Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

/is paper is concerned with a nonlinear viscoelastic Kirchhoff plate utt(t) − σΔutt(t) + Δ2u(t)− 􏽒 t 0 g(t − s)Δ 2u(s)ds � divF(∇u(t)). By assuming the minimal conditions on the relaxation function g: g′(t)≤ ξ(t)G(g(t)), where G is a convex function, we establish optimal explicit and general energy decay results to the system. Our result holds for G(t) � tp with the range p ∈ [1, 2), which improves earlier decay results with the range p ∈ [1, 3/2). At last, we give some numerical illustrations and related comparisons.


Introduction
In this paper, we consider the following nonlinear viscoelastic Kirchhoff plate equation: together with simply supported boundary condition and initial conditions where Ω ⊆ R n (n ≥ 1) is a regular and bounded domain; σ > 0 is a constant; and F: R n ⟶ R n is a vector field which will be assumed later. e relaxation function g(t) is a real function. From the physical point of view, systems (1)-(3) are related to classical theory for beams/plates appearing from materials with viscoelastic structures.
For viscoelastic wave equation, the problems are truly overworked. For example, for specific behavior of the relaxation function g, it is showed that the energy decays exponentially (polynomially) if g decays exponentially (polynomially) (see Cabanillas and Muñoz Rivera [1], Muñoz Rivera et al. [2,3], Santos [4], Cavalcanti and Oquendo [5], and so on). Messaoudi [6,7] considered two classes of system (4), by first introducing the assumption g′(t) ≤ − ξ(t)g(t), and established general decay results. Han and Wang [8], Liu [9,10], Messaoudi and Mustafa [11], Mustafa [12], and Park and Park [13] also used this assumption on g to get general decay of energy for problems related to (4). In [14], Lasiecka and Tataru introduced a more general assumption on g, which satisfies g ′ (t) ≤ − H(g(t)), where H is strictly increasing and convex function. ere are also so many stability results established by using this condition. We refer the reader to Cavalcanti et al. [15,16], Lasiecka et al. [17,18], Mustafa [19], Mustafa and Messaoudi [20], and Xiao and Liang [21]. Very recently, in [22,23], Mustafa considered two classes of wave equations and proved general and explicit decay results of energy by using a new general assumption on g: g ′ (t) ≤ − ξ(t)H(g(t)). For viscoelastic plate equation, Rivera et al. [24] studied the following equation: together with initial and dynamical boundary conditions. ey proved the energy decays exponentially (resp. polynomially) if the relaxation function g decays exponentially (resp. polynomially). Alabau-Boussouira et al. [25] considered together with Dirichlet-Neumann boundary conditions and established exponential and polynomial decay results for sufficiently small initial data. When f (u) � 0 in (6), Cavalcanti [26] considered the equation subject to nonlinear boundary conditions and established exponential decay of energy by assuming a nonlinear and nonlocal feedback acting on the boundary and provided that the relaxation function decays exponentially. In Andrade et al. [27], the authors considered a viscoelastic plate equation with p-Laplacian: where Δ p u � div(|∇u| p− 1∇u ), and they established an exponential decay of energy under the assumption g ′ (t) ≤ − k 1 g(t) When Δ p u is replaced by div(|∇u| p → (x,t)− 1 ∇u) in (7), Ferreira and Messaoudi [28] proved a general decay result of energy. Feng [29] established a general decay result of a plate equation with time delay. Recently, Gomes Tavares et al. [30] considered a class of nonlinear plate equations with memory; they proved, by using the methods developed by Lasiecka and Wang [18], the decay rates are expressed in terms of the solution to a given nonlinear dissipative ODE. In [31], the authors proved the well-posedness of solutions to problem (1)-(3) with the case σ > 0 and the case σ � 0. Under the assumptions on g(t): they established the general decay rates of energy of the form If the memory term is infinite, one can find some results on plate equation with history memory in [32][33][34][35][36][37][38] and so on.
In this paper, we continue to study (1)-(3), in which we consider σ � 1 for simplicity, with minimal conditions on the L 1 (0, ∞) relaxation function g (see (12)). We establish explicit and general energy decay results of systems (1)-(3) by using the idea of Mustafa [22,23] and some properties of convex functions developed in [18,39]. We point out that the decay results established here are optimal exponential and polynomial rates for 1 ≤ p < 2 when G(t) � t p , which improved the previous known results for 1 ≤ p < (3/2). Under this level of generality, the decay rates we get are optimal, and our results improve the stability results in previous works. At last, we give some numerical illustrations. e rest of this paper is as follows. In Section 2, we give some assumptions and results. e general decay result of the energy will be established in Section 3. In Section 4, we give some numerical illustrations.

