Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

Feng, Baowei; Zahri, Mostafa

doi:https://doi.org/10.1155/2020/6079507

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Abstract Introduction Results Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Geometric and Polynomial Approaches of Complex Systems and Control in Mathematics and Applied Sciences

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Research Article | Open Access

Volume 2020 | Article ID 6079507 | https://doi.org/10.1155/2020/6079507

Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

Baowei Feng¹and Mostafa Zahri²

Guest Editor: Jorge-Antonio Lopez-Renteria

Received02 Dec 2019

Accepted14 Feb 2020

Published16 Mar 2020

Abstract

This paper is concerned with a nonlinear viscoelastic Kirchhoff plate By assuming the minimal conditions on the relaxation function : , where is a convex function, we establish optimal explicit and general energy decay results to the system. Our result holds for with the range , which improves earlier decay results with the range . At last, we give some numerical illustrations and related comparisons.

1. Introduction

In this paper, we consider the following nonlinear viscoelastic Kirchhoff plate equation:together with simply supported boundary conditionand initial conditionswhere is a regular and bounded domain; σ > 0 is a constant; and is a vector field which will be assumed later. The relaxation function 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 , it is showed that the energy decays exponentially (polynomially) if 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 , 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 to get general decay of energy for problems related to (4). In [14], Lasiecka and Tataru introduced a more general assumption on , which satisfies , where H is strictly increasing and convex function. There 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 : .

For viscoelastic plate equation, Rivera et al. [24] studied the following equation:together with initial and dynamical boundary conditions. They proved the energy decays exponentially (resp. polynomially) if the relaxation function decays exponentially (resp. polynomially). Alabau-Boussouira et al. [25] consideredtogether 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_pu = div(|∇u|^p^−1∇u), and they established an exponential decay of energy under the assumption When Δ_pu is replaced by 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 :they established the general decay rates of energy of the formIf the memory term is infinite, one can find some results on plate equation with history memory in [32–38] 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 L¹(0, ∞) relaxation function (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 . 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.

The rest of this paper is as follows. In Section 2, we give some assumptions and results. The general decay result of the energy will be established in Section 3. In Section 4, we give some numerical illustrations.

2. Assumptions and Results

In the following, for simplicity, we write ‖⋅‖ instead of ‖⋅‖2. c > 0 is used to denote a generic constant. The positive constants λ1 and λ2 represent the embedding constantsfor .

For relaxation function , we assume the following:(A1) are nonincreasing C¹ functions satisfying In addition, there exists a C¹ function satisfying G(0) = G′(0) = 0. The function G(t) is linear or it is an increasing strictly convex function of class on (0, r], , such that where ξ(t) is a C¹ function satisfying With respect to F, we assume that(A2) is a C¹−vector field given by F = (F₁, F₂, …, F_n) such that where for j = 1, 2, …, n, the constants k_j > 0 and p_j satisfy Moreover, we also assume that where F is a conservation field with F = ∇f and 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 thatThe existence of global solutions has been proved in [31].

Theorem 1. Let (11) and (A2) hold. If the initial data , then problem (1)–(3) has a unique weak solution satisfying that for any T > 0,

The total energy of problem (1)–(3) is given bywhere

Now, we give the stability result of energy to problem (1)–(3).

Theorem 2. Let (A1) and (A2) hold. If , then the energy E(t) satisfieswhere the positive constant k₁ depends on the size of initial data, k₂ > 0, and

In particular, for G(t) = t^p in (12), we can getwhere , and k₁ 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 k₁ > 0 depends on the size of initial data.(2)Here, G₁ is strictly convex and decreasing on (0, r] with G₁(t) = +∞.

Remark 3. It follows from (A1) that . We know that there exists some t₁ ≥ 0 large enough such thatThen, we can get for every t ∈ [0, t¹],Therefore, there exist positive constants a and b,which yields for every t ∈ [0, t¹],We end this section by giving three examples.

Example 1. Let , (b > 0). We take a > 0 satisfying (11). If we choose , then . From (23)₁, we find

Example 2. Let , (b > 1). We take a > 0 satisfying (11). For a fixed positive constant ρ, we have with , (1, 2). From (23)₂, we get

Example 3. Let , (0 < b < 1). We take a > 0 satisfying (11). Then, with and ξ(t) = b(1 + t)b − 1. Then, we obtain from (23)₁ that

3. Optimal Decay

To prove Theorem 2, we need some lemmas.

3.1. Technical Lemmas

Lemma 1. It holds that for any t ≥ 0,

Proof. Multiplying L²(Ω) in equation (1) by u_t and using integration by parts and boundary conditions, we can get (31).
Let us define the functionals:

Lemma 2. The functional ϕ(t) satisfies for any t ≥ 0,for any , whereintroduced in [22, 23].

