Nonstandard Beam Overall Equation Posedness And Scattering Theory

In this dissertation, we mainly consider the nonstandard defocusing Beam equation utt+∑ni=1(?)4i+u+|u|p-1u=0in the subcritical case. We show the global well-posedness and the scattering results in low dimensions by the compactness-rigidity argument of Kenig and Merle [10,11].In chapter1, we first state the concept of scattering theory, taking Schrodinger equation as an example. Then we give some basic notations and introduce the Littlewood-Paley theory, as well as some important lemmas. In chapter2, we explain the physical background of the nonstandard Beam equa-tion, and state some fundamental properties of the equation. In addition, we give the main results of the dissertation.In chapter3, we construct the Strichartz estimates using the dispersive estimate built in [5]. In particular, we construct the special Strichartz estimate (3.2.14), which plays the crucial role in overcoming the difficulty caused by the derivative in classical Strichartz estimates. In addition, we prove the global well-posedness by Strichartz estimates and the fixed point argument.In chapter4, we prove the scattering theory in low dimensions2≤n≤8. We first reduce the scattering theory to the boundedness of a space-time norm of the solution. Then we use the contradictory argument inspired by Kenig and Merle [10] and Tao, Visan and Zhang [34], and utilize a profile decomposition inspired by Bahouri and Gerard [4], to find a critical element under the assumption. Finally, we use a Virial-type identity derived from Pausader [27] to get rid of the critical element, and consequently find a contradiction.