Some Problems To Nonlinear Evolution Equations

In this paper, we study the existence and uniqueness of global solutions for the initial boundary value problems to some classes of nonlinear evolution equations, and give the sufficient conditions of blowup of solutions for the above problems, we also discuss the time-periodic problems for two classes of nonlinear evolution equations of higher order and prove the existence and uniqueness of time-periodic solution or the existence of weak time- periodic solution. The main results include the following five aspects:In Chapter 2, we study the following initial boundary value problem for a generalized double dispersive equationThe existence and uniqueness of the generalized global solution and the classical global solution are proved by Galerkin method. Then the sufficient conditions of the nonexis-tence of global solution are given by constructing a new inequality. Moreover, we also consider the following double dispersion equation (DDE)and the general cubic DDE(CDDE)with the initial boundary value conditions (2),(3) respectively. The main results are stated as follows:has a unique global generalized solutionbounded below.generalized solution or a classical solution of theSuppose that the following conditions are satisfied:converges when d > 0, moreover, B < I; the integralhas a unique global generalized solution ube the generalized solution of the problemSuppose that the following conditions are satisfied:for some finite time t0In Chapter 3, we investigate the initial boundary value problem for the wave equation with nonlinear damping and source termswhich describes nonlinear vibration of elastic rods. The existence and uniqueness of the global solution are proved in case that the initial data are small by potential well method combined with Galerkin method, we also give the sufficient conditions of blowup of the solutions for the problem (6)-(8) in finite time.The main results are stated as follows:Theorem 6 Assume that the following conditions hold.here K4 > 0 and , > 0 are constants. Then the problem (6)-(8) has a unique global generalized soluition u(t, x) which satisfiesIn Chapter 4, we discuss the initial boundary value problem for a class of nonlinear wave equationwhich describes the water wave with surface tension. The existence and uniqueness of generalized local solution are proved by Galerkin method and compactness argument and the sufficient conditions of blow-up of the generalized solution are given, by concavity method.The main results are stated as follows:the problem (9)-(ll) has a unique local generalized solution u(x,t). where ?Theorem 9 Assume is the generalized solution of the problem (9)-(l 1) and the following conditions hold:Jo where u(x) is the first normalized eigenfunction of the eigenvalue problemLet = be the corresponding first eigenvalue . Then there is a T T such thatIn chapter 5, we study the time- periodic problem to a generalized Ginzburg-Landau model equation in population problemsThe existence and uniqueness of the time-periodic generalized solution and the time-periodic classical solution to the problem (12)-(14) are proved by Galerkin method. The main results are the following two theorems: Then the problem (12)-(14) has a generalized time-periodic solutionTheorem 11 Assume the conditions of Theorem 10 and the following conditions are satisfied.has a classical time-periodic solution u(x,t}. Moreover, if M is sufficiently small, the classical solution is unique.In chapter 6, we consider the time-periodic problem for a class of nonlinear wave equation of higher orderBy means of the critical point theory, we prove the existence of the weakly time-periodic solution to the problem (15)-(17). The main result is as follows:...