Large-time Behavior Of Solutions For The Dissipative Hyperbolic Equations

In this thesis, we consider the large-time behavior of solutions for the dissipativehyperbolic equations. The thesis is arranged as follows:In Chapter1, we review the physical background of the fractal Burgers equations,the two dimensional perturbed Hasegawa-Mima equation, the two dimensional viscousshallow water equations and the history on studying these equations. We also introducethe problems addressed in this study and summarized the main results.In Chapter2, we study the existence and large-time behavior of periodic solutionsfor the fractal Burgers equation with large initial data. In Section1, we obtained theasymptotic behavior of periodic solutions for fractal nonlinear Burgers equation. First-ly, we get the local existence of the solution directly by constructing a Cauchy sequence.Then, by the maximum principle and some important inequalities we obtain the expo-nential decay estimates of solutions for the fractal Burgers equations. At last, by usingthe exponential decay estimates we extend the local solution to be a global one and alsoget the continuity of the solution with respect to t at the same time. In Section2, we getthe the exponential decay estimates of solutions for the dissipative Quasi-Geostrophicequations. As the proof is similar to the one in Section1, we omit it here for brief.In Chapter3, we consider the pointwise estimates for the two dimensional per-turbed Hasegawa-Mima equation. In Section1, we give some known facts and Lemmaswhich will be used later. In Section2, we introduce the problems addressed in this studyand summarized the main results. In Section3, by the energy method, we obtain theexistence of global solution for the perturbed Hasegawa-Mima equation. In Section4,we obtain the pointwise estimates by the Green function method. First of all, we de-rive pointwise estimates of the Green function by studying the Green function problem. Moreover, by using the Duhamel principle and the pointwise estimates of the Greenfunction, we obtain pointwise estimates of the solutions when the initial perturbationcorresponding to a constant state is sufficiently small in Hs(R2).In Chapter4, we investigate the pointwise estimates of the two dimensional vis-cous shallow water equations. Since the nonlinear term of this viscous shallow waterequations can not be written as the divergence of a function, an essential difficulty willbe emerged when we close the estimates between the solutions for the nonlinear systemand the prior estimates. In order to overcome this difficulty, we turn to estimate of thevariational form of the original one. Then, we derive the pointwise estimates of thesolution for the nonlinear system by the classical Green function method. It is clearlyshown from our results that the move of the solution which is the right effect of thegeneralized Huygans' principle.