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November  2019, 18(6): 2905-2921. doi: 10.3934/cpaa.2019130

## On the decay rates for a one-dimensional porous elasticity system with past history

 Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  February 2018 Revised  July 2018 Published  May 2019

Fund Project: The author is supported by the National Natural Science Foundation of China (No. 11701465) and by the Fundamental Research Funds for the Central Universities (No. JBK1902026)

This paper studies a porous elasticity system with past history
 $\left\{\begin{array}{l}{\rho u_{t t}-\mu u_{x x}-b \phi_{x} = 0} \\ {J \phi_{t t}-\delta \phi_{x x}+b u_{x}+\xi \phi+\int_{0}^{\infty} g(s) \phi_{x x}(t-s) d s = 0}\end{array}\right.$
By introducing a new variable, we establish an explicit and a general decay of energy for the case of equal-speed wave propagation as well as for the nonequalspeed case. To establish our results, we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [Electron. J. Differ. Equa. 2012(2012), 1-45] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [C. R. Acad. Sci. Paris Ser. I, 347(2009), 867-872], Lasiecka and Tataru [Differ. Inte. Equa., 6(1993), 507-533]. In addition we remove the assumption that b is positive constant in [J. Math. Anal. Appl., 469(2019), 457-471] and hence improve the result.
Citation: Baowei Feng. On the decay rates for a one-dimensional porous elasticity system with past history. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2905-2921. doi: 10.3934/cpaa.2019130
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