Assumptions and Results
In the following, for simplicity, we write ‖·‖ instead of ‖·‖2. c > 0 is used to denote a generic constant. e positive constants λ1 and λ2 represent the embedding constants for u ∈ H 2 (Ω) ∩ H 1 0 (Ω). For relaxation function g, we assume the following: In addition, there exists a C 1 function G: R + ⟶ R + satisfying G(0) � G′(0) � 0. e function G(t) is linear or it is an increasing strictly convex function of class C 2 (R + ) on (0, r], r ≤ g(0), such that where ξ(t) is a C 1 function satisfying With respect to F, we assume that (A2) F: R n ⟶ R n is a C 1 − vector field given by F � (F 1 , where for j � 1, 2, . . ., n, the constants k j > 0 and p j satisfy 2 Complexity Moreover, we also assume that where F is a conservation field with F � ∇f and f: R n ⟶ R is a real valued function. Remark 1 (see [31]). Condition (14) implies that there exists a positive constant K � K(k j , p j , n), j � 1, 2, . . ., n, such that e existence of global solutions has been proved in [31].

Theorem 1. Let (11) and (A2) hold. If the initial data
has a unique weak solution satisfying that for any T > 0, e total energy of problem (1)-(3) is given by where Now, we give the stability result of energy to problem (1)-(3). (21) where the positive constant k 1 depends on the size of initial data, k 2 > 0, and

then the energy E(t) satisfies
In particular, for G(t) � t p in (12), we can get where k, k, and k 1 are positive constants.

Remark 2
(1) Since the constant cCα appearing within the estimate (47) depends on E(0), which leads to the final constant which will depend on the initial data, the constant k 1 > 0 depends on the size of initial data. (2) Here, G 1 is strictly convex and decreasing on (0, r] with lim t⟶0 G 1 (t) � +∞.
Remark 3. It follows from (A1) that lim t⟶0+∞ g(t) � 0. We know that there exists some t 1 ≥ 0 large enough such that en, we can get for every t ∈ [0, t 1 ], erefore, there exist positive constants a and b, which yields for every t ∈ [0, t 1 ], . (27) We end this section by giving three examples.
We take a > 0 satisfying (11). For a fixed positive constant ρ, we have We take a > 0 satisfying (11). en, g

Optimal Decay
To prove eorem 2, we need some lemmas.

Technical Lemmas
Lemma 1. It holds that for any t ≥ 0, Proof. Multiplying L 2 (Ω) in equation (1) by u t and using integration by parts and boundary conditions, we can get (31). Let us define the functionals: introduced in [22,23].
Proof. In view of (1), and integration by parts, we can obtain It follows from (16) that Hölder's inequality gives us which, together with Young's inequality, implies

Lemma 3. Assume that (A1) and (A2) hold, then the functional ψ(t) satisfies for any δ
Proof. Taking the derivative of ψ(t), using equation (1) and integration by parts, we can derive that By Young's inequality and (37), we shall see below, for any δ > 0, Complexity 5 Similarly, we can get for any δ > 0, With respect to I3, we have ds.

(43)
By using (A2), we derive that Following the same method as in [31], we can get where μ 1 and μ 2 are two positive constants and In view of Young and Hölder's inequalities and (37), we have for any δ > 0, us, we can find (39) from (27) and (40).

