We consider the nonlinear von Kármán equations with memory term. We show the exponential decay result of solutions. Our result is established without imposing the usual relation between g and its derivative. This result improves on earlier ones concerning the exponential decay.

1. Introduction

In this paper we consider the exponential decay rate of solutions for the nonlinear von Kármán equations with memory term:

()

and the boundary conditions

()

where Ω is an open bounded set of ℝ², with a sufficiently smooth boundary Γ = Γ₀ ∪ Γ₁. Here, Γ₀ and Γ₁ are closed and disjoint. The constants h, ρ > 0. Let us denote by ν = (ν₁, ν₂) the external unit normal to Γ and by η = (−ν₂, ν₁) the corresponding unit tangent vector. The von Kármán bracket is given by

()

Here, we are denoting by ℬ₁, ℬ₂ the following differential operators:

()

where B₁ and B₂ are given by

()

and the constant μ (0 < μ < 1/2) represents Poisson’s ratio.

This system describes the transversal displacement u(x, y, t) and the Airy stress function v(x, y, t) of a vibrating plate. The dissipation in (1) is due to the term −g*Δ²u, where g is positive real function and the convolution product * is given by . A material whose contained term is −g*Δ²u is called viscoelastic and is said to be “endowed with long-range memory” since the stress at any instant depends on the complete history of strain that the material has undergone.

Problems related to

()

are interesting not only from the point of view of PDE general theory, but also due to its applications in Mechanics. For instance, when the material density, f(u^′), is equal to 1, (8) describes the extensional vibrations of thin rods; see Love [1] for the physical details. When the material density f(u^′) is not constant, we are dealing with a thin rod which possesses a rigid surface and whose interior is somehow permissive to slight deformations such that the material density varies according to the velocity.

On the other hand, the problem of stability of the solutions to the following wave equation with memory was studied by many authors [2–6]:

()

Cavalcanti et al. [2] showed an exponential and polynomial decay for the viscoelastic wave equation (9) with F(u, u^′) = −γΔu^′ under the usual conditions

()

for some c_i, i = 1,2, 3. Han and Wang [3] proved the uniform decay for the nonlinear viscoelastic equation under condition

()

where c > 0. Park and Kang [7] studied the uniform decay for a nonlinear viscoelastic problem with damping. They obtained the exponential decay estimate under condition (11). Later, this assumption was relaxed by several authors. Messaoudi and Tatar [6] investigated exponential and polynomial decay for a quasilinear viscoelastic equation under condition on g such as

()

where ξ > 0, by choosing a suitable perturbed energy. Liu [5] showed exponential and polynomial decay for the system of two coupled quasilinear viscoelastic equation, under condition (12). Messaoudi and Tatar [8] proved the exponential decay rate for a quasilinear viscoelastic equation under the conditions

()

They improved some earlier results concerning the exponential decay. Han and Wang [4] studied the general decay rate for the nonlinear viscoelastic equations under the more general conditions on g such as

()

When ρ = 0, the problem of stability of the solutions to the viscoelastic system with memory has been studied by many authors. In [9, 10], the authors proved exponential and polynomial decay for the viscoelastic wave equation under conditions (10). Berrimi and Messaoudi [11] studied exponential and polynomial decay rates under condition (12). Messaoudi [12] investigated the general decay rate for the viscoelastic equations under general conditions (14). Guesmia and Messaoudi [13] obtained general stability for the Timoshenko system under weaker condition on g such as

()

where ξ is a nonincreasing and positive function. As for problem of stability of the solutions to a viscoelastic system under condition (15), we also refer the reader to [14–16] and references therein. These general decay estimates extended and improved on some earlier results—exponential or polynomial decay rates.

