Scattering theory for the nonlinear Klein-Gordon equations in the Sobolev critical case in two dimensions
speaker: Kenji Nakanishi (Department of Mathematics, Kyoto University)abstract: This is joint work with Slim Ibrahim, Mohamed Majdoub and Nader Masmoudi. We study asymptotic behavior for large time of solutions for the nonlinear Klein-Gordon equation with a defocusing exponential nonlinearity in two spatial dimensions. There is a certain threshold on the value of conserved energy below which the exponential potential energy is dominated by the kinetic energy by a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approach a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. This is a striking difference from the Sobolev critical case in higher dimensions, where the nonlinear term is dominated by some powers of the Strichartz norms with uniform constants on any time interval. Thus we have to trace the concentration of energy along time, in order to set up favorable nonlinear estimates, and then to use Bourgain's induction argument. The same result holds true for the nonlinear Schrodinger equation.
timetable:
Fri 7 Sep, 11:40 - 12:40, Aula Dini
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