General decay of solutions for a von Karman plate system with general type of relaxation functions on the boundary

1.   Introduction

The purpose of this work is to investigate the general decay of the solutions to the von Karman plate system with a memory condition on the boundary:

where $\Omega\subset {\mathbb R}^2$ is a bounded domain with smooth boundary $\Gamma$ and $x = (x_1, x_2).$ The constants $\alpha_1$ and $\alpha_2$ are positive. The von Karman bracket $[w, u]$ denotes the bilinear expression

Let us denote by $\nu = (\nu_1, \nu_2)$ the external unit normal vector on $\Gamma$ and by $\tau = (-\nu_2, \nu_1)$ the corresponding unit tangent vector. Denoting by the differential operators ${\mathcal A}_1$ and ${\mathcal A}_2$

where

and the constant $\lambda \in (0, \frac{1}{2})$ represents Poisson's ratio.

From the physical point of view, $w$ represents the transversal displacement, and $v$ is the Airy-stress function of the vibrating plate subjected to boundary viscoelastic damping. We see that the memory effect described in integral Eqs (1.4) and (1.5) can be caused by the interaction with another viscoelastic element. The problems (1.1)–(1.6) are interesting not only from the point of view of PDE general theory but also due to its applications in mechanics. This equation is used to predict the shape or deformation of the plate, for example, to understand and design the behavior of the plate in aircraft wings, the support structure of buildings or various mechanical systems. Furthermore, the von Karman plates with memory on the boundary is also widely used in communication and signal processing. In particular, understanding the energy decay that occurs when a signal passes through a boundary can contribute to improving the performance of the communication system.

Recently, some authors have applied the diffusion PDE model to study practical problems such as viral infection, cancer prevention and treatment and online game addiction, and they have achieved good results, see ^[1,2,3,4]. Fractional order differential equations (FODEs) have attracted considerable attention from researchers due to their applications in various scientific and engineering fields. Since most physical, technical and dynamical problems are influenced by certain boundary conditions, the study of boundary value problems is important. Shah et al. ^[5] established some sufficient conditions for the existence and uniqueness of solutions to impulsive FODEs with integral boundary condition. Recently, Shah et al. ^[6] investigated the existence and uniqueness of solutions to nonlinear impulsive FODEs under multipoint boundary conditions. Furthermore, fractional order delay differential equations (FODDEs) play crucial roles in modeling various physical and biological processes and phenomena. FODDEs have a wide range of applications in various fields, including electrodynamics, growth cells, quantum mechanics and astrophysics. Shah et al. ^[7] considered the existence and uniqueness of solutions to the boundary value problem of variable FODDEs.

In recent decades, many authors ^{[8,9,10,11,12,13]} have considered the mathematical analysis of Kirchhoff plates, investigating aspects such as global existence, uniqueness and stability under various boundary feedback conditions. Kang ^[14] studied the general decay of solutions to the Kirchhoff plates with a memory condition at the boundary. Mustafa and Abusharkh ^[15] established the general decay rate result for the plate equations with viscoelastic boundary damping. Recently, Al-Mahdi ^[16] proved the general and optimal decay rate result for the Kirchhoff plate equations with nonlinear damping. This result is a generalization of the work in ^[14,15,17].

On the other hand, the energy decay of the solutions for the von Karman system has been studied by many authors. In ^[18,19], the authors proved the stability of the solutions to a von Karman plate with nonlinear boundary feedback. Rivera and Menzala ^[20] showed the asymptotic behavior of the solution for the following von Karman plates with memory

where $h$ is a constant representing the thickness. Recently, Kang ^[21] investigated the general decay rates for the von Karman plate model (1.7) under the more general conditions

where $\zeta$ is a positive nonincreasing differentiable function and $G$ satisfies the suitable conditions. This result improved earlier results in ^[20,22,23]. Recently, Balegh et al. ^[24] established the general energy decay result for system (1.7) with nonlinear boundary delay term when $g$ satisfies condition (1.8). The general stability result of the viscoelastic equation, for relaxation function $g$ satisfying condition (1.8), has been investigated in ^[25,26,27].