The problem of stability of the solutions to a von Kármán system with dissipative effects has been studied by several authors. For example, in [17, 18] the authors studied the von Kármán equation in the presence of thermal effects. In [19–23] the authors considered the von Kármán system with frictional dissipations effective in the boundary. It is shown in these works that these dissipations produce uniform rate of decay of the solution when t goes to infinity. Rivera and Menzala [24] and Rivera et al. [25] studied the stability of the solutions to a von Kármán system for viscoelastic plates with memory and boundary memory conditions. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation function. Later, Santos and Soufyane [26] generalized the decay result of [24]. Raposo and Santos [27] considered the general decay of the solutions to a von Kármán plate model (1)–(4) for ρ = 0. They showed that the energy decays with a similar rate of decay of the relaxation function, which is not necessarily decaying in a polynomial or exponential fashion. Kang [28] investigated the general decay of the solution to a von Kármán system with memory and boundary damping. Recently, Kang [29] proved that solutions for a von Kármán plate with memory decay exponentially to zero as time goes to infinity in case g^′(t) + γg(t) ≥ 0 for all t ≥ 0 provided that [g^′(t) + γg(t)]e^αt ∈ L¹(0, ∞) for some α > 0.

In this paper, we establish an exponential decay of the solutions to the nonlinear von Kármán plate model (1)–(4) without assumption (15), which is the usual relation between g and its derivative. Instead of (15), we require the function e^αtg(t) to have sufficiently small L¹-norms on (0, ∞) for some α > 0. This result improves on earlier ones concerning the exponential decay of the solutions to the von Kármán equations.

The organization of this paper is as follows. In Section 2, we give some notations and introduce the relative results of Airy stress function and von Kármán bracket. In Section 3, we prove that the energy decreases exponentially. The construction of the Lyapunov function is inspired in multiplier techniques that was used in [8].

2. Preliminaries

In this section, we present some material needed in the proof of our result and state the main result. Throughout this paper we denote

()

and define

()

For a Banach space X, ∥·∥_X denotes the norm of X. For simplicity, we denote

by ∥·∥. We define for all 1 ≤ p < ∞

()

A simple calculation, based on the integration by parts formula, yields

()

where the bilinear symmetric form a(u, v) is given by

()

where dΩ = dx dy. Since Γ₀ ≠ ∅, we know that

is equivalent to the H²(Ω) norm; that is,

()

where c₀ and c₁ are generic positive constants. This and Sobolev embedding theorem imply that for some positive constants C_p and C_s

()

We establish the following hypotheses on the relaxation function g (see [8]). The relaxation function g : ℝ₊ → ℝ₊ is nonincreasing C¹ function satisfying

()

To simplify calculation in our analysis, we introduce the following notation:

()

From the symmetry of a(·, ·), we have that, for any v ∈ C¹(0, T; H²(Ω)),

()

Now, we introduce the relative results of the Airy stress function and von Kármán bracket [·, ·].

Lemma 1 (see [30].)Let u, w be functions in H²(Ω) and v in , where Ω is an open bounded and connected set of ℝ² with regular boundary. Then,

()

Lemma 2 (see [20], [31].)If u, v ∈ H²(Ω), then [u, v] ∈ L²(Ω) and satisfies

()

The energy of problem (1)–(4) is given by

()

The existence of solutions can be proved by the Faedo-Galerkin method; see [2, 9].

Theorem 3. Assume that the kernel g is a positive continuous function satisfying (23). Let (u₀, u₁) ∈ H⁴(Ω) × H²(Ω). Then, the system (1)–(4) has a unique weak solution u such that

()

3. Exponential Decay of the Energy

In this section we will prove the exponential decay rates. To demonstrate the stability of the system (1)–(4), the lemmas below are essential. The following result shows the dissipative property of the system (1)–(4). Multiplying (1) by u^′(t), we get the identity

()

Define the modified energy by

()

and applying (26) to (31), we have

()

This implies that F(t) is nonincreasing, and one easily sees that

()

Therefore, it is enough to obtain the desired decay for the modified energy F(t), which will be done below. The key point for showing our desired result is finding a Lyapunov functional L which is equivalent to F(t). First, we introduce three functionals and establish several lemmas. So, let

()

with

()

We define the modified energy by

()

for some positive constants γ_i is to be specified later.

