Artículos de revistas
Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes
Fecha
2022Registro en:
Journal of the European Mathematical Society (2022) 24: 2133-2167
10.4171/JEMS/1130
Autor
Kowalczyk, Michal Antoni
Martel, Yvan
Muñoz Cerón, Claudio
Institución
Resumen
In this paper, we consider the dynamics of even solutions of the one-dimensional nonlinear Klein-Gordon equation partial derivative(2)(t)phi - partial derivative(2)(x)phi + phi - vertical bar phi vertical bar(2 alpha) phi = 0 for alpha > 1, in the vicinity of the unstable soliton Q. Our main result is that stability in the energy space H-1(R) x L-2(R) implies asymptotics stability in a local energy norm. In particular, there exists a Lipschitz graph of initial data leading to stable and asymptotically stable trajectories. The condition alpha > 1 corresponds to cases where the linearized operator around Q has no resonance and no internal mode. Recall that the case alpha > 2 is treated by Krieger, Nakanishi and Schlag [Math. Z. 272 (2012)] using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed by Chang, Gustafson, Nakanishi and Tsai [SIAM J. Math. Anal. 39 (2007/08)].