This paper concerns the asymptotic behavior of the following thermoelastic porous-elastic system \(\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \rho U_{tt}- \mu U_{xx}-b Z_x+\beta \theta _x=0, \qquad & \text { in } (0, 1)\times (0, \infty )\\ JZ_{tt}-\delta Z_{xx}+ b U_{x}+\xi Z-m\theta +\tau Z_t =0,\qquad & \text { in } (0, 1)\times (0, \infty )\\ c\theta _{t}+k q_x+ \beta U_{tx}+m Z_t=0, \qquad & \text { in } (0, 1)\times (0, \infty )\\ \end{array}\right. } \end{aligned} \end{aligned}\) complemented with some boundary and initial conditions, where the heat flux q is given by (Coleman-Gurtin’s law): \(\begin{aligned} \tau q(t)+(1-\alpha )\theta _{x}+\alpha \int _{0}^{\infty } \Phi (s)\theta _{x}(x, t-s)ds=0, \end{aligned}\) where \(\alpha \in (0, 1)\) , \(\tau >0\) and \(\Phi :\mathbb {R}_+:=[0,\infty )\rightarrow \mathbb {R}_+\) is the thermal memory such that \(-\Phi '=g\) where where \(g: \mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) is called the relaxation function and it satisfies 1 \(\begin{aligned} g'(s)\le -\vartheta (s) g(s), ~\text {holds~for~almost~every}~ s > 0. \end{aligned}\) where \(\vartheta\) is a positive nonincreasing differentiable function. The Fourier’s law ( \(\alpha =0\) ) and the Gurtin-Pipkin’s law ( \(\alpha =1\) ) are special cases of the Coleman-Gurtin’s law. Using the multiplier method, we establish a general decay estimate depending on the arbitrary growth at infinity of the relaxation function g. This approach allows a wide class of relaxation functions, and the obtained result includes the particular exponential one. The stability result in this manuscript extends and improves earlier results in the literature, including those by Tijani and Almutairi [1], Dell’Oro and Pata [2], Fareh [3], Hanni et al. [4], Dell’Oro [5] and others.