For the case $\alpha_1 = \alpha_2 = 0$ in (1.1)–(1.6), Park and Park ^[28] studied the asymptotic behavior of the solutions, provided the resolvent kernels satisfy

for some positive constants $C_1$ and $C_2.$ Kang ^[29] considered the following generalized condition

where $H$ is a positive function, with $H(0) = H'(0) = 0,$ and $H$ is linear or it is strictly increasing and strictly convex on $(0, r],$ for some $0 < r < 1.$ The inequality in (1.10) has been introduced for the first time in ^[30]. These are weaker conditions on $H$ than those introduced in ^[30]. Thus, Kang ^[29] extended the decay result of ^[28]. Park ^[31], using the same assumption on the kernel in ^[29], obtained the general decay result of energy under $w_0 \neq 0$ on one part of the boundary. Recently, Feng and Soufyane ^[32] showed the general decay of the solution when the initial condition $w_0 = 0$ on one part of boundary and the resolvent kernels $k_i$ satisfy

where $\xi_i(t)$ are nonincreasing continuous functions and $G_i$ are positive functions, with $G_i(0) = G_i'(0) = 0,$ and the $G_i$ are linear or strictly increasing and strictly convex on $(0, r], \; r \geq -k_i'(0).$

For problems (1.1)–(1.6), Rivera et al. ^[33] proved that the solution decays exponentially, provided the resolvent kernels satisfy condition (1.9). Moreover, when the relaxation functions decay polynomially, they showed that the solution decays polynomially. Santos and Soufyane ^[34] improved the decay result of ^[33]. They assumed that the resolvent kernels satisfy

where $\eta_i : {\mathbb R}^+ \rightarrow {\mathbb R}^+$ are functions satisfying conditions

Motivated by their results, we prove the asymptotic behavior of the solution for system (1.1)–(1.6) when the initial condition $w_0 \neq 0$ on $\Gamma$ and the resolvent kernels $k_i$ satisfy condition (1.11). This condition is more general compared to the previous conditions (1.10) and (1.12). Therefore, these general decay estimates improve the earlier results of ^{[29,31,33,34]}. Moreover, using the multiplier method and some properties of convex functions, we obtain the general decay of solution for system (1.1)–(1.6) that depends on the functions $\xi_i$ and $G_i$.

The paper is organized as follows. In Section 2, we present some notations and assumptions needed for our work. In Section 3, we prove the general decay of the solutions for the von Karman plate system with memory condition on the boundary.

2.   Preliminaries

In this section, we present some material needed in the proof of our main result. Throughout this paper we denote $||\cdot||_{L^2(\Omega)}$ and $||\cdot||_{L^2 (\Gamma)}$ by $||\cdot||$ and $||\cdot||_{\Gamma},$ respectively. Let us define the bilinear form

We assume that there exists $x_0 \in {\mathbb R}^2$, such that

where $m(x) = x - x_0.$ The compactness of $\Gamma$ implies that there exists $\delta > 0$, such that

As shown in ^[33,34], we use the boundary conditions (1.4) and (1.5) to estimate the terms ${\mathcal A}_1 w$ and ${\mathcal A}_2 w.$ Differentiating (1.4) and (1.5) and applying Volterra's inverse operator, we have

where $\gamma_i = \frac{1}{h_i(0)}, (i = 1, 2)$ and the resolvent kernels $k_i, (i = 1, 2)$ satisfy

where $*$ is the convolution product. Thus, we use boundary conditions (2.2) and (2.3) instead of (1.4) and (1.5).

The following identity will be used later.

Lemma 2.1. (^[35]) For any $w\in H^4 (\Omega)$ and $u \in H^2 (\Omega),$ we have

We state the relative results of the Airy stress function and von Karman bracket $[\cdot, \cdot].$

Lemma 2.2. (^[8]) Let $w, u$ be functions in $H^2 (\Omega)$ and $v$ in $H^2_0(\Omega),$ where $\Omega$ is an open, bounded and connected set of ${\mathbb R}^2$ with regular boundary. Then,

By differentiating the term $h \square w,$ we obtain the following lemma.

Lemma 2.3. For $h, w \in C^1 ([0, \infty): {\mathbb R}),$ we have

where $(h \square w)(t) : = \int_0^t h(t-s) | w(t) -w (s) |^2 ds.$

As in ^[32,36], we consider the following assumptions on $k_i\ (i = 1, 2).$

(A) The resolvent kernels $k_i : {\mathbb R}_+ \rightarrow {\mathbb R}_+$ are twice differentiable functions, such that

and there exist positive functions $G_i \in C^1 ({\mathbb R}_+)$ and the $G_i$ are linear or strictly increasing and strictly convex $C^2$ functions on $(0, r], \ r < 1,$ with $G_i(0) = G_i'(0) = 0,$ such that

where $\xi_i : {\mathbb R}^+ \rightarrow {\mathbb R}^+$ are nonincreasing differentiable functions.

