Decay rate of a weakly dissipative viscoelastic plate equation with infinite memory

In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical illustrations are presented at the end. The numerical solution is computed using the popular finite element method in space, combined with time-stepping finite differences.

Assuming that the relaxation function g in (1.1) satisfies g (t) ≤ −δg(t), for t ≥ 0 and for some positive constant δ, Revira et al. [15] showed an exponential convergence of the energy decay. The main focus of this work was on investigating the energy decay of problem (1.1) but for a wider class of g, see (2.2) below. Owing to the presence of the weakly dissipative term, a second energy functional is introduced to achieve our goal. To make use of the convolution properties and the history condition, the viscoelastic (infinite memory) term is decomposed accordingly. For numerical illustrations of our theoretical finding, we provide a graphical illustration of the numerical energy decay using an approximate solution of problem (1.1) based on finite differences in time and finite elements in space.
Outline of the paper In the next section, we introduce some necessary notations and assumptions. For our decaying analysis, we state and prove a few technical lemmas. Section 3 is dedicated to show the decaying rates of the energy functional E, see Theorem 3.4. Having a weakly dissipative term in problem (1.1) leads us to introduce a second energy functional E [see (3.4)] to overcome the difficulties in proving the decay of E. For the sake of illustrating the theoretical decaying rate of E numerically, we develop a fully discrete numerical method in Sect. 4. To avoid dealing with C 2 numerical methods in the spatial variable (which is often not convenient on various physical domains Ω) due to the presence of the biharmonic operator in problem (1.1), we rewrite (1.1) as a coupled system that involves a second-order elliptic operator instead. Then, we apply the C 0 Galerkin finite element method to discretize in space. In the time variable, a second central difference is used to handle the second time derivative, while the other terms are approximated appropriately. We show the decaying of both, the numerical solution of problem (1.1) and also the approximation of the energy functional E.

Preliminaries
For ≥ 0, H (Ω) is the standard Sobolev space which reduces to L 2 (Ω) space when = 0. On this space, ·, · is the usual inner product and · is the associated L 2 (Ω)-norm. An application of the Poincaré inequality and using the elliptic regularity property, In the decaying energy analysis (including Lemmas 2.1 and 2.2), we assume that the relaxation function g ∈ C 1 (R + ) and satisfies where ξ is a positive non-increasing C 1 function, μ 0 = ∞ 0 g(s)ds, and c 0 is a positive constant. For later use, by (2.1) and the second inequality in (2.2), we have, for t ≥ 0, For convenience, we introduce the following notations: for t ≥ 0, and for 0 < ε < 1, where the function η t is the relative history of u [5], defined as η t (s) = u(t) − u(t − s). The next two lemmas will be used in the forthcoming decaying analysis section:

Decay
In this section, we find the energy functional E of problem (1.1). As a starting point, taking the inner product of (1.1) with u , and then applying Green's formula (twice for the second term and once for the third term) and using the fact that u = Δu = 0 on ∂Ω, yield the following weak formulation of (1.1): Using Lemma 2.2 with ϕ = ∇u, this equation can be rewritten as follows: where E is the first energy functional given by where the non-negative property of E follows from (2.3). Now, taking the inner product of (1.1) with −Δu , following the above steps, we get where E is the second energy functional, defined as follows: and its non-negativity follows also from (2.3). In the remaining analysis, c is a generic positive constant. We estimate in the next two lemmas the time derivative of the following functionals: Lemma 3.1 Along the solution of (1.1), we have , using (1.1) and Green's formula, we get Young's inequality, Lemma 2.1, and the inequalities in (2.1) imply that Combining the above results completes the proof.

Lemma 3.2
Along the solution of (1.1) and for δ > 0, we have Proof Differentiating I 2 and using the differential equation in (1.1), we get where By Green's formula, Young's inequality and Lemma 2.1, we have and in addition, using To estimate I 2,3 (t), we perform integration by parts and then make use of Young's and Hölder's inequalities, the fact that g = εg − h ε and Lemma 2.1. So, we obtain Inserting the obtained estimates of I 2,1 (t), I 2,2 (t), and I 2,3 (t) in (3.7) and using (by Young's and Hölder's inequalities), in addition to the elliptic regularity property of the operator −Δ, the desired bound followed.
for a sufficiently large N (the proof is in [12]). Moreover, there exist positive constants α 1 and α 1 , such that Proof To prove the estimate in (3.8), we differentiate L and use (3.1) and (3.3), in addition to Lemmas 3.1 and 3.2 . Hence, Recalling that g (t) = εg(t) − h ε (t), then rearranging the terms in the above equation, and noting that h ε > 0, we arrive at Choosing ε = 1 N . Since (which is valid for ε 1 and ε 2 are sufficiently small), we notice from the above equation that Choosing δ < l 8 μ 0 < 1 8 μ 0 (because 0 < l < 1) and ε 1 = 3 8 μ 0 ε 2 , after some calculations, we have and consequently, Finally, recalling the definition of E and η t , and using the above bound, we obtain the desired estimate.
We are ready now to estimate the energy functional associated with the problem (1.1).

Numerical study
This section is devoted to illustrate numerically the achieved theoretical decaying results in Theorem 3.4 on a two-dimensional test problem of the form (1.1) with space variables x and y. We develop a numerical scheme for problem (1.1) using finite differences for the time discretization combined with the continuous Galerkin finite element method in space. Applying Galerkin method to problem (1.1) directly forces us to deal with C 2 polynomial approximations, which is definitely not convenient owing to the complexity in constructing the basis functions on various physical domains, and also increase the cost of computations. To avoid this, we rewrite (1.1) as a coupled system of lower order elliptic problems: Taking the inner product of the first two equations in (4.1) with φ, ψ ∈ H 1 0 (Ω), respectively, then, applying Green's formula and using u = w = 0 on ∂Ω. This implies Motivated by the above formulation, our fully discrete numerical scheme is defined as follows: Therefore, the fully discrete scheme (4.2) has the following matrix representation: with g n+1, j := Therefore, at each time level t n+1 , the numerical coupled system (4.2) reduces to a finite square linear system, where the unknown is the column vector b n+1 . Whence the coefficient vector b n+1 is computed, c n+1 can be determined by solving Mc n+1 = Ab n+1 , that is, c n+1 = M −1 Ab n+1 . This will be used to compute the numerical energy function E h (t n ). To solve linear system in (4.3) for b n+1 , successively, the column vectors b 0 and b 1 need to be determined first. In other words, we have to compute the approximate solutions U 0 h and U 1 h first. We consider U 0 h to be the Ritz projection of u 0 on the finite dimensional space V h . However, motivated by the Taylor series expansion of u in time about t = 0, we choose U 1 h to be the Ritz projection u 0 + t 1 u 1 on V h . For sake of computing efficiently the spatial integrals in the linear system in (4.3), on each cell of our two-dimensional partition, the associated integral is approximated using 4-point (that is, 2-point in each direction) Gauss cubature rule.
In our numerical example, choose T = 150, u 0 (x, y, t) = t 2 sin(π x) sin(π y) and u 1 (x, y) = 0. We run our computer program with M = 20 (that is, 400 cells in space) and N = 120000. We choose g(t) = e −t ; then g (t) = −ξ(t)g(t) with ξ(t) = 1. Thus, by Theorem 3.4, we expect our energy to decay monomially, that is, t E(t) ≤ c for a sufficiently large t. This is confirmed in Fig. 1. In addition, the plot of the numerical solution U h in Fig. 2 shows its convergence to zero as the time t getting far away from 0.