Lemma 4. Assume that g satisfies (23) and (24). For N > 0 large enough, there exist α₁ > 0 and α₂ > 0 such that

()

Proof. From Young inequality, we deduce

()

Considering the embedding and taking (22) into account, it holds that

()

where C comes from the inequality ∥u∥_ρ+2 ≤ C||∇u|| for all

. On the other hand, by Young inequality, Hölder inequality and (22) can be estimated as

()

Thus, from (40) and (41) we obtain

()

where c₀ is a positive constant depending on γ₂, γ₃, h, ρ, l, C, and C_s. Choosing N > 0 large, we complete the proof of Lemma 4.

Lemma 5. For each t₀ > 0 and sufficiently large N > 0, there exists positive constant c₂ such that

()

Proof. By differentiating Φ₁(t) and using Young inequality, we get

()

where

, and δ > 0. Using (1)–(4), we have

()

We use the following inequality:

()

then we obtain

()

where η > 0. Similarly we deduce

()

Now, we estimate the terms in the right hand side of (48). The Young and Hölder inequalities and (22) give that

()

where a₀ = C^2(ρ+1)(2h⁻¹E(0)) ^ρ > 0. From Lemmas 1 and 2 and (22), we obtain

()

Summarizing these estimates with (48), we deduce that

()

Since g is continuous and positive, for any t ≥ t₀ > 0 we have

()

Thus, making use of (52) and combining (33), (37), (44), (47), and (51), we obtain

()

We first take γ₂ > 0 and δ > 0 so small that

()

respectively. And then, we choose η > 0 and γ₃ > 0 so small that

()

respectively. We then pick γ₁ large enough so that

()

Finally, taking N > 0 large enough and by (53), we conclude that

()

for some c₂ > 0.

Our main result reads as follows.

Theorem 6. Suppose that g satisfies (23) and (24). Then, for each t₀ > 0, there exist two positive constants C₁ and β such that

()

Proof. From (38) and (43), we have

()

Integrating this over (t₀, t), we obtain

()

with β = c₂/α₂. Consequently, (34), (38), and (60) yield the result in Theorem 6.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2012R1A1A3011630).

