On the decay rates for a one-dimensional porous elasticity system with past history. (English) Zbl 1480.35035
Summary: This paper studies a porous elasticity system with past history
\[
\begin{cases}
\rho u_{tt}-\mu u_{xx}-b \phi_x = 0 \\
J \phi_{tt}-\delta \phi_{xx}+b u_x+\xi \phi+\int_0^{\infty} g(s) \phi_{xx} (t-s) ds = 0
\end{cases}
\]
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 nonequal-speed case. To establish our results, we mainly adopt the method developed by A. Guesmia et al. [Electron. J. Differ. Equ. 2012, Paper No. 193, 45 p. (2012; Zbl 1295.35088)] and some properties of convex functions developed by F. Alabau-Boussouira and P. Cannarsa [C. R., Math., Acad. Sci. Paris 347, No. 15–16, 867–872 (2009; Zbl 1179.35058)], I. Lasiecka and D. Tataru [Differ. Integral Equ. 6, No. 3, 507–533 (1993; Zbl 0803.35088)]. In addition we remove the assumption that \(b\) is positive constant in [T. A. Apalara, J. Math. Anal. Appl. 469, No. 2, 457–471 (2019; Zbl 1402.35042)] and hence improve the result.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L53 | Initial-boundary value problems for second-order hyperbolic systems |
| 35R09 | Integro-partial differential equations |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 93D20 | Asymptotic stability in control theory |
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