Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Abstract

Based on the endpoint Strichartz estimates for the fourth order Schrodinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schrodinger operator. J. Funct. Anal., 274, 605–658 (2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader, B.: The cubic fourth-order Schrodinger equation. J. Funct. Anal., 256, 2473–2517 (2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity $$1 + \frac{8}{n} < p < 1 + \frac{8}{{n - 4}}$$ in dimensions n ≥ 7.