References

1 Love A. E. H., A Treatise on the Mathematical Theory of Elasticity, 1944, Dover, New York, NY, USA, MR0010851.
Google Scholar
2 Cavalcanti M. M., Cavalcanti V. N. D., and Ferreira J., Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Mathematical Methods in the Applied Sciences. (2001) 24, no. 14, 1043–1053, https://doi.org/10.1002/mma.250, MR1855298.
10.1002/mma.250
Web of Science® Google Scholar
3 Han X. and Wang M., Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Analysis. (2009) 70, no. 9, 3090–3098, https://doi.org/10.1016/j.na.2008.04.011, MR2503053, ZBL1173.35579.
10.1016/j.na.2008.04.011
Web of Science® Google Scholar
4 Han X. and Wang M., General decay of energy for a viscoelastic equation with nonlinear damping, Mathematical Methods in the Applied Sciences. (2009) 32, no. 3, 346–358, https://doi.org/10.1002/mma.1041, MR2484178, ZBL1161.35319.
10.1002/mma.1041
Web of Science® Google Scholar
5 Liu W., Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Analysis. (2009) 71, no. 5-6, 2257–2267, https://doi.org/10.1016/j.na.2009.01.060, MR2524433, ZBL1167.35318.
10.1016/j.na.2009.01.060
Web of Science® Google Scholar
6 Messaoudi S. A. and Tatar N. E., Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Analysis. (2008) 68, no. 4, 785–793, https://doi.org/10.1016/j.na.2006.11.036, MR2382297, ZBL1136.35013.
10.1016/j.na.2006.11.036
Web of Science® Google Scholar
7 Park J. Y. and Kang J. R., Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Applicandae Mathematicae. (2010) 110, no. 3, 1393–1406, https://doi.org/10.1007/s10440-009-9516-3, MR2639177, ZBL1191.35061.
10.1007/s10440-009-9516-3
Web of Science® Google Scholar
8 Messaoudi S. A. and Tatar N. E., Exponential decay for a quasilinear viscoelastic equation, Mathematische Nachrichten. (2009) 282, no. 10, 1443–1450, https://doi.org/10.1002/mana.200610800, MR2571706, ZBL1179.35062.
10.1002/mana.200610800
Web of Science® Google Scholar
9 Cavalcanti M. M., Cavalcanti V. N. D., Filho J. S. P., and Soriano J. A., Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations. (2001) 14, no. 1, 85–116, MR1797934, ZBL1161.35437.
10.57262/die/1356123377
Google Scholar
10 Cavalcanti M. M., Cavalcanti V. N. D., and Soriano J. A., Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic Journal of Differential Equations. (2002) 44, 1–14, 2-s2.0-52449085317.
Google Scholar
11 Berrimi S. and Messaoudi S. A., Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis. (2006) 64, no. 10, 2314–2331, https://doi.org/10.1016/j.na.2005.08.015, MR2213903, ZBL1094.35070.
10.1016/j.na.2005.08.015
Web of Science® Google Scholar
12 Messaoudi S. A., General decay of solutions of a viscoelastic equation, Journal of Mathematical Analysis and Applications. (2008) 341, no. 2, 1457–1467, https://doi.org/10.1016/j.jmaa.2007.11.048, MR2398540, ZBL1145.35078.
10.1016/j.jmaa.2007.11.048
Web of Science® Google Scholar
13 Guesmia A. and Messaoudi S. A., General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Mathematical Methods in the Applied Sciences. (2009) 32, no. 16, 2102–2122, https://doi.org/10.1002/mma.1125, MR2571867, ZBL1183.35036.
10.1002/mma.1125
Web of Science® Google Scholar
14 Messaoudi S. A., General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis. (2008) 69, no. 8, 2589–2598, https://doi.org/10.1016/j.na.2007.08.035, MR2446355, ZBL1154.35066.
10.1016/j.na.2007.08.035
Web of Science® Google Scholar
15 Messaoudi S. A. and Mustafa M. I., On convexity for energy decay rates of a viscoelastic equation with boundary feedback, Nonlinear Analysis. (2010) 72, no. 9-10, 3602–3611, https://doi.org/10.1016/j.na.2009.12.040, MR2606805, ZBL1185.35025.
10.1016/j.na.2009.12.040
Web of Science® Google Scholar
16 Park J. Y. and Park S. H., Decay rate estimates for wave equations of memory type with acoustic boundary conditions, Nonlinear Analysis. (2011) 74, no. 3, 993–998, https://doi.org/10.1016/j.na.2010.09.057, MR2738648, ZBL1202.35032.
10.1016/j.na.2010.09.057
Web of Science® Google Scholar
17 Bisognin E., Bisognin V., Perla Menzala G. P., and Zuazua E., On exponential stability for von Kármán equations in the presence of thermal effects, Mathematical Methods in the Applied Sciences. (1998) 21, no. 5, 393–416, https://doi.org/10.1002/(SICI)1099-1476(19980325)21:5<393::AID-MMA958>3.3.CO;2-A, MR1608076, ZBL0911.35022.
