On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model

<p style='text-indent:20px;'>In this paper, we are concerned with the long-time asymptotic behavior of the two-dimensional temperature-dependent tropical climate model. More precisely, we obtain the sharp time-decay of the solution of the system with the general initial data belonging to an appropriate Sobolev space with negative indices. In addition, when such condition of the initial data is absent, it is shown that any spatial derivative of the positive integer <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula>-order of the solution actually decays at least at the rate of <inline-formula><tex-math id="M2">\begin{document}$ (1+t)^{-\frac{k}{2}} $\end{document}</tex-math></inline-formula>.</p>