Communications in Mathematical Sciences

Volume 19 (2021)

Number 8

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

Pages: 2119 – 2138



Nangao Zhang (School of Mathematics, South China University of Technology, Guangzhou, China)

Changjiang Zhu (School of Mathematics, South China University of Technology, Guangzhou, China)


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.


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