The Dynamics Of A Class Of Nonlinear Wave Equations

Nonlinear wave equations have been widely used in optics, plasma andother fields of physics. This dissertation focuses on the research of the global dynamicsof a class of nonlinear wave equations with energy conservation and specific initial data.Our research will be helpful to explain and predict the related physical phenomena.This dissertation is devoted to three typical wave equations: Schrodinger equation,Zakharov system and Klein–Gordon–Zakharov system.Firstly, we proved the scattering of the Schrodinger equation with exponential non-linearity (eλ|u|~2-1)u in2D energy space when0<λ <4π. This type of Schrodingerequation describes the self-focusing of electromagnetic waves in plasma. In2008,Ibrahim, Majdoub, Masmoudi and Nakanishi [26] proved the energy scattering of thetwo-dimensional Schrodinger equation with nonlinearity (e~λ|u|21λ|u|2)u. Theysubtracted the cubic term λ|u|2u from the nonlinearity in order to avoid the L2crit-ical exponent related to the decay property of solutions. This dissertation improvestheir result and proves the scattering of the two-dimensional Schrodinger equation withnonlinearity (eλ|u|~2-1)u in energy space. Especially, we give a new linear profile de-composition for H1sequence in2D space which does not depend on a specific equationand can be used to analyze the scattering of other Schrodinger equation with initial datain H1(R~2).Secondly, we study the global dynamics below the ground state energy for theKlein–Gordon–Zakharov system in the3D radial case, and proved its scattering andfinite time blow-up. Klein–Gordon–Zakharov system describes the interaction betweenLangmuir waves and ion sound waves in a plasma. Before this paper, there have not anyresults for scattering and blow up below ground state energy about this system (even forsmall data). This dissertation will combine normal form technique and radial-improvedStrichartz estimates to prove small energy scattering by contracting mapping principle,and then prove the scattering and finite time blow-up below the ground state energy byconcentration compactness method and virial identity. Especially, we prove the radial-improved Strichartz estimates for Klein–Gordon operator in this part. These estimateshave more regularity than the classical ones. Since the radial-improved Strichartz esti-mates are proved for Klein–Gordon equation, they can be used for the research or the scattering and blow up of other systems which contain Klein–Gordon equation.Finally, we prove the global dynamics below the ground state energy for the Za-kharov system in the3D radial case, and prove its scattering and infinite time blow-up,i.e. grow-up. Zakharov system was introduced as a mathematical model for the Lang-muir turbulence in unmagnetized ionized plasma. The energy scattering of this systemhas not been solved until now. In2012, Guo and Nakanishi [22] proved its small en-ergy scattering in radial case. This dissertation further prove the scattering and infinitetime blow-up below the ground state energy by concentration compactness method andvirial identity.