The Overall Solution Of The Nonlinear Evolution Equations And Asymptotic Behavior

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 sciences such as physics,mechanics,material science and biology.The asymptotic behavior of global solutions to nonlinear evolution equations,especially the research on convergence of the trajectory to an equilibrium as time goes to infinity attracts a lot of interest from many mathematicians and scientists in nonlinear sciences.For the one-dimension case,significant progress has been made by Zelenyak[91] and Matano[60]respectively using different methods.They proved that,if the global solution of a nonlinear parabolic equation is bounded with respect to time, then it will converge to an equilibrium as time goes to infinity.However,the situation is much more complicated when the space dimension in larger or equal to 2. Unlike the 1-d case,the set of equilibria is in general not discrete and can form a continuum.For instance,P.Polacik & K.P.Rybakowski[68]have 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.In 1983,L.Simon[74]proved that if the nonlinearity of a nonlinear parabolic equation is analytic in the dependent variable,then the convergence holds.He extended a lemma by Lojasiewicz on analytic functions defined on R~n to the infinite dimension case and developed a method for the study of convergence to the stationary solution when the nonlinear term is analytic,i.e.,the so-called Lojasiewicz-Simon's approach.Since then,there have been many research papers on this direction.The arrangement of the thesis is 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 problem under consideration.Some basic materials and frequently used inequalities are also presented.Chapter 2 is concerned with the phase-field system under Cattaneo heat flux law.This section is divided into two parts.In the first part,we consider the coupled parabolic-hyperbolic type phase field problem subjected to homogeneous Neumann boundary condition for the phase function x and no-flux boundary condition for the heat flux vector q.Existence and uniqueness of a global(strong) solution is obtained.Then we prove the convergence result and obtain an estimate of the decay rate.In the second part,the phase-lag effect of the phase function is taken into account and our system becomes a fully hyperbolic one.We prove the well-posedness of this system subject to homogeneous Neumann boundary condition and no-heat flux boundary condition.Then,we prove that the global(weak) solution of this problem converges to an equilibrium as time goes to infinity.We also obtain an estimate of the decay rate to equilibrium.We would like to mention that,since the problems under consideration are of hyperbolic type,the lack of smooth property of the solutions to hyperbolic type equations causes a lot of difficulties.Firstly, the proof of the precompactness of the trajectories is not trivial.Secondly,the Lojasiewicz-Simon inequality can not been applied directly,one needs to construct an auxiliary functional based on the Lyapunov functional of the system.Chapter 3 is devoted to the convergence to an equilibrium for a chemotaxis model with volume-filling effect subject to homogeneous Neumann boundary conditions. We would like to mention that the energy functional related to our problem is not analytic on the underlying space and only belongs to C~1.Consequently, Lojasiewicz-Simon's approach must be considerably modified.Observing that the gradient of the functional is of type"monotone operator+linear compact perturbation", we establish a non-smooth version of the Lojasiewicz-Simon inequality and finally prove the convergence result and convergence rate.