well as the Lojasiewicz- Simon inequality which plays a crucial role fail to apply. So far, those problems have never been considered in the existing literature.The present thesis is devoted to the study of convergence of global solutions for a series of nonlinear evolution equations with complex boundary conditions such as dynamical boundary condition etc.. Under the basic assumption that the nonlinearity is analytic with respect to the unknown dependent variable, we develop several new Lojasiewicz-Simon type inequalities which vary from problem to problem and obtain the required convergence result as well as the convergence rate. All the results obtained in this thesis have never been found in the previous literature.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 associated mathematical difficulties of the problems under consideration. Some basic materials and frequently used inequalities are also presented.Chapter 2 is concerned with the Cahn-Hilliard equation with dynamical bound-ary condition. The global existence and uniqueness, maximal regularity and existence of global attractor have been considered before ( [107,108] ). Based on the previous results, we show the convergence to equilibrium.Chapter 3 is devoted to the convergence to equilibrium for the damped semilinear wave equation with critical growth exponent and dissipative (dynamical) boundary condition.Chapter 4 is concerned with the convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition.The final Chapter 5 is divided into two parts. In the first part we consider the convergence to equilibrium for a nonlinear parabolic-hyperbolic phase-field system with Neumann boundary conditions for both unknown dependent variables (relative temperatureθand phase-fieldχ) while the second part deals with the convergence to equilibrium for such a system with Neumann boundary condition forθand dynamical boundary condition forχ.We briefly point out the new features, mathematical difficulties of the problems considered in this thesis and our main contributions.First, we treat the dynamical boundary condition which is important from phys-ical point of view, and it turns out that for the corresponding elliptic operator, it yields a non-homogeneous boundary condition. The Lojasiewicz-Simon type inequal-ity for homogeneous (Dirichlet) boundary condition in the previous literature fail to apply. We prove several extended Lojasiewicz-Simon type inequalities with boundary term with which we are able to get the expected convergence results.Second, in order to apply Simon's idea to prove the convergence result, for the hyperbolic equation as well as parabolic-hyperbolic system, we have to construct new auxiliary functionals (adding proper perturbation terms to the original Lyapunov functional) which vary from problem to problem.Third, by delicate energy estimates, we prove the rate of convergence to equilib-rium for all the problems considered in this thesis. In particular, we obtain the rate of convergence to equilibrium for Cahn-Hilliard equation as well as parabolic-hyperbolic Phase-Field system for the first time in the literature.Fourth, the growth of nonlinearity in our hyperbolic equation is critical which implies that one could not use the result by Webb [125] to obtain the precompactness.It is worth mentioning that the new techniques and Lojasiewicz-Simon type inequalities developed in this thesis can be further applied to the study of convergence properties of other nonlinear evolution equations.