Optimal decay rates of the solution of the linearized $M_1$ model

In this paper, we are concerned with the optimal decay rates of the solution to Cauchy problem on the system of linearized $M_1$ model in whole space $\mathbb{R}^n$ for any spatial dimension $n \geq 1$. The time-decay rates of perturbed solutions and its derivatives in $L^q$ space are obtained when initial data are around a constant equilibrium state. The proof is mainly based on both the energy method and the $L^p - L^q$ estimates from the detailed analysis of the Green’s function of the linearized system. The decay estimates thus obtained will play a key role in discussing the decay structure of nonlinear $M_1$ model in the future.

Keywords

linearized $M_1$ model, Green’s function, optimal decay rates

2010 Mathematics Subject Classification

35B40, 35F10, 65M80, 85A25

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The research was supported by the National Natural Science Foundation of China #11771150, 11831003, 11926346 and Guangdong Basic and Applied Basic Research Foundation #2020B1515310015.

Received 6 December 2020

Accepted 26 April 2021

Published 7 October 2021