Limiting behavior of solutions of multidimensional Landau-Lifshitz equations with second approximation of effective field (I)

TL;DR

This paper investigates the limiting behavior of solutions to multidimensional Landau-Lifshitz equations with a second approximation of the effective field, establishing convergence properties and solution behavior in different regions.

Contribution

It introduces a novel approach using $ ext{delta}$-viscosity supersolutions and subsolutions to analyze the solutions' limits in multidimensional Landau-Lifshitz equations.

Findings

Solutions tend to (0,1,0) in one region

Solutions tend to (0,-1,0) in another region

The limiting behavior is characterized on compact subsets

Abstract

By a maximum principle under global control map from $R^{n+1}$ to$S^2,$ $\delta $-viscosity supersolution and subsolution of multidimensional Landau-Lifshitz equations with second approximation of effective field are built. Utilizing the $\delta $-viscosity supersolution and subsolution, the viscosity solution of the equations is showed and the limiting behavior of the solution of the equations is obtained. Exactly, there exist two disjoint open subsets such that the solutions tend to the point $(0,1,0)$ on arbitrary compact subsets in one of them and tend to the point $(0,-1,0)$ on arbitrary compact subsets in the other of them, respectively.