Well-posedness and general decay of a nonlinear damping porous-elastic system with infinite memory

Abstract

In the present work, we consider a one-dimensional porous-elastic system with infinite memory and a nonlinear damping term. We establish the well-posedness of the system using semigroup theory and show the general decay for the case of nonequal speeds of wave propagation. Introducing some conditions on the kernel of the infinite memory term helps estimate the nonequal speed term even if this complementary control is not strong enough to stabilize the system exponentially. Our result is an extension of many other works in this area.

Well-Posedness and Stability Results for a Nonlinear Damped Porous–Elastic System with Infinite Memory and Distributed Delay Terms

“…Proof. Multiplying (16) 1 by u t and (16) 2 by φ t , then integration by parts over (0, 1) and using (17), we get…”

“…The authors proved the global well-posedness and stability results of (6), which has been extended in [17] for the case of nonequal speeds of wave propagation. Very recently, one-dimensional equations of an homogeneous and isotropic porous-elastic solid with an interior time-dependent delay term feedbacks was treated by Borges Filho and M. Santos in [1].…”

Stability of Multi-dimensional Nonlinear Piezoelectric Beam with Viscoelastic Infinite Memory

“…Mustafa et al in mus [16] proved an explicit and general decay result of a class of nonlinear Timoshenko beam system. Kh [11] studied the decay rates of a one-dimensional porous-elastic beam with infinite memory under nonlinear damping mechanism. freitas [7] studied the long time dynamics of a kind of nonlinear piezoelectric beams with fractional damping and thermal effects, and fre [8] considered a nonlinear piezoelectric system with delay effect, in which the global attractor and exponential attractors are studied.…”

General stability result for a porous thermoelastic system with infinite history and microtemperatures effects

“…The main problem concerning the stability in the presence of this complementary control is determining the largest class of kernels g which guarantee the stability and the best relation between the decay rates and the solutions of the considered system. However, it remains with great importance in the study of the asymptotic behavior of the solution for the different types of problems such that Timoshenko system, 12–18 von Kármán, 19 wave equation, 20–22 and porous‐elastic system 18,23‐25 …”