Well-Posedness and Asymptotic Stability of Solutions to the Bresse System under Cattaneo ’ s Law with Infinite Memories and Time-Varying Delays

In this paper, we study a one-dimensional Bresse-Cattaneo system with infinite memories and time-dependent delay term (the coefficient of which is not necessarily positive) in the internal feedbacks. First, it is proved that the system is well-posed by means of the Hille-Yosida theorem under suitable assumptions on the relaxation functions. Then, without any restriction on the speeds of wave propagations, we establish the exponential or general decay result by introducing suitable energy and Lyapunov functionals.


Introduction
The Bresse system is known as the circular arch problem (see [16] for details) and is given by the following equations: where denote the axial force, the shear force and the bending moment, respectively, The functions w, ϕ and ψ are the longitudinal, vertical and shear angle displacements, respectively.We use ρ for the density, E for the elastic modulus, G for the shear modulus, k for the shear factor, A for the cross-sectional area, I for the second moment of area of the cross-section, R for the radius of curvature of the beam, F i (i = 1, 2, 3) for the external forces.The arch with elastic structure is widely used in the fields of engineering, architecture, ocean engineering, aviation and others.In particular, the free vibration of elastic structure is a function of its natural property, and it is an important research subject in engineering and Mathematics.In the field of mathematical analysis is interesting to know properties which relate the behavior of the energy associated with the respective dynamic model.For feedback laws, for example, We can ask what conditions of the kinetic model can be obtained from the decay of the energy of the solution.In this sense, the property of stabilization has been studied for dynamic problems in elastic structures translated in terms of partial differential equations.
For stabilization of Timoshenko systems via heat effect, there are some results in recent time.Almeida Júnior [2] considered 1-D thermoelastic Timoshenko beam of the form with two types of boundary (Dirichlet-Dirichlet-Dirichlet or Dirichlet-Neumann-Neumann) conditions and established both exponential and polynomial stability results depending on the wave speeds and the initial data.In the above system, the heat equation is governed by Fourier's law of heat conduction.It is well known that the model using the classic Fourier's law leads to the physical paradox of infinite speed of heat propagation.That is, any local thermal disturbance can have an instantaneous effect everywhere in the medium.However, experiments showed that heat conduction in some dielectric crystals at low temperatures propagates with a finite speed.This phenomenon in dielectric crystals is called second sound.To overcome this drawback, a number of modifications of the basic assumption on the relation between the heat flux and the temperature have been made.One of which is the second sound effects observed experimentally in materials at a very low temperature.This theory suggests replacing the classic Fourier's law where q is the heat flux and γ is the coefficient of thermal conductivity by a modified law of heat conduction called Cattaneo's law τ q t + q + γθ x = 0, where τ > 0 represents the relaxation time describing the time lag in the response of the heat flux to a gradient in the temperature.On the basis of the above theory, Santos et al. [28] studied the Timoshenko beam model with second sound of the form The authors obtained exponential decay result when the stability number χ τ = 0. Otherwise, the polynomial decay result is obtained.Moreover, they showed that the rate is optimal.For more papers related to the second sound, we refer the reader to [4,6,25] and the references therein.
In recent years, the PDEs with time delays have become an active area of research and arise in many practical problems.The presence of delay may act as a source of instability.In [8], the authors showed that a small delay in a boundary control can destabilize a system which is uniformly asymptotically stable in the absence delays.To stabilize a hyperbolic system involving input delay terms, additional control will be necessary [22,29].Kirane and Said-Houari [20] considered a viscoelastic wave equation with a linear damping and a delay of the form where µ 1 and µ 2 are positive constants.They established a general decay result under the condition that µ 2 ≤ µ 1 .Later, Liu [15] improved this result by considering the equation with a time-varying delay term, with not necessarily positive coefficient µ 2 of the delay term.Moreover, some researchers considered the Timoshenko and Bresse systems with delay term.For instance, Said-Houari and Laskri [26] considered the following Timoshenko system with a constant time delay of the form They established an exponential decay result for the case of equal-speed wave propagation under the assumption µ 2 < µ 1 .More recently, Kirane el al. [19] considered the following Timoshenko system with a time-varying delay of the form where τ (t) represents the time-varying delay, 0 < τ 0 ≤ τ (t) ≤ τ and µ 1 , µ 2 are positive constants.Under the assumpyions µ 2 < √ 1 − d 1 µ 1 and τ (t) ≤ 1, they proved the exponential decay result.Motivated by the above results, we investigate in this paper system (1.1) under suitable assumptions and prove the well-posedness and the asymptotic stability of system.
Using (1.1) 5 and the boundary condition, we can easily verify that Consequently, we obtain L 0 q(x, t)dx = e − t τ L 0 q 0 (x)dx.
The main difficulty in carrying out this paper is the simultaneous appearance of the infinite memories, heat effect and time-varying delay.To overcome this difficulty, we have two key points in the proofs.On the one hand, to create the negative counterparts of the terms in the energy, we combine the fireworks of [4], [14] and [19] with necessary modifications.On the other hand, to estimate the infinite integral terms in (4.20) below, we use the approach which was first proved by Guesmia [10] and used by many researchers (see [11,14]).
This paper is organized as follows.In section 2, we present some assumptions needed for our work and state the main results.In section 3, we prove the well-posedness of problem (1.1).In section 4, we prove the asymptotic stability of problem (1.1).

