STABILITY RESULT FOR A WEAKLY NONLINEARLY DAMPED POROUS SYSTEM WITH DISTRIBUTED DELAY

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. Received June 5th, 2021; accepted June 30th, 2021; published October 28th, 2021. 2010 Mathematics Subject Classification. 35B35, 35B40, 93D20.

Back to system (1.1), it is to be noted that when µ 1 = µ 2 = 0 and replacing the term α (t) g (φ t ) by the term t 0 g(t − s)u xx (x, s) ds then (1.1) is equivalent to the well-known Timoshenko system of memory type which is exponentially stable depending of the relaxation function g and provided that the wave speeds of the system are equal (See [1,15]).
Messaoudi and Fareh [16] investigated the following system: and established, using the energy method, an exponential decay result. For more results on the subject, we refer the reader to [5,10,11,19].
Concerning the weight of the delay, we assume that and establish the well-posedness as well as the exponential stability results of the energy E (t), defined by

Preliminaries
In this section, we present some materials needed in the proof of our result. We assume α and g satisfy the following hypotheses: (H1) α : R + → R + * is a non-increasing differentiable function; (H2) g : R → R is a non-decreasing C 0 -function such that there exist positive constants c 1 , c 2 , η and G ∈ C 1 ([0, ∞)) , with G (0) = 0, and G is linear or strictly convex C 2 −function on (0, η] such that * 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. Furthermore, they proved that the monotonicity and continuity of g guarantee the existence of the function G defined in (H2).
For completeness purpose we state, without proof, the existence and regularity result of system (1.1). First, we introduce the following spaces: and Moreover, if U 0 ∈ H, then the solution satisfies This result can be proved using the theory of maximal nonlinear monotone operators (see [8]).

Technical Lemmas
In this section, we state and prove our stability results for the energy of system (1.1) by using the multiplier technique. To achieve our goal, we need the following lemmas.

Stability Result
ds and G 0 (s) = tG (η 0 t) . Proof. We multiply (3.15) by α (t) to get Now, we discuss two cases: Case I: G is linear on [0, η]. In this case, using (H2) and Eq.(3.1), we deduce that which can be rewritten as Using (H1), we obtain By exploiting (3.14), it can easily be shown that So, for some positive constant λ 1 , we obtain The combination of Eq. (4.3) and (4.4), gives Case II: G is nonlinear on [0, η]. In this case, we first choose 0 < η 1 < η such that Using (H2) along with fact that g is continuous and |g (s)| > 0, for s = 0, it follows that To estimate the last integral in Eq. (4.2), we consider the following partition of (0, 1):