Stability Result for a Weakly Nonlinearly Damped Porous System with Distributed Delay

Khoudir Kibeche,Abdelhak Djebabla,Fahima Hebhoub,Lamine Bouzettouta

doi:10.28924/2291-8639-19-2021-812

Khoudir Kibeche, Abdelhak Djebabla + Show 2 more

Open Access

https://doi.org/10.28924/2291-8639-19-2021-812

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Abstract

In this paper, we consider a one-dimensional porous system damped with a single weakly nonlinear feedback and distributed delay term. Without imposing any restrictive growth assumption near the origin on the damping term, we establish an explicit and general decay rate, using a multiplier method and some properties of convex functions in case of the same speed of propagation in the two equations of the system. The result is new and opens more research areas into porous-elastic system.

Highlights

Apalara in [2] considered the following on-dimensional porous system damped with a single weakly nonlinear feedback ρutt − μuxx − bφx = 0, x ∈ (0, 1), t > 0
Hypothesis (H2) implies that sg (s) > 0 for all s = 0. * According to our knowledge, hypothesis (H2) with η = 1 was first introduced by Lasiecka and Tataru [13]. They established a decay result, which depends on the solution of an explicit nonlinear ordinary differential equation
If U0 ∈ H, the solution satisfies u ∈ L∞ R+; H∗2 (0, 1) ∩ H∗1 (0, 1) ∩ W 1,∞ R+; H∗1 (0, 1) ∩ W 2,∞ R+; L2∗ (0, 1), φ ∈ L∞ R+; H2 (0, 1) ∩ H01 (0, 1) ∩ W 1,∞ R+; H01 (0, 1) ∩ W 2,∞ R+; L2 (0, 1) Remark 2.2. This result can be proved using the theory of maximal nonlinear monotone operators

Summary

Introduction

We consider the following porous system: ρutt − μuxx − bφx + μ1ut +. Problem (1.1) is equivalent to μ2(s)z (x, 1, t, s) szt (x, ρ, s, t) + zρ (x, ρ, s, t) = 0, x ∈ (0, 1) , ρ ∈ (0, 1) , ρ ∈ (τ 1, τ 2) , t>0. Apalara in [2] considered the following on-dimensional porous system damped with a single weakly nonlinear feedback ρutt − μuxx − bφx = 0, x ∈ (0, 1), t > 0, jφtt − δφxx + bux + ξφ + α (t) g (φt) = 0,. = 0, in (0, 1) × (0, ∞), and established, using the energy method, an exponential decay result. Τ1 and establish the well-posedness as well as the exponential stability results of the energy E (t), defined by

Preliminaries

Stability Result