Large time behavior of solutions to the 2D damped wave-type magnetohydrodynamic equations

TL;DR

This paper investigates the long-term behavior of solutions to a 2D damped wave-type magnetohydrodynamic system, focusing on how a specific term affects decay rates and establishing optimal decay estimates in Sobolev spaces.

Contribution

It provides the first analysis of the large time decay rates for the 2D damped wave MHD system, especially considering the influence of the $eta ightarrow eta$ term, with explicit decay rates depending on $eta$.

Findings

Decay rates for solutions and derivatives are obtained with explicit dependence on $eta$.

Decay rates match those of the standard MHD system, indicating optimality.

The influence of the $eta ightarrow eta$ term on long-term behavior is clarified.

Abstract

In this paper, we are concerned with the 2D damped wave-type magnetohydrodynamic system (abbreviated as MHD-wave system). The purpose of this paper is to study the large time behavior of solutions to the MHD-wave system, espesically to investigate the influence of the bad term $\gamma \partial_{tt}b$ on the large time behavior. Rates of decay are obtained for both the solutions and higher derivatives in different Sobolev spaces with explicit rates of $\gamma$, which shows that the decay rates closely align with that of the MHD system under the same norm, for any fixed $\gamma>0$. In this sense, these decay rates are optimal.