Preliminaries and main results
In this section, we shall introduce some notation, basic definitions and main results which will be needed in the course of this paper.
Next, let us consider the following variables (see [7]): Then, it is easy to check that ) Therefore, problem (1.1) takes the form: (2.9) The above system is subject to the following initial and boundary conditions Then problem (2.9)-(2.10)can be written as where time varying operator A is defined by . Now, we consider the following space and define the functional spaces of U as follow: where Then the domain of A is defined by where For the relaxation functions g i , motivated by [14], we have the following assumptions: (G1) g i : R + → R + (i = 1, 2, 3) are non-increasing differentiable functions and integrate on (0, +∞) such that there exists a positive constant k 0 satisfying, for any (ϕ, ψ, w) Remark 1 It is easy to check that there exists a positive constant k0 such that, for any (ϕ, ψ, w) ∈ Therefore, let ) On the other hand, due to Poincaré's inequality, there exists a positive constant k0 such that, for Remark 2 Under hypothesis (G1), H * i and H are Hilbert spaces, respectively, with the inner products that generate the norms where ζ i (i = 1, 2, 3) are positive constants such that (2.17) (G2) For i = 1, 2, 3, there exist positive constants δ i (i = 1, 2, 3) such that or there exists an increasing strictly convex function Now, we state the well-posedness result of problem (2.9) Theorem 2.1 Assume that (G1) holds.Let U 0 ∈ H , then problem (2.9) has a unique weak solution The energy associated with problem (2.9) is defined by (2.21) Our decay results read as follows: Theorem 2.2 Let U 0 ∈ H be given.Assume that (G1) and (G2) hold, (i) If for all i = 1, 2, 3, (2.18) holds, then there exist positive constants c , c such that then there exist positive constants c , c and 0 such that where 3 Proof of the well-posedness In this section, we prove the well-posedness of the solution of problem (2.11).For this purpose, we will follow the method used in [4], [14] and [23] with the necessary modification imposed by the nature of our problem.
Proof of Theorem 2.1.In order to prove result stated in Theorem 2.1, first, we prove that the operator A is dissipative.A simple computation implies that, for any U ∈ D(A ), Integrating by parts, using (G1) and the boundary conditions in (2.10), we get Then, using Young's inequality, we obtain Notice the fact that, for i = 1, 2, 3, the kernel g i is non-increasing and using (2.3), (2.17), we obtain Hence, A is a dissipative operator.
Next, we prove that the operator Id − A is surjective.For this purpose, Given we prove that there exists The last equation of (3.5) yield On the other hand, by using (2.6), we can find z 1 , z 2 , z 3 as Following the same approach in [23] and using (3.4) 9 -(3.4) 11 , we obtain Hence, from (3.5), we obtain and Similarly, we get ρ 0 f 11 (x, σ)e στ 3 (t) dσ, if τ 3 (t) = 0 (3.12) and where From (3.8)-(3.13),we have (3.15) where (3.17) (3.18) It is clear from the above formula that z 10 , z 20 and z 30 depend only on f i , i = 1, 2, 3, 9, 10, 11.Next, (3.4) 12 -(3.4)14 and (3.5) imply By solving above three differential equations and noticing that v 9 (0) = v 10 (0) = v 11 (0) = 0, we get By using (3.5)-(3.6)and (3.20), it can be shown that v 1 , v 2 , v 3 and v 5 satisfy where where the operator ã is defined by the same formula (3.23).Now, we introduce the Hilbert space It is clear that a and ã are bounded.Furthermore, from (2.13), we find that there exists a positive constant c such that which implies that a is coercive.
From the above, we obtain that a is a bilinear continuous coercive form on V × V , and ã is a linear continuous form on V .Therefore, using the Lax-Milgram theorem [24], we obtain that Next, it remains to show that Recalling (3.5) and using (3.21), we have Then, by the L 2 theory for the linear elliptic equations, we obtain that In the same way, we obtain Similarly, recalling (3.6) and using (3.21), we have consequently, we get v 5 ∈ H 1 * (0, L).Then, using the classical regularity theory of linear elliptic equations, we obtain a unique solution V ∈ D(A ) which satisfies (3.3).Hence, the operator Id − A is surjective.
Finally, from above, we get A is a maximal monotone operator.Then, by using the Hille-Yosida theorem [5], we obtain that if U 0 ∈ D(A ), then U ∈ C(R + ; D(A )) ∩ C 1 (R + ; H ).Moreover, it is easy to see that D(A ) is dense in H .At last, basing on the above analysis, the well-posedness result stated in Theorem 2.1 follows from the Hille-Yosida [5,17].