Proof. In view of (1), and integration by parts, we can obtainIt follows from (16) thatHölder’s inequality gives uswhich, together with Young’s inequality, impliesReplacing (36) and (38) in (35), we can get the desired estimate (33).

Lemma 3. Assume that (A1) and (A2) hold, then the functional ψ(t) satisfies for any δ > 0,

Proof. Taking the derivative of ψ(t), using equation (1) and integration by parts, we can derive thatBy Young’s inequality and (37), we shall see below, for any δ > 0,Similarly, we can get for any δ > 0,With respect to I3, we haveBy using (A2), we derive thatFollowing the same method as in [31], we can getwhere μ₁ and μ₂ are two positive constants andIn view of Young and Hölder’s inequalities and (37), we have for any δ > 0,Thus, we can find (39) from (27) and (40).

Lemma 4. Let (A1) and (A2) hold, the functional θ(t) defined bywhere , satisfies

Proof. First we know that σ′(t) = −g(t). Then,By using Young and Hölder’s inequalities, we will show thatSince σ(t) ≤ σ(0) = 1 − l and , we can obtain (49).

Our argument in the following is based on the choice of a suitable Lyapunov function bywhere N, N₁ and N₂ are positive constants. Clearly, for N large, there exist β₁ > 0 and β₂ > 0 such that

Lemma 5. It holds that for any t ≥ t₁,

Proof. Let . Combining (31), (33), and (39), taking , and noting , we can infer that for any t ≥ t₁,Firstly, we take N₁ large enough so thatand then we choose N₁ so large thatSince , using the Lebesgue dominated convergence theorem, we can getHence, there exist some α₀ (0 < α₀ < 1) such that if α < α₀, thenAt last, for any fixed N₁ and N₂, we choose N large enough and choose α satisfyingand thenThis ends the proof.

3.2. Proof of Theorem 2

Taking into account (27) and (31), we find that for any t ≥ t₁,which, along with (54), gives us for some constant m > 0 and for all t ≥ t₁,

Define . Then, we find from (63),

We consider the following two cases.

Case 1. The particular case G(t) = tp.(I). 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 c¹, c² > 0,(II)1 < p < 2. Define by It follows from (49) and (54) that and there exists a positive constant β such that for any t ≥ t₁, Then, there exists a certain constant β₁ > 0,This gives usHence,Defineand we know thatWithout loss of generality assuming t₁ so large that I(t₁) > 0, thenUsing 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 deduceBy Young’s inequality, we have for any ɛ₁ > 0,Taking , we concludeDefine . Multiplying (79) by ξ(t), we haveThen, there exists a certain constant q₀ > 0 such thatfrom which we obtainwhere c₃ is a positive constant.
Combining (I) and (II), we can get (35).

Case 2. The general case.
Define I(t) byIt follows from (72) that we can choose a constant 0 < q < 1 so that for all t ≥ t₁,Without loss of generality, we assume that I(t) > 0 for all t ≥ t₁; otherwise, (64) implies an exponential decay. In addition, we define λ(t) byClearly, λ(t) ≤ −cE′(t). Since G(t) is strictly convex on (0, r] and G(0) = 0, thenprovided 0 ≤ ν ≤ 1 and x ∈ (0, r]. By using (12), (84), and Jensen’s inequality, we can obtainHere, is an extension of G, which is strictly convex and strictly increasing C² function on (0, ∞). We can get from (87) thatwhich, together with (64), implies for any t ≥ t¹,Now, for ɛ0 < r, we define the function bywhich is equivalent to E(t). In view of E′(t) ≤ 0, , and , and using (89), we conclude thatAs in Arnold [40], we define the conjugate function of by , which satisfiesFor and , and using (91), we getMultiplying (93) by ξ(t), we see thatwhere we used the fact that as , .
Define the functional byIt is easy to obtain that , i.e., there exist two positive constants β₃ and β₄ such thatWe choose suitable ɛ0 to get from (94) that for a certain constant k > 0,where G₂(t) = tG′(ɛ₀t). Denote . It follows from (96) thatSince G′₂(t) = G′(ɛ₀t) + ɛ₀tG″(ɛ₀t), then, using the strict convexity of G on (0, r], we know that G′₂(t), G₂(t) > 0 on (0, 1]. By (97), we get for some k₁ > 0 and for all t ≥ t₁,Integrating (99) over (t₁, t), we havewhich, noting , defined byis strictly decreasing on and , we getThen, (21) follows from (98) and (102). The proof is done.

4. Numerical Tests

In this section, we present various tests in order to illustrate our theoretical results proved in Theorem 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 belowFor the following values of the parameter , 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 g₁(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 , 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. The 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:

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. The 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 and .

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

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 , i = 1, 2 or the positive parameter σ.

5. 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. 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 . 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

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (no. 11701465).

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Copyright © 2020 Baowei Feng and Mostafa Zahri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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