On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

1.   Introduction

A coupled system of two Kirchhoff plate equations is considered:

where $\Omega$ is a bounded domain of $\mathbb{R}^2$ with a smooth boundary $\Gamma = \partial \Omega = \Gamma_0 \cup \Gamma_1$, such that $\overline{\Gamma}_{0}\cap\overline{\Gamma}_{1} = \emptyset$, the initial data $y_{0}$, $y_{1}$, $z_{0}$ and $z_{1}$ lie in appropriate Hilbert space.

The symbols $y_t$ and $y_{tt}$ refer, respectively, to first order and second order derivatives (with respect to $t$) of $y$, while $\Delta$ and $\Delta^2$ are the Laplacian and Bilaplacian operators. The functions $h_i$ and $f_i$ (for $i = 1, 2$) verify some assumptions that will be given in the next section. $\rho$ is a positive constant, ${{\boldsymbol{x}}} = (x_1, x_2)$ is the space variable, and the operators ${\bf{B}}_{1}$ and ${\bf{B}}_{2}$ are defined by

and

where the constant $0 < \mu < \frac{1}{2}$ is the Poisson coefficient. Here, $\partial_{{{{\boldsymbol{\nu}}}}}$ stands for normal derivative, ${{{{\boldsymbol{\nu}}}}} = (\nu_{1}, \nu_{2})$ is the unit outer normal vector to $\Gamma$ and ${{\boldsymbol{\tau}}} = (-\nu_{2}, \nu_{1})$ is a unit tangent vector.

Model (1.1) describes the interaction of two viscoelastic Kirchhoff plates with rotational forces, which possess a rigid surface and whose interiors are somehow permissive to slight deformations, such that the material densities vary according to the velocity ^[1]. Each one of these two plates is clamped along $\Gamma_0$, and without bending and twisting moments on $\Gamma_1$. The analysis of stability issues for plate models is more challenging due to free boundary conditions and the presence of rotational forces, etc. ^[2]. Moreover, in our case the source term competes with the dissipation induced by the viscoelastic term only. Therefore, it will be interesting to study this interaction ^[3].

We start off by reviewing some works related to quasi-linear wave equation and plate equation. Cavalcanti et al. ^[1] considered the following equation

and proved the global existence of weak solutions and a uniform decay rates of the energy in the presence of a strong damping, of the form $-\gamma \Delta u_t$ acting in the domain and assuming that the relaxation function decays exponentially. Messaoudi and Tatar ^[3] studied (1.2) but without a strong damping ($\gamma = 0$). They showed that the memory term is enough to stabilize the solution. The global existence and uniform decay for solutions of (1.2), provided that the initial data are in some stable set, are obtained in ^[4] with the presence of a source term and with $\gamma = 0$. Later, in ^[5], for $\gamma = 0$, the authors investigated the general decay result of the energy of (1.2) with nonlinear damping. In ^[6], the author investigated (1.2) with weakly nonlinear time-dependent dissipation and source terms, and he established an explicit and general energy decay rate results without imposing any restrictive growth assumption on the damping term at the origin. For other related results for quasi-linear wave equations, we refer the reader to ^[7,8,9,10]. For quasi-linear plate equations, we mention the work of Al-Gharabli et al. ^[11] where the authors studied the well-posedness and asymptotic stability for a quasi-linear viscoelastic plate equation with a logarithmic nonlinearity. Recently, Al-Mahdi ^[12] studied the same problem as in Al-Gharabli et al. ^[11], but with infinite memory. With the imposition of a minimal condition on the relaxation function, he obtained an explicit and general decay rate result for the energy. Very recently, in ^[13], the authors considered a plate equation with infinite memory, nonlinear damping, and logarithmic source. They proved explicit and general decay rate of the solution.

The stability of coupled quasi-linear systems has been discussed by many authors. Liu ^[14] considered two coupled quasi-linear viscoelastic wave equations. He showed that the viscoelastic terms' dissipations guarantee that the solutions decay exponentially and polynomially. Later on, with more general relaxation functions and specific initial data, He ^[15] extended the result of Liu ^[14]. Recently, Mustafa and Kafini ^[16] considered the same problem and improved earlier results for a wider class of relaxation functions. In ^[17], the authors studied the same problem, but with nonlinear damping, and showed a general decay rate estimates of energy of solutions. Very recently, Pişkin and Ekinci ^[18] generalized and improved earlier results by considering a degenerate damping. Finally, let's mention the recent works of Fang et al.^[19] and Zhu et al.^[20] that relate to our problem.

