Asymptotic Behavior of the Initial-boundary Value Problem of Landau-Lifshitz-Schrödinger Type

Abstract

This paper is concerned with the asymptotic behavior of the classical solutions of a Landau-Lifshitz-Schrödinger-type problem with initial-boundary values when the parameter $\varepsilon$ goes to zero. We establish several uniform estimates of $u_{\varepsilon}$ by a conservation result and the standard parabolic method. Based on these results, we obtain parabolic behavior in the dissipative case and non-parabolic behavior of the semi-classical limits of those solutions respectively.