Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

abstract: In this paper, we investigate the well-posedness as well as optimal decay rate estimates of the energy associated with a Kirchhoff-Carrier problem in ndimensional bounded domain under an internal finite memory. The considered class of memory kernels is very wide and allows us to derive new and optimal decay rate estimates then those ones considered previously in the literature for Kirchhoff-type models.


Introduction
The nonlinear vibrations of an elastic string are written in the form of partial integro-differential equations by for 0 < x < L and t ≥ 0, where u is the lateral deflection, x is the space coordenate variable while t denotes the time variable, E represents the Young's modulus, ρ designates the mass density, L indicates the string's lengh, h represents the cross section, p 0 denotes the axial tension, f represents an external force.
The model (1.1) has been introduced by Kirchhoff [15] in the study of the oscillations of stretched strings and plates, so that equation (  Guesmia, Messaoudi and Webler equation of Kirchhoff type until now.It is worth mentioning that, when p 0 = 0, the model (1.1) is called degenerate, and when p 0 > 0, we denominate it as a non-degenerate model.
There is a large literature regarding the Kirchhoff equation.In the sequel, we would like to mention some important works on this subject.Regarding the wellposedness of problem (1.1), the analytic case is rather known in general dimensions, as, for instance, [8], [9] and [25].In what concerns solutions for (1.1) lying in Sobolev spaces and, as far as we know, the results presented in the literature are only local in time, as, for example, [1] and [24].However, when equation (1.1) is supplemented by some type of dissipative mechanism, which allows us, roughly speaking, to derive decay rate estimates for the solutions of the linearized problen of (1.1), it is possible to recover the global solvability in time.Consequently, deriving global solutions in time deeply depends on the decay structure of the solutions to the corresponding linearized problem of (1.1).Therefore, we are led naturally to consider the Kirchhoff equation subject to a dissipative term which guarantees the decay properties of the linearized problem.When the dissipation is given by a frictional mechanism, like g(∂ t u), there is a large body of works in the literature, see, for instance, [2], [10], [4], [13], [16], [17], [18], [24], [21], [23] and a long list of references therein.
In this paper, we investigate the well-posedness as well as optimal decay rate estimates of the energy associated with the following Kirchhoff-Carrier problem with memory: where Ω is a bounded domain in R n , n ∈ N * , with smooth boundary ∂Ω := Γ.While there is a great number of papers regarding the Kirchhoff equation subject to a frictional damping, in contrast, there is just a few number of papers concerned with the Kirchhoff equation subject to a dissipation given by a memory term.We are aware solely the paper [22], where stronger conditions were considered on the kernel of the memory term.The assumption given in (1.7), firstly introduced in [20], is much more general and allows us to consider a wide class of kernels, and consequently, get new and optimal decay rate estimates then those ones considered previously in the literature for the linear viscoelastic wave equation.In the present paper, we combine techniques given in [20] with new ingredients inherent to the nonlinear character of the Kirchhoff equation (1.2).
The following assumptions are made on the function M : (1.5) We shall assume the following assumptions on the kernel g: Assumption 1.2.The function g : R + → R + belongs to the class g ∈ C 1 (R + ), g ′ ≤ 0 and, in addition Moreover, there exists a differentiable non increasing function ξ : R Now, we are in a position to state our main result.
Theorem 1.3.Assume that Assumption 1.1 and Assumption 1.2 are in place.
Then, there exists an open unbounded set S in (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) which contains (0, 0) such that, if (u 0 , u 1 ) ∈ S, and, in addition, the initial data are taken in bounded sets of Furthermore, we have the following decay estimates for the energy E given in (2.10): where θ and c are positive constants independent of the initial data.
Our paper is organized as follows: in Section 2, we prove the general stability (1.9).The Section 3 is devoted to the proof the well-posedness (1.8).

General stability
In what follows, let us consider the Hilbert space L 2 (Ω) endowed with the inner product and the corresponding norm and the Banach space L p (Ω), for p ≥ 1, endowed by the norm Let −∆ be the operator defined by the triple , where We recall that the Spectral Theorem for self-adjoint operators guarantees the existence of a complete orthonormal system (ω ν ) of L 2 (Ω) given by the eigenfunctions of −∆.If (λ ν ) are the corresponding eigenvalues of −∆, then We denote by V m the subspace of viscoelastic Kirchhoff-Carrier equation

