A General Stability Result for Viscoelastic Equations with Singular Kernels

with smooth kernel, has attracted a great deal of researchers and several existence and stability results have been established. See, for instance, the works of [3-19] where the relaxation function was assumed to be either of polynomial or of exponential decay. Messaoudi [20] studied (1.1) for more general decaying kernels and established some general decay results, from which the usual exponential and polynomial rates are only special cases. After that a series of papers have appeared, using similar techniques, and obtaining similar general decay results. See, among others, [21-25].


s H g s s
Where H is a given continuous positive increasing convex function such that H (0) = 0, and developed an intrinsic method for determining optimal decay rates.
In all the above mentioned works, kernels are assumed to be regular on %[0, +∞).
However, Kinetic theories for chain molecules as mentioned in [27] and some experimental data [28] suggests that this a realistic possibility, at least for some viscoelastic materials like dilute solutions of coiling polymers. Contrary to the regular kernel case, only very few results related to singular (at the origin) kernels have been established. For instance, Hrusa and Renardy [29] studied a model equation in nonlinear viscoelasticity and proved local and global existence theorems, allowing the memory function to have a singularity. To achieve their result, they approximated the equation by equations with regular kernels and then used the energy estimates to prove convergence of the approximate solutions. In [30], the authors showed that a singular kernel may yield smoothing effects for the solution of an evolution prob-lem, though the gain in regularity cannot be derived without specifying the kind of singularity [31]. Gentili considered a linear viscoelastic material with a relaxation function which may exhibit an initial singularity. He used the Laplace transform to study existence, uniqueness and asymptotic behavior of the solution to the dynamic problem. To provide these results, the author required the relaxation function to satisfy only restrictions deriving from Thermodynamics. He also used the energy method to establish a stability theorem and obtained a regularity result for a class of singular kernels which ensures the asymptotic stability of the solution. Tatar considered and proved an exponential decay result. Notice that the kernel here exhibits an initial singularity, summable, and decays exponentially at infinity. This type of kernels appears mostly in fractional calculus [32]. Wu [33]  with , 0 ρ > p and g in [34]. We refer the reader to Carillo et al. [35][36][37] for more recent results regarding viscoelastic problems with singular Kernels.
In this paper we are concerned with the following viscoelastic problem Where Ω is a bounded domain of R n (n ≥ 1) with a smooth boundary ∂Ω and the relaxation function g is a positive non increasing function which can exhibit a singularity at 0. Our aim in this work is to show that, with a slight modification in the arguments of [20], we extend the general decay result, established for regular relation functions, to singular kernels. As we show later, the exponential decay results of Tatar [31] and Wu [27], among others, are only special cases. We also give some examples and present some numeric to illustrate our decay results. The paper is organized as follows. In section 2, we present some hypotheses and technical lemmas and our main decay result. In section 3, the proof of our main result, as well as some illustrating examples will be given. Section 4 is devoted to the numerical setting and tests of our decay results. We finish our paper with some concluding remarks [11].

Preliminaries
In this section we state our hypotheses, give, without proof a standard existence theorem, and state our main decay result. So, we assume (H1) g: (0, +∞) → (0, +∞) is a differentiable integrable function satisfying Remark 2.1: These conditions allow a larger class of functions than that considered in [14,20] and others. In particular it allows singular integrable functions such as Notice, also, that [14,20] is no longer required. Now, we introduce the energy functional ( ) For completeness, we state the following existence result, which can be proved using the Galerkin method [7]. For more about existence, see [38] and [29].
Our main stability result is Theorem 2.2: Assume that (H1) and (H2) hold. Then, for any t 0 >0, there exist two positive constants k and K such that the solution of (1.2) satisfies

Main Result
In this section we prove our main decay result. We will use c, throughout this paper, to denote a generic positive constant. We start with the following lemmas. Satisfies, along the solution, the estimate Proof: By using equation (1.2), we easily see that We then estimate the third term in the right-hand side of (3.3), as in Lemma 3.4 [20], to obtain By combining (3.3) and (3.4), we arrive at

g t u t u d dx g s ds u dx
Similarly to (3.3), we estimate the right-hand side terms of (3.7). So, by Lemma 2.3, Young's inequality and the fact that Similarly to (3.4), the second term of (3.7) can be estimated as follows As for the third and the fourth terms of (3.7) we have By combining (3.7)-(3.10), the assertion of the lemma is established.

Numerical Test
In this section, we present a two dimensional case of system (1.1) in order to illustrate our theoretical decay result.

Numerical scheme
For computational purposes, we rewrite (1.1) as follows ( ) We consider a square domain [0, 1]×[0, 1] meshed uniformly in N x ×N y grids with the space steps ∆x=∆y= 1/max(N x , N y ). We chose a time interval [0, T] subdivided uniformly into N t = T /∆t sub-intervals with a time step ∆t = α∆x. The solution u(x, y, t) approximated at each point of the mesh (x i , y j ) = (i∆x, j∆y) and at any time t n = n∆t is given by , , At each time t n the interval [0, t n ] is subdivided uniformly in n sub-intervals using the same time step ∆t where the function g at each time s k = k∆t is given by g(s k ).
The full discretization of (1.1) in time and space is given by If we compute the variation of the discrete energy, we obtain  The quantity ∆E is numerically strictly negative as expected.

Decay behavior of the discrete energy
In order to show the non-increasing behavior of g and the decay relation E(t n ) ≤ Eps(t n ), we consider the examples stated in Section 3 and choose the following parameters: α=0.3, Nx=Ny=50, T=10, k=0.01, K=4, c 0 =0.01, a=0.1, b=1.1.
The space step is ∆x = ∆y = 0.1 and the time step ∆t = 0.033 (Figure 1).

Conclusions
In this work, we have the following conclusions • The general decay result, known for viscoelastic problems with regular kernels, has been extended successfully to problems with singular kernels. • The decay result is established with weaker conditions on the function ξ(t).
• The exponential decay results of [33,34] are only special cases.
• The numerical tests presented for the four types of relaxation functions are in accordance with our theoretical result.