10.1002/(SICI)1099-1476(19980325)21:5<393::AID-MMA958>3.0.CO;2-J
Web of Science® Google Scholar
18 Chueshov I. and Lasiecka I., Long time dynamics of von Karman evolutions with thermal effects, Boletim da Sociedade Paranaense de Matemática. (2007) 25, no. 1-2, 37–54, MR2485638, ZBL1171.35336.
Google Scholar
19 Bradley M. E. and Lasiecka I., Global decay rates for the solutions to a von Kármán plate without geometric conditions, Journal of Mathematical Analysis and Applications. (1994) 181, no. 1, 254–276, https://doi.org/10.1006/jmaa.1994.1019, MR1257968, ZBL0806.35180.
10.1006/jmaa.1994.1019
Web of Science® Google Scholar
20 Favini A., Horn M. A., Lasiecka I., and Tataru D., Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Differential and Integral Equations. (1996) 9, no. 2, 267–294, MR1364048, ZBL0847.35070.
10.57262/die/1367603346
Google Scholar
21 Horn M. A. and Lasiecka I., Uniform decay of weak solutions to a von Kármán plate with nonlinear boundary dissipation, Differential and Integral Equations. (1994) 7, no. 3-4, 885–908, MR1270110, ZBL0806.35181.
10.57262/die/1370267712
Google Scholar
22 Horn M. A. and Lasiecka I., Global stabilization of a dynamic von Kármán plate with nonlinear boundary feedback, Applied Mathematics and Optimization. (1995) 31, no. 1, 57–84, https://doi.org/10.1007/BF01182557, MR1296101, ZBL0823.93046.
10.1007/BF01182557
Web of Science® Google Scholar
23 Puel J. and Tucsnak M., Boundary stabilization for the von Kármán equations, SIAM Journal on Control and Optimization. (1996) 33, 255–273.
10.1137/S0363012992228350
Web of Science® Google Scholar
24 Rivera J. E. M. and Menzala G. P., Decay rates of solutions to a von Kármán system for viscoelastic plates with memory, Quarterly of Applied Mathematics. (1999) 82, no. 1, 181–200.
10.1090/qam/1672191
Google Scholar
25 Rivera J. E. M., Oquendo H. P., and Santos M. L., Asymptotic behavior to a von Kármán plate with boundary memory conditions, Nonlinear Analysis. (2005) 62, no. 7, 1183–1205, https://doi.org/10.1016/j.na.2005.04.025, MR2154103, ZBL1132.35085.
10.1016/j.na.2005.04.025
Web of Science® Google Scholar
26 Santos M. L. and Soufyane A., General decay to a von Kármán plate system with memory boundary conditions, Differential and Integral Equations. (2011) 24, no. 1-2, 69–81, MR2759352, ZBL1240.35548.
10.57262/die/1356019045
Web of Science® Google Scholar
27 Raposo C. A. and Santos M. L., General decay to a von Kármán system with memory, Nonlinear Analysis. (2011) 74, no. 3, 937–945, https://doi.org/10.1016/j.na.2010.09.047, MR2738643, ZBL1202.35033.
10.1016/j.na.2010.09.047
Web of Science® Google Scholar
28 Kang J. R., Energy decay rates for von Kármán system with memory and boundary feedback, Applied Mathematics and Computation. (2012) 218, no. 18, 9085–9094, https://doi.org/10.1016/j.amc.2012.02.053, MR2923007, ZBL1245.74041.
10.1016/j.amc.2012.02.053
Web of Science® Google Scholar
29 Kang J. R., Exponential decay for a von Kármán equations with memory, Journal of Mathematical Physics. (2013) 54, no. 3, 033501, https://doi.org/10.1063/1.4791694, MR3087361.
10.1063/1.4791694
Web of Science® Google Scholar
30 Lagnese J. E., Boundary Stabilization of Thin Plates, 1989, 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, Studies in Applied and Numerical Mathematics, https://doi.org/10.1137/1.9781611970821, MR1061153.
10.1137/1.9781611970821
Google Scholar
31 Chueshov I. and Lasiecka I., Global attractors for von Karman evolutions with a nonlinear boundary dissipation, Journal of Differential Equations. (2004) 198, no. 1, 196–231, https://doi.org/10.1016/j.jde.2003.08.008, MR2037755, ZBL1041.37038.
10.1016/j.jde.2003.08.008
Web of Science® Google Scholar

All articles

Exponential Decay for Nonlinear von Kármán Equations with Memory

Abstract

1. Introduction

2. Preliminaries

3. Exponential Decay of the Energy

Acknowledgment

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Exponential Decay for Nonlinear von Kármán Equations with Memory

Abstract

1. Introduction

2. Preliminaries

3. Exponential Decay of the Energy

Acknowledgment

References

References

Related

Information