General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay

In this paper we consider a viscoelastic modified nonlinear Von-Karman system with a linear delay term. The well posedness of solutions is proved using the Faedo-Galerkin method. We use minimal and general conditions on the relaxation function and establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.

1. Introduction. In this paper, we are concerned with the following nonlinear modified Von-Kármán system with time delay and a memory term, where D = (I − h 2 12 ∂ 2 ∂ 2 x ) and the interval I = (0, L) is the segment occupied by the beam. The unknowns ψ = ψ(x, t), and η = η(x, t) represent, respectively, the vertical displacement, and the longitudinal displacement at time t of the cross section located x units from the end-point x = 0.
In (1), subscripts mean partial derivatives and h > 0 is a parameter related to the rotational inertia of the beam.
When α 1 = α 2 = 0, this system describe approximately the planar motion of a uniform prismatic beam of lenght L with memory term. Here, h and ρ are two strictly positive constants represent respectively the thickness and the mass density per unit volume of the beam. In the system (1), α 1 η t represents a frictional damping. The time delay is given by α 2 η t (t − τ ), where α 1 , α 2 , τ are positive constants.
In (1), (g * f )(t) is defined by This integral term or the viscoelastic damping term that appears in the equations describes the relationship between the stress and the history of the strain in the beam, according to Boltzmann theory. The function g represents the kernel of the memory term or the relaxation function.
The main purpose about problems (1) − (3) is to deal with the well posedness and asymptotic behavior of solutions. Before stating and proving our results, let us recall some other results related to our work.
Several authors have studied the Mindlin-Timoshenko system of equations. This Model is a widely used and fairly complete mathematical model for describing the transverse vibrations of beams. It is a more accurate model than the Euler-Bernoulli one, since it also takes into account transverse shear effects. The Mindlin-Timoshenko system is used, for example, to model aircraft wings (see, e.g., [23]).
For a beam of length L > 0, this one-dimensional nonlinear system reads as where Q = (0, L) × (0, T ) and T is a given positive time. Here, the unknown φ = φ(x, t) represent the angle of rotation. The parameter k is the so called modulus of elasticity in shear. It is given by the expression k = kEh/2(1 + ), where k is a shear correction coefficient, E is the Young's modulus and is the Poisson's ratio, 0 < < 1/2. For Mindlin-Timoshenko system, there is a large literature, addressing problems of existence, uniqueness and asymptotic behavior in time when some damping effects are considered, as well as some other important properties (see [18,30,32] and references therein).
When one assumes the linear filament of the beam to remain orthogonal to the deformed middle surface, the transverse shear effects are neglected, and one obtains, from the Mindlin-Timoshenko system of equations, the following von Kármán system (see [32]).
Lagnese and Leugering [31] considered a one-dimensional version of the von Kármán system describing the planar motion of a uniform prismatic beam of length L. More precisely, in [31] the following system was considered: In [31], suitable dissipative boundary conditions at x = 0, x = L and initial conditions at t = 0 were given and the stabilization problem was studied.
In [4], Araruna et al. have showed how the so called von Kármán model (6) can be obtained as a singular limit of a modified Mindlin-Timoshenko system (4) when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k −→ ∞, the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.
Remark 1. Since k is inversely proportional to the shear angle, we note that neglecting the shear effects of the beam is formally equivalent to considering the modulus k tending to infinity in the Mindlin-Timoshenko system.
The subject of stability of von Kármán system has received a lot of attention in the last years. It is important to mension that the authors in [17,25,33] proved uniform decay rates for the von Kármán system with frictional dissipative effects in the boundary. The stability for a von Kármán system with memory and boundary memory conditions was treated in [25,44,53,38]. They proved the exponential or polynomial decay rate when the relaxation function decay is at the same rate. As for the works about general decay for viscoelastic system, we refer the reader to [14,55] and references therein.
Delay effects are very important because most natural phenomena are in many cases very complicated and do not depend only on the current state but also on the past history of the system. The presence of delay can be a source of instability. In recent years, the stabilization of PDEs with delay effects has draw attention for many author and become an active area of research (see [28,14,22,51,52,60,62,64,63]).
In [9], Benaissa et al. in studied a system of viscoelastic wave equations with a linear frictional damping term and a delay where Ω is a bounded domain in R n , n ∈ N, with a smooth boundary ∂Ω = Γ, g is a positive non-increasing function defined on R n , h 1 and h 2 are two functions, τ > 0 is a time delay, α 1 and α 2 are positive real numbers and the initial data (u 0 , u 1 , f 0 ) belong to a suitable function space.
In the case g = 0, Cavalcanti et al. [13]) studied (7) for h 2 ≡ 0 and with a linear localised frictional damping a(x)u t . This work was later improved by Berrimi and Messaoudi [10] by introducing a different functional which allowed them to weaken the conditions on g.