From (A), we easily see that there exists $t_0 > 0$ large enough such that

and

where $G = \min\{G_1, G_2\}.$

As $\xi_i(t)$ and $-k_i'(t)$ are positive nonincreasing continuous functions and $G_i(t)$ is a positive continuous function, there exist positive constants $a_i$ and $b_i$ such that

Therefore, for all $t\in [0, t_0],$ we obtain

where $c_i = -\frac{a_i }{k_i'(0)}.$

The well-posedness of von Karman system plates with boundary conditions of memory type is given by the following theorem.

Theorem 2.1. (^[33]) Let $k_i (i = 1, 2) \in C^2 ({\mathbb R}_+)$ be such that $k_i, \ - k_i', \ k_i'' \geq 0.$ If the initial conditions $(w_0, w_1)\in (H^4 (\Omega)\cap H^2 (\Omega)) \times H^2 (\Omega)$ satisfy the compatibility conditions

then the solution of (1.1)–(1.6) has the following regularity

The energy function of system (1.1)–(1.6) is given by

To get a general stability result, the following is needed.

Remark 2.1. 1) If $G_i (i = 1, 2)$ are strictly convex on $(0, r]$ and $G_i(0) = 0,$ then,

2) Let ${G}^*$ be the convex conjugate of ${ G}$ in the sense of Young ^[37], then

and ${ G}^*$ satisfies the following Young's inequality

3) Let $F$ be a convex function on $[c, d],$ and if $\ \varrho :\Omega \rightarrow [c, d]$ and $p$ are integrable functions on $\Omega$ such that $p(x)\geq 0$ and $\int_\Omega p (x) d x = p_0 > 0,$ then Jensen's inequality states that

3.   General decay

In this section, we study the asymptotic behavior of the solutions for system (1.1)–(1.6). To show the general decay property, we first prove the dissipative property. Multiplying (1.1) by $w_t$ and using (2.4), (2.7), Young's inequality and the boundary conditions (2.2) and (2.3), we obtain the following.

Lemma 3.1. (^[33]) The energy function $E(t)$ satisfies

Since $w_0\neq 0$ on $\Gamma,$ Lemma 3.1 says that $E(t)$ may not be nonincreasing. So, we introduce the modified energy functional $\mathcal E (t)$ by

Then, from (3.1), we have

For suitable choice of $N_1$ and $N_2,$ let us introduce the Lyapunov functional

where

It is not difficult to see that $L(t)$ satisfies $q_0 E(t) \leq L(t) \leq q_1 E(t),$ for some positive constants $q_0$ and $q_1.$

Lemma 3.2. Under the assumption (A), the functional $\Upsilon(t)$ satisfies

for any $0 < \delta_i < 1\ (i = 1, 2),$ where

Proof. According to ^[33,34], from (2.5) and (2.6), we obtain

Applying Young's inequality, we get

where $\epsilon$ is a positive constant. Using the trace theory and the fact $m\cdot \nu \geq \delta$ on $\Gamma,$ we obtain

where $\lambda_0$ is a positive constant. Noting that

the boundary condition (2.3) can be written as

Similarly, we can show that

Using (3.6)–(3.11), we arrive at

Using the Cauchy-Schwarz inequality and (3.5), we have (see details in ^[25,27])

and

Substituting (3.13) and (3.14) into (3.12), we have (3.4). □

Next, we define the functionals

where $f_i(t) = \int_t^\infty (-k_i'(s)) ds, \ i = 1, 2.$

Lemma 3.3. Under the assumption (A), the functionals $K_1(t)$ and $K_2(t)$ satisfy the estimates

Proof. Taking the derivative of the functional $K_2(t)$ and using the fact $f_2'(t) = k_2'(t),$ we find that

Using Young's inequality and (2.8), we obtain

From (3.17) and (3.18), we get the estimate (3.16). Similarly, we can obtain the estimate (3.15). □

Lemma 3.4. Suppose that the assumption (A) holds. Then, for $N_1, N_2 > 0$ large enough, there exist positive constants $\beta_1$ and $\beta_2$, such that

where $t_0$ was introduced in (2.10).