Proof the Stability
In this section, we prove Theorem 2.2.Our method builds on a suitable Lyapunov functional that can be obtained by the energy method.
Before proving our main results, we will state and prove some useful lemmas in advance.
As in [14], let us define the functionals: and Lemma 4.2 The functionals I i (i = 1, 2, 3) satisfy, for any δ > 0, and where g 0 i is defined by (2.15) and c δ is a positive constant depending on δ.
Proof.Differentiating I 1 with respect to t, using the first equation of problem (2.9), integrating by parts and using the fact that we obtain Using Young's, Poincaré's and Hölder's inequalities for the last eight terms of this equality, we get By using (2.14) to estimate L 0 ϕ 2 x dx and choosing ) is established.Similarly, using the second and the third equations of problem (2.9), we obtain (4.5) and (4.6).
Lemma 4.3 There exist positive constants c 1 and c 2 such that the functional Proof.By exploiting first three equations of problem (2.9) and integrating by parts, we have Using Young's Poincaré's and Hölder's inequalities for the last ten terms in the right hand side of (4.8), we get, for any > 0, there exists a positive constant c such that Inserting this inequality into (4.8) and using (2.13), we get Proof.Taking the derivative of I 5 with respect to t, using the fourth and fifth equations of problem (2.9) and integrating by parts, we get Using Cauchy-Schwarz's and Young's inequalities with ε 3 > 0, we get (4.10).
Lemma 4.5 The functionals

.13)
Proof.Taking the derivative of I 6 with respect to t, we get By using (2.8), the last term in (4.14) can be rewritten as follows Also, one can see that from which immediately follows (4.11).Similarly, we get (4.12) and (4.13) in the same way.Now, let N 1 , N 2 , N 3 > 0, and First, taking the derivative of L(t) with respect to t, using (4.4)-(4.6)with δ = 1 where c N 2 = N 2 c δ + c 2 .At this point, we choose N 3 large enough so that Then, we choose ε small enough and N 2 large enough (note that g i is continuous non-negative and g i (0) > 0) so that Finally, we choose N 1 large enough so that From the above, we deduce that there exist positive constants c 3 and c 4 such that (4.In order to finish the proof of the stability, we need to estimate the last three terms in the right hand of (4.20).Inspired by [14], we have the following lemma.where κ > 0. Using the fact that G ( 0 E(t)) is non-increasing and (4.25), we get F (t) ≤ −c 5 τ E(t)G ( 0 E(t)).
Then, choosing c = c 5 τ (note that s → sG ( 0 s) is non-decreasing), we arrive at