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where u m is the solution of the approximate Cauchy problem By standard methods in differential equations, we can prove the existence of a solution to (2.2) on some interval [0, t m ).Then, this solution can be extended to the interval R + by using of the first estimate below.(2.3) and since then we deduce, taking (2.3) and the last identity into account, (2.5) Using the binary notation (2.6) Then substituting (2.6) in (2.5) yields and using 1 2 we get (2.7) On the other hand, the hypothesis (1.3) implies that (2.8) Combining (2.7) and (2.8), and observing that g > 0 and g ′ ≤ 0, we deduce where L 1 does not depend neither on m ∈ N nor on t ≥ 0. This implies that the approximated solution u m exists globally in the topologies given in (2.9).
Defining the energy E associated to problem (1.2) by then, in view of (2.7), it is non increasing function.In addition, as a consequence of (2.7), the following identity of the energy holds: Energy decay estimate.Define and Proof.Applying Hölder inequality and Fubini theorem, we have ✷ From now on, for short notation, we shall drop the parameter "m" in u m .We have the following useful lemma: Lemma 2.2.Let u be a solution to the approximated problem (2.2) corresponding to initial data taken in bounded sets of H 1 0 (Ω)×L 2 (Ω).Then, we have the following decay rate estimate: for some positive constants c and θ which do not depend on m ∈ N.
Proof.From (2.2), we have, Recovering the potential energy.
Substituting w = u in (2.12), multiplying by ξ(t) and integrating over [0, T ], we can write Having in mind that Now, we will estimate separately the last terms on the right hand side of (2.15).We have, using Cauchy-Schwarz and Young's inequalities, On the other hand, because ξ ′ ξ is bounded, we see that, for any ǫ 0 > 0, where From (2.15), (2.16) and (2.17) we arrive at Recovering the kinectic energy.Substituting w = g ⋄ u ∈ V m in (2.12) and multiplying by ξ(t), it results that Integrating the last identity over (0, T ), we obtain, Let t 0 > 0 such that g(t 0 )t 0 > 0. This is possible in vertue of Assumption 1.2.Then one has t 0 g(s) ds ≥ g(t 0 )t 0 > 0, ∀t ≥ t 0 . ( On the other hand, it is convenient to observe that dt. Combining (2.23) and (2.24) we infer, for all T ≥ t 0 , Next, we shall analyse the terms on the right hand side of (2.25).
Estimate for We have, .
(2.26) Thus, having in mind lemma 2.1, the definition of the energy in (2.10) and that ξ is non increasing, we deduce Therefore for some C > 0, which, from now on, will represent various constants do not depend on T and m ∈ N, which is crucial in the proof.
Estimate for Employing lemma 2.1 and the property ξ(t) ≤ ξ(0), one has where ε is an arbitrary positive constant.
Similarly, because ξ ′ ξ is bounded, we have where ε is an arbitrary positive constant.

Well-posedness
Lemma 3.1 (H 2 (Ω) a priori bounds).Suppose that u is a local solution on [0, T [ such that sup for some K > 0 and T > 0.Then, the following estimate holds: Proof.Taking w = −∆u ′ ∈ V m in the approximate problem (2.2) yields  2 are satisfied, for example, if g converges to zero at infinity faster than 1 t d , for any d > 0, like g 1 (t) = a 1 e −b1(t+1) q 1 and g 2 (t) = a 2 e −b2(ln(t+e q 2 −1 )) q 2 , where a i , b i , q 1 > 0 and q 2 > 1 such that a i are small enough so that (1.6) holds.For these two particular examples, ξ is given, respectively, by Howover, when g converges to zero at infinity slower than 1 t d , for some d > 0, like where a 3 > 0 and q 3 > 1, Assumption 1.2 is satisfied with provided that a 3 is small enough so that (1.6) holds.But (3.8) is not always satisfied, since (3.8) is equivalent to 1 2 (2α + 1)θq 3 > 1. Assume that Assumption 1.1, Assumption 1.2 and (3.8) hold, let K > 0 and set and Recalling Lemma 2.2, one can assert that u (the approximate solution constructed by Galerkin method) and u ′ exist globally in R + .Suppose that (u 0 , u 1 ) ∈ S K for some K > 0. Thus, we would like to prove that which contradicts (3.13).Thus, we have shown (3.11).As a consequence, we can repeat the continuation procedure indefinitely and we can conclude that, if (u 0 , u 1 ) ∈ S, the solution u can be continued globally on R + and (u(t), u ′ (t)) ∈ S, for all t ≥ 0.  Taking the inner product in L 2 (Ω) of the first equation of the above system with w ′ , we deduce 204 viscoelastic Kirchhoff-Carrier equation 211 and, using (1.3) from (2.14), we find