AMMAR KHEMMOUDJ AND YACINE MOKHTARI
For a wider class of relaxation functions, Messaoudi [41,42] considered for γ > 0 and b = 0 or b = 1, and the relaxation function satisfies where ζ is a differentiable nonincreasing positive function. He established a more general decay result, from which the usual exponential and polynomial decay results are only special cases. Such a condition was then employed in a series of papers, see for instance [26,46,47,55].
Recently, Mustafa and Messaoudi [50] studied the problem (8) with b = 0 for the relaxation functions satisfying where H is a nonnegative function, with H(0) = H (0) = 0 and H is strictly increasing and strictly convex on ]0, k[ for some k 0 > 0. The authors showed a general relation between the decay rate for the energy and that of the relaxation function g without imposing restrictive assumptions on the behavior of g at infinity.
On the other hand, a condition of the form (10) where H is a convex function satisfying some smoothness properties, was introduced by Alabau-Boussouira and Cannarsa [3] and used then by several authors with different approaches. We refer to [37] where not only general but also optimal result was established by Lasiecka and Wang. Our purpose in this paper is to give a global solvability in Sobolev spaces and energy decay estimates of the solutions to the problem (1) for linear damping, time delay terms and finite memory. We would like to see the influence of frictional and viscoelastic dampings on the rate of decay of solutions in the presence of linear degenerate delay term.
Our aim is to investigate (1) for relaxation functions g of more general type than the ones in (9) and (10). We consider the condition where H is increasing and convex without any additional constraints on H and the coefficients, and establish energy decay results that address both the optimality and generality. The energy decay rates are optimal in the sense that they decay qualitatively the same as the viscoelastic kernels g do.
To obtain global solutions of problem (1)−(3), we use the Galerkin approximation scheme (see Lions [39]) together with the energy estimate method. The technique based on the theory of non linear semigroups used in Nicaise and Pignotti [51] does not seem to be applicable in our case.
To prove decay estimates, we use a perturbed energy method and some properties of convex functions. In order to accomplish this goal, we shall pursue a strategy based on an adaptation of non linear differential inequalities technique developed in [45,48,49]. Arguments of convexity were introduced and developed by many authors [12,20,35,36,40,24,2].
We observe that our problem is set in a context where: a: ) The memory damping is defined only on the equation for the vertical displacement.
b: ) The presence of a frictional damping and a time delay on the equation for the longitudinal displacement. c: ) Energy decay estimates under a nonlinear tention. Our work is organized as follows. In the next section, we prepare some material needed in the proof of our result, like some lemmas (Poincaré's and Young's inequalities) and some useful notations. We introduce the different functionals by which we modify the classical energy to get an equivalent useful one. In Section 3, we state and prove the well-posedness of the problem. Finally, in Section 4, we will prove our main results concerning the exponential decay of the energy associated to the solutions of the problem.

