In this paper, we are concerned with the asymptotic behavior of solution to the Cauchy problem of M 1 M₁ model. Under some smallness conditions, we show that the solution to this system exists globally and tends time-asymptotically to the corresponding equilibrium state at the optimal rate. The proof is based on both the time-weighted energy method and the L p − L q L^p-L^q estimates from the detailed analysis of the Green’s function. To the best of our knowledge, this may be the first result regarding the optimal decay of solution to this model in the multi-dimensional case.
Nangao Zhang (Mon,) studied this question.