In this work, we investigate a one-dimensional porous-elastic system with thermoelasticity of type III. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical results.

<p style='text-indent:20px;'>In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} u_{tt}+\Delta^2 u+ u+ \int_0^t g(t-s)A u(s)ds = 0, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with <inline-formula><tex-math id="M3">\begin{document}$ A = \Delta $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ A = -Id $\end{document}</tex-math></inline-formula>. We use the energy method in the Fourier space to establish several general decay results which improve many recent results in the literature. We also present two illustrative examples by the end.</p>

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In this paper, we investigate the general decay rate of the solutions for a class of fractional Laplace viscoelastic equations in the whole space ℝn,0.1emn≥1$$ {\mathbb{R}}^n,n\ge 1 $$, given by utt−normalΔu+u−∫0tgfalse(t−sfalse)false(−normalΔfalse)αufalse(sfalse)ds=0,$$ {u}_{tt}-\Delta u+u-{\int}_0^tg\left(t-s\right){\left(-\Delta \right)}^{\alpha }u(s) ds=0, $$ with 0<α<1$$ 0<\alpha <1 $$ and g$$ g $$ satisfying g′false(tfalse)≤−ηfalse(tfalse)gfalse(tfalse)$$ {g}^{\prime }(t)\le -\eta (t)g(t) $$. We prove the existence of a solution formula then use the energy method in the Fourier space to establish a general decay result which depends on α$$ \alpha $$ and η$$ \eta $$. We also present three illustrative examples by the end.