Assumptions.
To state and prove our result, we use the following assumptions: A 1 ) To preserve the hyperbolicity of our system, we assume that the kernel is such that g ∈ L 1 (R + ) ∩ C 2 (R + ) with g(0) > 0 and A 2 ) There exists a C 1 function H : (0, ∞) −→ (0, ∞) wich is linear or strictly increasing and streactly convex where ζ is a positive nonincreasing differentiable function. Now, we prepare some notations and hypotheses which will be needed in the proof of our result. Let L 2 (0, L) be the usual Hilbert space with the inner product (·, ·) and the inner product induced norm · . Throughout this paper, we define From the Poincaré inequality, it follows that . V1 and . W are equivalent to the standard norms of H 2 (0, L) and H 4 (0, L) in the spaces H 1 * (Ω) and W , respectively. Also we have v x is equivalent to v H 1 * (Ω) . C and c denote some general positive constants, which may be different in different estimates.
The following inequality will be used repeatedly in the sequel.
We shall use the following inequalities. and where . is the norm in L 2 (0, L).
Proof. We multiply the third equation in (12) by ξ τ z and integrate the result over (0, L) × (0, 1) with respect to p and x, respectively, to get which gives (20).
Throughout this paper, we denote by • and the binary operators defined by and where u ∈ C([0, T ]; L 2 (0, L)).
We define the energy associated with the solution of system (12) − (14) by where ξ is a positive constant such that with α 1 and α 1 satisfying The next lemma gives an identity for the convolution product.
3. Global well-posedness. In this section we show the existence and regularity of solutions of the the one dimensional viscoelastic Von-Karman system (12) − (14). The existence and uniqueness result of problem (12) − (14) is stated as follows.
Theorem 3.1 (Well-posedness). Assume that the initial datum satisfy with the compatibility condition f (., 0) = η 1 . Morever, assume that the Hypotheses (A 1 ) , (A 2 ) hold. Then problem (12) − (14) admits a unique weak solution Proof. The proof will be organized as follows. First, using the Galerkin method, we prove the existence of weak solutions, then using elliptic regularity and second order estimates we show the regularity of the solution.
Let T > 0 be fixed and and denote by V m and W m the spaces generated by (e i ) 1≤i≤m and (σ i ) 1≤i≤m , and where the sets e k , k ∈ N and w k , k ∈ N are basis of spaces H 2 0 (I) and H 1 * (I) respictively, that is V m = span {e 1 , e 2 , .., e m } , W m = span {σ 1 , σ 2 , .., σ m } .
Our starting point is to construct the Galerkin approximation (ψ m , η m , z m ) of the solution where u m i , v m i , and w m i , i = 1, 2, ..., m, are determined by the following ordinary integro-differential equations: with initial conditions where ψ m,j (0) = ψ 0 , e j , ψ m,j By virtue of the theory of ordinary differential equations, the system (33) − (34) has a unique local solution which is extended to a maximal interval [0, T m [ (with 0 < T m ≤ ∞), by Zorn lemma since the nonlinear terms in (33) are locally Lipschitz continuous. Note that u m i (t) , v m i (t) are from the class C 2 . In the next step, we obtain a priori estimates for the solution such that it can be extended beyond [0, T m [ to obtain a single solution defined for all t > 0.
In order to use a standard compactness argument for the limiting procedure, it suffices to derive some a priori estimates for u m i (t), v m i (t). The first estimate. Since the sequences (ψ m 0 ),(η m 0 ) and (z m 0 ) converge, standard calculations, using (33) − (34), similar to those used to derive (28), yield where Integrating (36) over (0, t) yields for some positive constant C independent of m ∈ N. Also, to get an apriory estimate for η, we use the Poincaré's-Wirtinger inequality and the boundedness of E to obtain where C is a positive constant independent of m ∈ N. These estimates imply that the solution (ψ m , η m , z m ) of the system ( for any T > 0. The second estimate. We have to estimate ψ m tt (0), ψ m ttx (0) and η m tt (0) in L 2 norm.
Considering t = 0 in the first equation of (33), then multiplying it by (u m i ) (0) and summing up over i from 1 to m, it follows that Integrating by parts and using Young's inequality, we get Similarly Using the embedding H 1 (I) → L ∞ (I), we estimate the second term of the right hand side of (43) as follows From the embedding H 1 (I) → L ∞ (I) and Young's inequality, we deduce After choosing a suitable δ, we infer from (34)-(35) and ( 41)-(45) that there exists a positive constant C independent of m such that Next, multiplying the second equation of (33) by (v m i ) (0), choosing t = 0 and summing up over i from 1 to m, we get Appying Young's and Poincaré's inequalities, using the embedding H 1 (I) → L ∞ (I) and the fact that E is nonincreasing, we conclude that Then we use Young's inequality to obtain, for any δ > 0, and Hence, from (34)- (35) and with a suitable choice of δ, there exists a positive constant C independent of m such that η m tt (0) ≤ C.
Next, differentiating the first equation of (33) with respect to t, using We have Making use of Young's inequality, the embedding H 1 (I) → L ∞ (I) and (39), we conclude that Making use of the embedding H 1 (I) → L ∞ (I), we get |g | exist for all T < ∞.
Using Cauchy-Shwarz and Young's inequalities produce the estimates and Employing Young's inequality, combining (53) − (57), then integrating (52) over (0, t), we obtain where C is a positive constant independent of m but depends on T and the initial data.
The term t 0 s 0 ψ m xx 2 dzds can be estimated as follows Next, Multiplying the second equation of (33) by (v m i ) and summing up over i from 1 to m, we arrive at Now, applying Young's and Poincaré's inequalities, we get A differentiation with respect to t of the third equation of (33) implies Myltiplying (62) by (w m i ) (t), integrating by parts and adding from i = 1 to m, we obtain Integrating (63) over (0, 1) with respect to p, taking the sum of (60) and (63) and integrating over (0, t), we obtain Combining (58) and (64) with a suitable choice of δ, then using Gronwall's lemma, we arrive at where C is independent of m ∈ N. Replacing φ i by −φ ixx in the third equation of (33), multiplying the resulting equation by w m i (t), summing over i from 1 to m, leads to where C independent of m.
Combining (68) − (71) and (73), it follows that f = η x + 1 2 ψ 2 x , and As a consequence of (74) we have The convrgence (72) − (75) allows us to pass to the limit in (33) − (34). Thus, the problem (12) − (14) admits a global weak solution (ψ, η, z). Uniqueness can be proved by the straightforward methods and Gronwall's inequality. 4. General decay. In this section we consider a wider class of kernel functions, and we establish a general decay result, which contains the usual exponential and polynomial decay rates as special cases. The main result of general decay is the following.
To prove Theorem 4.1, we first proceed to prepare a series of useful lemmas.
Lemma 4.2. The following inequalities hold where g ψ is given by (22).
Let F be the functional defined by where N 1 and N 2 are positive constants that will be chosen later. Let λ > 0 and define Y(t) by where E(t) is defined in (23).
The following proposition gives the equivalence between E(t) and the functional Y(t).
Proposition 1. Assume that (A 1 ) hold, then there exists two positive constants δ 1 , δ 2 such that Proof. To compare Y(t) with E(t), we have to estimate the terms of the right hand side of (79) and show that.
In order to proof the main theorem, we need some additionals lemmas.
Proof. Using (12) and (77), we have d dt where Integrating I 1 and I 2 by parts twice, and using the boundary conditions, we obtain A integration by parts in I 3 , leads to Substituting (92)-(93) in (91), we get d dt Substituting (94) in (94), we obtain d dt Making use of Young's and Cauchy-Schwarz's inequalities for the lsecond term in the right-hand side of (94), we get, for any ε > 0, From (94)-(94) and (15), we infer that which proves the lemma 4.3.
Applying Young's inequality, we obtain Keeping in mind (87), the proof follows.
Lemma 4.5. Suppose that (ψ, ψ t , η, η t , z) is the solution of (12) − (14). Then the derivative of the functional K(t) satisfies where ε is an arbitrary positive constant and Proof. Differentiating K and using (12), we obtain AMMAR KHEMMOUDJ AND YACINE MOKHTARI Next, we shall analyze the terms such as J 1 − J 5 of the right hand side of (97).
Estimate for J 1 Applying Young's inequality and (77), we get where ε is an arbitrary positive constant.
Estimate for J 2 We follow the previous steps, with applying Poincaré's inequality, we get On the other hand, we know that H 1 (Ω) → L ∞ (Ω), so This implies that Estimate for J 3 We inveoke (77) to get Estimate for J 4 Applying Young's, Poincaré's inequalities and using (78), we conclude that Estimate for J 5 Finally, for J 5 , invoking (78), we obtain: Combining (98) − (102), we arrive at the proof of (4.5).
Lemma 4.6. Suppose that (ψ, ψ t , η, η t , z) is the solution of (12) − (14). Then the time derivative of the functional L(t) satisfies Proof. Keeping in mind that z t (p) = − 1 τ z p (p), we infer d dt Proposition 2. Assume that (A 1 ) and (A 1 ) hold, then there exists two positive contants β 1 , β 2 such that d dt Proof. By using (79) , (84) and combining (90) − (103), we get d dt We want to impose suitable conditions on the different parameters so that the coefficients on the right hand side of (105) are all strictly negative. That is to obtain the following inequalities