Proof. Combining (3.1), (3.4) and (3.5), we see that

where $R = \max\{m(x)\cdot \nu(x): x \in \Gamma \}.$ We first fix $\epsilon > 0$ small such that

Next, we apply the fact ${\lim_{t \to \infty}k_i(t) = 0 \ (i = 1, 2)}$ and choose $N_2$ large enough so that

for all $t > t_0.$ From (2.8), (2.9) and (3.5), we have

Integrating (3.20) and using (2.8), we obtain

By the Lebesgue dominated convergence theorem, we find that $\delta_i C(\delta_i) \ \rightarrow 0 \ $ as $\delta_i \rightarrow 0.$ Then, there exists $0 < \delta_0 < 1$ such that, if $\delta_i < \delta_0,$ then $4\delta_i\gamma_i C_\epsilon C(\delta_i)N_2 < \frac{1}{8} \ (i = 1, 2).$ Finally, taking $N_1$ large enough so that

and choosing $\delta_i = \frac{1}{2N_1} < \delta_0 \ (i = 1, 2),$ we have the estimate (3.19). □

Now, we are ready to prove our main result.

Theorem 3.1. Suppose that the assumption (A) holds. Then there exist positive constants $\epsilon_0, \sigma_1, \sigma_2, \kappa_1$ and $\kappa_2$ such that the energy functional satisfies, for all $t\geq t_0,$

where $H_1(t) = tG'(\epsilon_0 t), \; G(t) = \min\{ G_1(t), G_2(t) \}$ and $\xi(t) = \min\{ \xi_1(t), \xi_2(t)\}.$

Proof. From (2.13) and (3.19), there exist positive constants $\beta_3$ and $\beta_4$ such that, for $t\geq t_0$

Applying (2.12) and (3.1), we see that, for all $t \geq t_0$

where $c_0 = \min\{ c_1, c_2\}$ and $\gamma_0 = \min\{ \gamma_1, \gamma_2\}.$

Let $\Phi(t) = L(t) + \frac{2\beta_4}{c_0 \gamma_0} E(t),$ which is equivalent to $E(t).$ Using (3.23) and (3.24), we obtain for all $t\geq t_0$

where $\beta_5 = \max\{ \beta_2+ \frac{\beta_4\gamma_1}{c_0 \gamma_0}, \ \beta_2+ \frac{\beta_4\gamma_2}{c_0 \gamma_0} \}.$

We consider the following two cases.

1) $G$ is linear: Multiplying (3.25) by the nonincreasing function $\xi(t)$ and using (2.9) and (3.1), we have

where $\beta_6 = \max \{\beta_5 \xi_0 +\frac{\beta_4\gamma_1}{\gamma_0}, \ \beta_5 \xi_0 +\frac{\beta_4\gamma_2}{\gamma_0} \}$ and $\xi(t)\leq \xi_0,$ for some $\xi_0 > 0.$ This gives

where $\xi(t)$ is a nonincreasing function. Hence, using the fact that $I(t) = \xi(t)\Phi(t) + \frac{2\beta_4}{\gamma_0} E(t) \sim E(t),$ we deduce that

where $\beta_7$ is a positive constant. We introduce

From (3.26), we have

Integrating this over $(t_0, t),$ we obtain

Using the fact that $I(t) \sim E(t)$ and (3.27), we get the estimate (3.21).

2) $G$ is nonlinear: First, we construct the functional

which is nonnegative. From (2.13), (3.15), (3.16) and (3.19), we obtain

where $\rho_0$ is some positive constant. Integrating this over $(t_0, t),$ we arrive at

Therefore, from (2.8), we conclude that

Then, we define $\zeta_1(t)$ and $\zeta_2(t)$ by, for constants $\theta_1$ and $\theta_2 \in (0, 1),$