As I know, there is no work regarding quasi-linear plate equations. This paper seems to be the first that deals with this problem.

The structure of this paper is shown as follows: In Section 2, we present some presumptions that are necessary for the proof of essential results. The third section provides the proof of well-posedness of our system. The general energy decay result is stated and established in Section 4. The fifth section provides two examples that illustrate explicit formulas for the energy decay rates. A concluding section is given at the end.

2.   Preliminaries

This part is devoted to give some necessary materials and assumptions for the proof of our key results. We define

and

Denoting $dx = dx_1 dx_2$, we define the bilinear form $b: V\times V \rightarrow \mathbb{R}$ by:

Firstly, we must recall Green's formula (see ^[2]):

and a weaker version of it (see Theorem 5.6 in ^[21]) in the following form:

We need the following lemma.

Lemma 2.1. (^[22]) For any $y\in C^1(0, T; H^2(\Omega))$, we get

where

In this paper, we suppose that:

(A1): The two non-increasing $C^1$ functions $h_i: [0, +\infty)\to (0, +\infty)$ (for $i = 1, 2$) such that

(A2): There are a positive $C^1$ functions $G_i:(0, +\infty)\to (0, +\infty)$, that are linear or strictly increasing and strictly convex $C^2$ on $(0, r], (r < 1)$, with $G_i(0) = G_i^{\prime}(0) = 0$, satisfying for all $t\geq 0$

where $\xi_1$ and $\xi_2$ are positive non-increasing differentiable functions.

(A3): $f_i:\mathbb{R}^2\to\mathbb{R}$ (for $i = 1, 2$) are $C^1$ functions and there exists a positive function $F$, such that

and

for some constant $d > 0$ and $\beta_{ij}\geq 1$ for $i, j = 1, 2.$

Remark 2.1. 1.The condition ${\bf (A1)}$ guarantees the hyperbolicity of the first two equations in the system (1.1).

2. By (2.6) and the mean value theorem, we have for some positive constant $d_1$

and

for all $(x_1, x_2), (u_1, u_2)\in \mathbb{R}^2$ and $i = 1, 2$.

The energy functional is defined by

Here,

represent, respectively, the kinetic and the elastic potential energy of the model.

We have the following dissipation identity:

Proposition 2.1.

Proof. Multiplying $(1.1)_1$ by $y_t$ and $(1.1)_2$ by $z_t$, summing the resultant equations and integrating over $\Omega$ to get

Inserting (2.3) in (2.11), we get the desired result.

Throughout this paper, $c$ denotes a generic positive constant, and not necessarily the same at different occurrences.

3.   Global existence

We begin this part by defining a weak solution of the system (1.1).

Definition 3.1. A couple of functions $(y, z)$ defined on $[0, T]$ is a weak solution of the problem (1.1) if $y\in C([0, T], V)\cap C^1([0, T], W), \; z\in C([0, T], V)\cap C^1([0, T], W),$ and satisfies

and

for a.e. $t\in [0, T]$ and all test functions $u, v\in V$.

Theorem 3.1. Let $(y_0, y_1), (z_0, z_1)\in V\times W$. Assume that assumptions (A1)–(A3) are true. Then, the system (1.1) has at least a local weak solution. Moreover, this solution is global and bounded.

Proof. With the help of the Faedo-Galerkin approach, the existence is demonstrated. In order to achieve this, let $\{w_j\}_{j = 1}^{\infty}$ be a basis of $V$. Define $E_m = {\text span}\{w_1, w_2, ..., w_m\}$. On the finite dimensional subspaces $E_m$, the initial data are projected as follows:

such that

Considering the following solution

which satisfies the following approximate problem in $E_m$:

This leads to a system of ordinary differential equations (ODEs) for unknown functions $p_k$ and $q_k$. Hence, from the standard theory of system of ODEs, a solution $(y^m, z^m)$ of (3.2) exists, for all $m\geq 1$, on $[0, t_m)$, with $0 < t_m\leq T, \; \forall\; m\geq 1$.

A priori estimate 1: Let $w = y^m_t$ in $(3.2)_1$ and $w = z^m_t$ in $(3.2)_2$. Combining the resultant equations and integrating on $\Omega$ to obtain

where

Noting, by (3.1), that

Then, by integrating (3.3) over $(0, t), 0 < t < t_m$, we get a constant $M_1 > 0$ that doesn't depend on $t$ and $m$, satisfying

Hence, $t_m$ can be replaced by some $T > 0$, for all $m\geq1$.