AMMAR KHEMMOUDJ AND YACINE MOKHTARI
We observe that (106) and (107) will be satisfied if we choose ε > 0 small enough and such that To make (108) and (109) hold we can choose Concerning (110), (111) and (112), we pick This completes the proof.
We consider the following two cases.
for some β 1 > 0. Then for some γ 2 > 0. Furthermore, using the continuity and boundedness of E(t) in [0, t 1 ], we get Case II. H(t) is nonlinear: Next, with f (t) = ∞ t g(s)ds, we use the functional Lemma 4.7. Assume that (A) and (B) hold. The functional K satisfies, for any ε > 0, the estimate d dt Proof. By Young's inequality and the fact f (t) = −g(t), we see that Combining (114) and (115), we obtain (113).
Let us introduce the functional where κ is a positive constant. Then we have

Y(t) ∼ E(t).
Therefore, it is always possible to pick N 1 (in 105) and κ large enough to get d dt Y(t) ≤ −CE(t).
Integrating over (t 0 , ∞), we get Next, let us define the functional L(t) where q > 0. Thanks to (116), we can always choose q such that L(t) < 1, ∀t ≥ t 0 .

So (104) becomes
Let 0 < r, using the fact that E ≤ 0, H > 0, H > 0, we observe that the functional N defined by is equivalent to E. Using (120), we find that N satisfies Let us denote by G * the conjugate function of the convex function G defined by G * (s) = Sup t∈R + (st − G(t)), then st ≤ G * (s) + G(t), and, thanks to the arguments given in [5,12,20 in (122), then making use of (121), (122) and (123), we arrive at Next, multiplying (124) by ξ(t) and using the fact that 0 where c is a positive constant. Now, let us define the functional N N (t) = N (t)ξ(t) + E(t).
It is not difficult to see that there exist positive constants ρ 1 and ρ 2 for which we have Consequently, with an appropriate choice of 0 , then there exists a positive constant k such that where H 2 (s) = sH ( 0 s).
Defining now