On the stability of a double porous elastic system with visco-porous damping

In this paper, we focused on a one-dimensional elastic system with a double porosity structure and frictional damping acting on both porous equations. We introduce two stability numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>χ</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {χ₀} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>χ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {χ₁} and prove that the solution of the system decays exponentially provided that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>χ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {χ₀=0} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>χ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {χ₁≠ 0} . Otherwise, we prove the absence of exponential decay. Our results improve the results of [N. Bazarra, J. R. Fernández, M. C. Leseduarte, A. Magaña and R. Quintanilla, On the thermoelasticity with two porosities: Asymptotic behaviour, Math. Mech. Solids 24 2019, 9, 2713–2725] and [A. Nemsi and A. Fareh, Exponential decay of the solution of a double porous elastic system, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 83 2021, 1, 41–50].