□
Our argument in the following is based on the choice of a suitable Lyapunov function L(t) by where N, N 1 and N 2 are positive constants. Clearly, for N large, there exist β 1 > 0 and β 2 > 0 such that Lemma 5. It holds that for any t ≥ t 1 , (54) (31), (33), and (39), taking δ � 1/(2N 2 ), and noting g ′ � ag − h, we can infer that for any t ≥ t 1 , Firstly, we take N 1 large enough so that and then we choose N 1 so large that Since ((αg 2 (s))/(αg(s) − g ′ (s))) < g(s), using the Lebesgue dominated convergence theorem, we can get Hence, there exist some α 0 (0 < α 0 < 1) such that if α < α 0 , then (59) At last, for any fixed N 1 and N 2 , we choose N large enough and choose α satisfying and then is ends the proof.  (27) and (31), we find that for any t ≥ t 1 , which, along with (54), gives us for some constant m > 0 and for all t ≥ t 1 , Define F(t) � L(t) + cE(t) ∼ E(t). en, we find from (63), We consider the following two cases.
(I) p � 1. Multiplying (64) by ξ(t) and using (31) and (A2)-(A3), we have Since ξ(t) is a nonincreasing continuous function and ξ′(t) ≤ 0 for a.e. t, then In view of ξF + cE ∼ E, we obtain that there exist two positive constants c 1 , c 2 > 0, Complexity 7 It follows from (49) and (54) that E(t) ≥ 0 and there exists a positive constant β such that for any t ≥ t 1 , (69) en, there exists a certain constant β 1 > 0, is gives us Define and we know that Without loss of generality assuming t 1 so large that I(t 1 ) > 0, then (75) Using Jensen's inequality and (12), we can derive from (64) that for some constant q > 0, We multiply (76) by E p− 1 (t) and use (31) to deduce By Young's inequality, we have for any ε 1 > 0, Taking ε 1 < (1/2)q, we conclude en, there exists a certain constant q 0 > 0 such that from which we obtain 8 Complexity where c 3 is a positive constant. Combining (I) and (II), we can get (35).

Numerical Tests
In this section, we present various tests in order to illustrate our theoretical results proved in eorem 2. We solve problem (1) using the nonlinear Lax-Wendroff method in time and space in the space-time domain [0, 1] × [0, 5]. Moreover, for all partial derivatives of problem (1), we used a second-order discretization in time and space, and we consider the vector field below For the following values of the parameter σ � 0.1, 1, 5, we simulate six tests of the decay of the energy (19) (for similar constructions, we refer to [41,42]). Test 1: in the first three tests, we present the decay case using the exponential function g 1 (t) = e − 2t , the vector field (103), and the parameters σ = 0.1, 1, 5 (Test 1.1, 1.2, and 1.3). Test 2: in the second three numerical tests, we examine the energy decay (19) using the polynomial function g 2 (t) � (1/(1 + t)), the vector field (103), and the parameters σ = 0.1, 1, 5 (Test 1.1, 1.2, and 1.3).
In order to ensure the scheme stability, we use Δt � 0.0005 < dx � 0.005 satisfying the stability of the Courant-Friedrichs-Lewy (CFL) inequality, where Δt represents the time step and dx represents the spatial step. e spatial interval [0, 1] is subdivided into 200 subintervals, where the temporal interval [0, 5] is deduced from the stability condition above. We run our code for 10000 time steps using the following initial conditions:

Complexity
In Figure 1, we show the results of the first three tests, namely, Test 1.1 for σ = 0.1, Test 1.2 for σ = 5, and Test 1.3 for σ = 5. We present the cross section cuts at x = 0.25, x = 0.5, and x = 0.75. e damping behavior is demonstrated for all experiments. Moreover, it should be stressed that for larger σ, the pseudoperiod decreases within decaying envelope. But for smaller σ (tending to 0), the pseudoperiod increases within same decaying envelope. Under similar initial and boundary conditions, we present in Figure 2 the results obtained for the Test 2.1 for σ = 0.1, Test 2.2 for σ = 5, and Test 2.3 for σ = 5. In Figure 3 we can clearly compare the energy decay obtained in Test 1 and in Test 2. We remarked that the energy decay is not affected by the choice of the memory functions g 1 (t) � e − 2t and g 2 (t) � 1/(1 + t).
Finally, it should be stressed that solving problem (1) using linear vector field functional F leads to similar damping behavior of the waves and similar decay results, either for the choice of the type of the function g i , i � 1, 2 or the positive parameter σ.

Conclusion
In this paper we investigate a nonlinear Kirchhoff viscoelastic plate. Under suitable assumptions on relaxation function, we establish a more general decay result of energy, by introducing suitable energy and perturbed Lyapunov functionals. e decay results established here are optimal exponential and polynomial rates for 1 ≤ p < 2 when G(t) � t p , which improved the previous known results for 1 ≤ p < (3/2). Under this level of generality, the decay rates we get are optimal, and our results improve the stability results in previous works. At last, we give some numerical illustrations and related comparisons.

Data Availability
e data used to support the findings of this study are included within the article.