WELL-POSEDNESS AND EXPONENTIAL STABILITY OF SWELLING POROUS ELASTIC SOILS WITH A SECOND SOUND AND DISTRIBUTED DELAY TERM

Abstract. In this paper we consider a one-dimensional swelling porous-elastic system with second sound and distributed delay term. We prove that the combination of the frictional damping with the heat flux effect is strong enough to provoke an exponential decay of the energy even if the delay is a source of destabilization.


INTRODUCTION
Using the second law of thermodynamics, Eringen [9] developed general and linear constitutive equations of mixtures of viscous liquids, elastic solids and gas. Then established a relation between the continuum theory of swelling porous elastic soils and the classical diffusion theories. For more discussion on continuum theories that have been developed to model mixtures we refer the reader to [3]. As discussed deeply in [14], expansive soils cause minor to major structural damages to buildings, that includes floor slab on grade cracking, buckling of pavements and cracking of buried pipes. Thus to deal with this problematic soil, it is essential to evaluate its swelling potential, and then propose several techniques to prevent structural damages, such as reducing the swelling, using sufficiently strong structures and isolating the structure from the swelling soil (see [22]). For more practical applications of the theory in architecture and civil engineering, we mention for example Handy [11], Hung [12], see also ([10], [13]).
The linear theory of swelling porous elastic soils as considered in [23] and [24] is given by the system where ϕ represent the displacement of the fluid with density ρ 1 and ψ is the elastic solid material with density ρ 2 . The functions (P 1 , P 2 ) represent the partial tension, (G 1 , G 2 ) the internal body forces, and (H 1 , H 2 ) the external forces, acting on the displacement and on the elastic solid, respectively. Moreover, the partial tensions (P 1 , P 2 ) are given by where A is the positive definite matrix Wang and Guo [26] considered (1.1) by taking where γ(x) is an internal viscous damping function with positive mean. By using the spectral method they established an exponential stability result. More recently, in [7], the authors studied (1.1) with different conditions they used the multiplier method to establish a general decay result.
A system which is asymptotically stable may be destabilized under the effects of time delay, that make it a property of practical and theoretical importance for many physical systems. As mentioned in [21], by a change of variable, distributed delay can be regarded as a memory acting only on the time interval (t − τ 2 ,t − τ 1 ) , for more discussions (see [8], [27]). Models governed by the Fourier's law of heat conduction leads to an infinite speed of heat propagation, which means that any thermal disturbance at one point has an instantaneous effect somewhere else. By replacing Fourier's law β q + θ x = 0 with a wave propagation process described by Cattaneo's law τq t + β q + θ x = 0, the problem of the infinite speed of heat propagation is eliminated (see [6], [25]).
In the present work, we consider (1.1) with distributed delay term on the elastic solid is in the form of distributed delay term, that is: Thus, we are concerned with the following thermoelastic system of swelling porous elastic soils with a linear frictional damping and an internal distributed delay acting on the transverse displacement, where the heat flux is given by Cattaneo's law: where the functions (ϕ, ψ, θ , q) are the transverse displacement of the beam, the rotation angle, the difference temperature, the heat flux, respectively. The coefficients, ρ 1 , ρ 2 , ρ 3 , a 1 , a 2 , a 3, β , δ , µ 1 , τ are positive constants. τ 1 and τ 2 are two real numbers with Finally, ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , θ 0 , q 0 , f 0 are the initial data and f 0 is history function, belong to an appropriate functional spaces.
The purpose of this paper is to study the well-posedness and the asymptotic behavior of the solution of (1.2) regardless of the speeds of wave propagation.
For any regular solution of (2.2), we define the energy by We assume (1.3) holds and establish the well-posedness as well as the exponential stability results of the energy.

WELL-POSEDNESS OF THE PROBLEM
In this section, we prove the existence and uniqueness of solutions for (2.2) using semigroup theory. Introducing the vector function φ = (ϕ, u, ψ, v, θ , q, z) T , and the two new dependent variables u = ϕ t and v = ψ t , then the system (2.2) can be written as Where the operator A is defined by We have reserved the following spaces and H is the energy space given by We will show that A generates a C 0 Semi-group on H .
H is the Hilbert space. In this box, the inner product above is equivalent to the natural inner product set to H .

The domain of A is
Clearly, D(A ) is dense in H . Now we can give the following existence and uniqueness result.
Proof. To obtain the above result, we will prove that A : D(A ) → H is a maximal monotone operator. For this purpose, we need the following two steps.
Step 1: In this step, we prove that the operator A is monotone. Let φ ∈ D(A ), Using Young's inequality, the last term in (3.3), we have Step 2: We prove that operator A + I is surjective. That is to say, for everything G = (g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 ) ∈ H , we are looking for φ ∈ D (A) satisfying Suppose we have found u and v. Therefore, the first and the third equation in (3.5) given It is clear that u ∈ H 1 0 (0, 1), v ∈ H 1 0 (0, 1). And we can find Following the same approach as in [20], we obtain by using equations for z in (3.6), Then by (3.7) and (3.8), From (3.5) 6 we have then θ (0) = θ (1) = 0. Using (3.6) and (3.10) in (3.5), we get The variational formulation associated with (3.11) takes the form (y) dy dx, which is equivalent to To solve the problem (3.11), it suffices to show that B is continuous and coercive, and that F is continuous, we can therefore easily see that B and F are bounded, and moreover, we have for a c > 0 : therefore, according to the Lax-Milgram theorem, the system (3.11) admits a unique solution ϕ ∈ H 1 * (0, 1), ψ ∈ H 1 0 (0, 1) and q ∈ L 2 * (0, 1).
Hence, from Lumer-Phillips theorem the result of Theorem 1 follows.

EXPONENTIAL STABILITY
In this section, we state and prove our stability result for the energy of the solution of system (2.2), using the multiplier technique. To achieve our goal, we need the following lemmas.
Lemma 4. Let (ϕ, ψ, θ , q, z) be the solution of (2.2). Then the functional Proof. By differentiating F 3 (t), taking in account the second and the third equations in (2.2), and integrating by parts, we obtain Finally, the estimate (4.19) is established, using Young's and Poincaré inequalities.  Proof. Differentiating F 4 (t), and using the equations in (2.2), we obtain
Next, we define a Lyapunov functional L and show that it is equivalent to the energy functional E (t).

Lemma 7.
For N sufficiently large, the functional defined by where N 1 , N 2 , N 3 and N 4 are positive real numbers to be chosen appropriately later, satisfies for two positive constants c 1 and c 2 .
Exploiting Cauchy-Schwarz's and Young's and Poincaré inequalities, gives Which yields Consequently, By choosing N large enough. We obtain estimate (4.23). Now, we are ready to state and prove the main result of this section.
Next, we carefully choose our constants so that the terms inside the brackets are positive.