The Asymptotic Behavior Of The Global Solutions For Nonlinear Evolution Equations

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,mechanics and material science. The complexity and challenges in the theoretical study of nonlinear evolution equations have attracted a lot of interests from many mathematicians for a long time.The present thesis is devoted to the study of the asymptotic behavior as time tends to infinity of global solution of nonlinear evolution equations.The asymptotic behavior of global solution of nonlinear evolution equations,which include convergence to a certain equilibrium as time goes to infinity and the study for the related infinite-dimensional dynamical system,has become two of the main concerns in the field of nonlinear evolution equation since 1980s.We usually view the solution of the nonlinear evolution equations as an orbit in a certain Sobolev space,starting from the initial datum u₀.Now we are concerned with the asymptotic behavior of the single orbit starting from an arbitrary,but fixed initial datum and the existence of global attractor of a family of orbits starting from initial data varying in any bounded set of a Sobolev space.The study of global attractor for the infinite-dimensional dynamical system arises from continuum physics,continuum mechanics,material sciences has become a hot topic in research since the 1980s.There is a lot of work has been done in this direction,such as the reaction-diffusion equations from chemical dynamics and biological sciences,Cahn-Hilliard equations and Phase-Field equations from material science,the incompressible Navier-Stokes equations in space dimension 2, the dissipative wave equations etc(see the books Zheng[72],Temam[41],Babin and Vishik[5],Hale[22],Sell and You[55]and references therein). This thesis is devoted to the study of one-dimensional nonlinear thermoviscoelastic systems arise from the study of phase transitions in shape memory alloys. Based on the global existence and uniqueness of the solutions,we further obtain the existence of global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions and the convergence to equilibrium for the nonlinear thermoviscoelastic systems with constant temperature boundary conditions respectively.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 in the literature,but also illustrate the main idea of the proof of the existence of global attractor.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 Ginzburg-Landau thermoviscoelastic system with hinged boundary conditions.Overcome the mathematical difficulties due to the nonlinearity and high order derivative,we obtain the existence and uniqueness of the global solution,the asymptotic behavior of the solution as time tends to infinity and the compactness of the orbit.Furthermore,we investigate dynamics of the system and prove the existence of global attractor.Chapter 3 is concerned with the nonlinear thermoviscoelastic system with constant temperature boundary conditions,we show the global existence and uniqueness of the weak solution and the convergence to a steady state as time tends to infinity, which can be considered to be an extension of Pego[42]result on isothermal case to the non-isothermal case.We briefly point out the new features,mathematical difficulties of the problems considered in this thesis and our main contributions.First,in chapter 2 we consider the same model with Hoffmann and Zochowchi [25].Different with[25],we derive delicate uniform a prior estimates independent of T in the proof,which is essential for the study of the asymptotic behavior of the solution as time tends to infinity.Second,the existing setting of the global solution in chapter 2 is Sobolev space H which is not complete.On the other hand,there exists an energy conservation in H,which means there can be no global attractor for initial data varying in the whole space.In order to solve these problems,Zheng,Shen & Qin([62],[63],[70], [71]) introduced subspaces defined by some parametersβ_i,i.e.,H_βi,and proved the existence of global attractor in H_βi.Motivated by these results,we consider the dynamics in closed subspace H_{β1,β2,β3} in chapter 2.The difference is that here we use the constraintθ≥β₁＞0 instead ofθ＞0,which brought H_{β1,β2,β3} to be closed and complete.Overcome the mathematical difficulties arise from the constraintθ≥β₁＞0,we prove the existence of global attractor in complete subspaces H_{β1,β2,β3} for the first time.Third,one of the main contributions in chapter 3 is to handle the boundary termθ_x|x=0,1 appears in the integration by parts due to the non-homogeneous Dirichlet boundary conditionθ|x=0,1=T₀＞0,which seems to have been a major obstacle for people working on such problems.Fourth,all the estimates obtained in chapter 3 does not depend on T,we prove that as time goes to infinity,θwill converge toθ_T uniformly,the strain u will almost pointwise converge to an equilibrium function u_∞.Such results are still open for the other initial boundary problems for this system.