Global Attractor Of Nonlinear Wave Equations

It is welt known that the nonlinear Kirchhoff equations, the nonlinear Schr(?)dinger equationsand the nonlinear Benjamin-Bona-Mahony equations are three kinds of imPortant wave equation.The nonlinear Kirchhoff equations origin in the mathematical description of small amplitudevibrations of an elastic string[54]. The nonlinear SchrSdinger equations are the basic mathematicmodel in the quantum mechanics[40]. The nonlinear Benjamin-Bona-Mahony equation wasproposed in [9] as a model for propagation of long waves which incorporates nonlinear dispersiveand dissipative effects. There exists a large body of literature regarding three kinds of equationand a series of important advances are achieved on the mathematical studies. Especially forexistence of global solution and their asymptotic behavior of initial boundary value problem,plentiful and substantial results are got, for example, see [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15,16, 17, 18, 28, 29, 30, 33, 48, 49, 50, 59, 61, 64, 65, 70, 71, 72, 73, 74, 75, 77, 78, 81, 82, 83, 84,85, 95, 96, 97].From the view point of study of quality for partial differential equations, the key point isto obtain some prior estimate of solutions when t→∞in some sense[43]. So the infinitedimensional dynamical systems defined by the nonlinear evolution equations is important aspectfor research of partial differential equations.In the present paper, our first main goal is to show the existence of global attractors forKirchhoff equation with three different boundary condition respectively; Our second main goal isto show the existence and regularity of global attractors for Schr(?)dinger equation with nonlinearboundary conditions; At last, we will investigate the existence of global attractors for the dampedBenjamin-Bona-Mahony equation defined on R¹ by harmonic analysis.The thesis consists of six chapters.The first chapter is devote to the summary of the dissertation. First of all, we present somebasic notions of infinite dimensional dynamical systems. Secondly, we present our goal and themain results of the paper. In chapter 2, 3 and 4, we shall show the existence of global attractors for Kirchhoff equationswith three different boundary condition respectively. At present, there are many full resultsfor infinite dimensional dynamical systems defined by the nonlinear evolution equations withhmomgeneous boundary condition [7, 19, 20, 21, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44,45, 46, 47, 53, 56, 57, 58, 62, 67, 76, 79, 86, 87, 88, 89, 90, 91, 92, 93, 94, 98, 99]. But fornonhmomgeneous boundary condition, there are many problems need to answer [22, 23, 24, 25].This is why we consider ldrchhoff equation with nonhmomgeneous boundary condition. It iswell known that a compact global attractor exists if and only if the semigroup has a boundedabsorbing set and is asymptotically compact[88]. We meet the first difficulty is nonhmomgeneousboundary conditions when proving existence of bounded absorbing set. In order to overcomethis difficulty, we shall combine the perturbed energy method used in [100, 55] with techniquesfrom [66]. Secondly, for the proof of asymptotically compact, one usually decompose the solutionoperator into a compact part and a asymptotically small part. However, because M(||â–½u||²) isnonlinear, there are additional difficulties when proving asymptotically compact. To overcomethis difficulty, we shall utilize a new decomposition for the solution operator which is differ from[74, 75, 76] to verify the asymptotic compactness.In chapter 2, we study the existence of solutions and global attractors for Kirchhoff equationwith nonlinear boundary conditions. Here, we discuss this problem divide into two cases.Case (1): M(||â–½u||²)≥m₀＞0,Case (2): M(||â–½u||²)≥0,In chapter 3, we consider the existence of global attractors for Kirchhoff equation with memory boundary condition:In chapter 4, we investigate the existence global attractors for the following Kirchhoff equa-tion with nonlinear damping and memory term at boundary. Concretely, we shall investigatethe following two problems respectively:andIn chapter 5, we are interested in the long time behavior of solutions to the weakly dampednonlinear Schr(o|¨)dinger equation with a nonlinear boundary condition as follows:Here we shall prove that the existence and regularity of a global attractor. The issue of theexistence and regularity of the attractor is classical in the study of infinite dimensional dissipativesystems, for example, see [37, 38, 88, 93]. We follow the methods of Goubet [38, 39], where theauthor has established the asymptotic effect for the nonlinear Schr(o|¨)dinger equations and for thenonlinear Kdv equations with periodic condition, and Zhang [99] also deals with the equationof shallow water type with this methods. In chapter 6, we will investigate the asymptotic behavior of the solutions of the followingdamped Benjamin-Bona-Mahony (BBM) equation defined on R¹:When the equation is defined in a bounded domain, there exists finite dimensional global attrac-tor [19, 93, 94]. Note that when the domain of the equation is unbounded there are additionaldifficulties when proving the existence of attractors because, in this case, the Sobolev embeddingsare not compact. There are several methods which can be used to show the existence of attrac-tors in standard Sobolev spaces when the equations are defined in unbounded domains. Onecan use energy equation technique to show that the weak asymptotic compactness is equivalentto the strong asymptotic compactness or decompose the solution operator into a compact partand a asymptotically small part. A third method is to prove that the solutions axe uniformlysmall for large space and time variables by a cut-off function or by a weight function.Follow the third method, Stanislavova [86, 87] present a new necessary and sufficient con-dition to verity the asymptotic compactness of an evolution equation defined in an unboundeddomain, which involves the Littlewood-Paley projection operators. Here we show that thereexists a global attractor in H¹ (R¹). Moreover the attractor is in fact smoother and it belongsto H^3/2-ε for everyε＞0...