Using (2.9), (2.14), (2.17) and the fact that $\xi_1(t)$ is a positive nonincreasing function, we find that

where $\xi(t) = \min\{\xi_1(t), \xi_2(t)\}.$

Similarly, we can prove that

Combining (2.8), (3.2), (3.25), (3.28) and (3.29), we see that, for all $t\geq t_0,$

where $\beta_8 = \max\{ \beta_5 k_1(0) + \frac{\beta_3\gamma_1}{2}\int_0^\infty k_1(s)ds, \ \beta_5 k_2(0) + \frac{\beta_3\gamma_2}{2}\int_0^\infty k_2(s)ds \}.$ Now, for $\epsilon_0 < r,$ we define the functional

where ${\mathcal E}(t)$ is the modified energy given in (3.2). Using (2.11), (2.15), (2.16), (3.1), (3.3), (3.30) and the fact that ${\mathcal E}'\leq 0, \ G' > 0$ and $G'' > 0,$ we obtain for all $t\geq t_0$

where $\theta_0 = \min\{ \theta_1, \ \theta_2\}$ and $\gamma_0 = \min\{ \gamma_1, \ \gamma_2\}.$ Choosing $\epsilon_0$ such that $\rho_1 = \beta_3 {\mathcal E }(0)-\big(\beta_8\big|\big|\frac{\partial w_0}{\partial \nu}\big|\big|^2_\Gamma +\beta_8 ||w_0 ||^2_\Gamma +\frac{2\beta_4}{\theta_0} \big)\epsilon_0 > 0,$ we have

Then, multiplying this by $\xi(t),$ we get

Taking ${\mathcal F}(t) = \xi(t) R(t) + \frac{2\beta_4}{\theta_0\gamma_0}{\mathcal E}(t)$ and using (3.31) and $\xi'\leq 0,$ we arrive at

where $H_1(t) = tG'(\epsilon_0 t).$ Applying (3.32) and the fact that $\xi'\leq 0, \ {\mathcal E}'\leq 0$ and $H_1' \geq 0,$ we find that

Integrating this over ($t_0, t),$ we see that

where $\rho_2 = \max\{ t_0 \xi(t_0) H_1 \big(\frac{{\mathcal E}(t_0)}{{\mathcal E }(0)}\big)+\frac{1}{\rho_1} {\mathcal F}(t_0), \ \frac{\beta_8}{\rho_1}\}.$ Therefore, we conclude that

Hence, applying (3.2), (3.22) is established. □

Examples. We provide examples to explain the decay of energy ^[32,36].

1) For $k_1'(t) = k_2'(t) = -e^{-t^q}$ with $0 < q < 1,$ we obtain $k_i''(t) = G_i(-k_i'(t)) (i = 1, 2)$, where $G_1(t) = G_2 (t) = \frac{qt}{[\ln(\frac{1}{t})]^{\frac{1}{q}-1}}.$ Since

the functions $G_1$ and $G_2$ satisfy the condition (2.9) on $(0, r]$ for any $0 < r < 1.$

2) Let $k_i'(t) = -\frac{a_i}{(1+t)^2},$ where $a_i > 0, (i = 1, 2),$ be chosen so that assumption (A) holds. We choose $a = \min\{a_1, a_2 \},$ then $k_i''(t) = b_i G_i(-k_i'(t)).$ We select $b = \min\{b_1, b_2\}, \ G = \min\{ G_1, G_2 \}$ and $\xi(t) = \min\{ \xi_1 (t), \xi_2 (t) \},$ then $G(t) = t^{\frac{3}{2}}, \xi(t) = b.$

4.   Conclusions

The von Karman plates with memory on the boundary is also widely used in communication and signal processing. In particular, understanding the energy decay that occurs when a signal passes through a boundary can contribute to improving the performance of the communication system. This type of equation plays an important role in explaining various physical phenomena in the real world. In this paper, we study the von Karman plate system with general type of relaxation functions on the boundary. Here, we consider the resolvent kernels $k_i(i = 1, 2)$, namely $k_i''(t) \geq - \xi_i(t) G_i(-k_i'(t))$, where $G_i$ are convex and increasing functions near the origin and $\xi_i$ are positive nonincreasing functions. Using some properties of convex functions without the assumption that initial value $w_0 \equiv 0$ on the boundary, we prove the general decay rate result. These general decay estimates improve earlier results in the literature.

Use of AI tools declaration

The author declares she has not used Artificial Intelligence(AI) tools in the creation of this article.

Acknowledgments

The author would like to thank the referee for a careful reading of the manuscript and valuable comments. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1I1A3042239).

Conflict of interest

The author declares that she has no conflict of interest.