A priori estimate 2: Let $w = y^m_{tt}$ in $(3.2)_1$ and $w = z^m_{tt}$ in $(3.2)_2$, adding the resultant equations, integrating on $\Omega$, and using Young's inequality to obtain for all $\eta > 0$

Using Hölder's inequality, Sobolev's embedding, (2.7) and (3.4), one has for some $M_2 > 0$,

Similarly, we obtain that

From (3.5)–(3.7), we infer that

Integrating (3.8) on $(0, T)$, and using (3.4) gives us

Choosing $\eta$ small enough, such that

and so that

Consequently, (3.9) becomes

Then, we have

for some constant $M_3 > 0$.

From (3.4) and (3.10), we conclude that

and

Hence, we can extract subsequence of $(y^m)$ and $(z^m)$, still denoted by $(y^m)$ and $(z^m)$ respectively, such that

and

Analysis of the non-linear terms:

1. Term $f_i(y^m, z^m)$: We have that $(y^m)$ and $(z^m)$ are bounded in $L^{\infty}(0, T; V)$. This shows, by the use of the embedding of $V\subset L^{\infty}(\Omega) (\Omega \subset \mathbb{R}^2)$, the boundedness of $(y^m)$ and $(z^m)$ in $L^2(\Omega\times (0, T))$. Likewise, $(y^m_t)$ and $(z^m_t)$ are bounded in $L^2(\Omega\times (0, T))$. Hence, by the use of the Aubin-Lions Theorem, we get, up to a subsequence, that

Then,

and, therefore, from ${\bf{(A3)}}$,

On the other hand, we have $(y^m)$ and $(z^m)$ that are bounded in $L^{\infty}(0, T; L^2(\Omega))$, then, by using (2.7) and (3.4), we get that $f_i(y^m, z^m)$ is bounded in $L^{\infty}(0, T; L^2(\Omega))$. This fact and (3.17) leads to

2. Terms $\vert y^m_t\vert^{\rho}y^m_t$ and $\vert z^m_t\vert^{\rho}z^m_t$: We recall that $(y_t^m)$ and $(z_t^m)$ are bounded in $L^{\infty}(0, T; W)$, which gives that $(y_t^m)$ and $(z_t^m)$ are bounded in $L^{\infty}(\Omega \times (0, T))$, and so in $L^{2}(\Omega \times (0, T))$. By the same, we deduce that $(y_{tt}^m)$ and $(z_{tt}^m)$ are bounded in $L^{2}(\Omega \times (0, T))$. Now, using Aubin-Lions theorem, we conclude, up to a subsequence, that

and

Using (3.4), we see that

and similarly

where $C_*$ is a positive constant satisfying $\Vert u\Vert_2 \leq C_* \Vert {{\boldsymbol{\nabla u}}}\Vert_2$, for all $u\in W$.

Then, the sequences $(\vert y^m_t\vert^{\rho}y^m_t)$ and $(\vert z^m_t\vert^{\rho}z^m_t)$ are bounded in $L^2(\Omega\times (0, T))$. Combining (3.18), (3.19) and (3.20) and using Lion's lemma ^[23], one derives

Next, by integrating (3.2) on $(0, t)$, one obtains

Letting $m\to+\infty$, the aforementioned convergence results give that

for all $w\in V$.

Since the terms in the right hand side of $(3.23)_1$ and $(3.23)_2$ are absolutely continuous, then (3.23) is differentiable for a.e. $t\geq0$, and, therefore, one has for all $w\in V$

Regarding the initial conditions, we recall that

Consequently, the use of Lion's Lemma ^[23] leads to

Hence, $y^m(x, 0)$ and $z^m(x, 0)$ make sense and $y^m(x, 0)\to y(x, 0)$, $z^m(x, 0)\to z(x, 0)$ in $L^2(\Omega)$. Recalling that

we obtain that

Besides, multiplying (3.2) by $\phi\in C_{0}^{\infty}(0, T)$ ^[24] and integrating on $(0, T)$, to get

and

As $m\to +\infty$, we have for any $w\in V$ and any $\phi\in C_{0}^{\infty}(0, T)$

and

This means that (see ^[24])

Since $y_t$ and $z_t \in L^2(0, T; L^2(\Omega))$, we deduce that $y_t$ and $z_t \in C(0, T; V)$.

So, $y^m_t(x, 0)$ and $z^m_t(x, 0)$ make sense and

But

Hence,

Consequently, the proof of local existence of weak solutions is complete. Besides, it is easy to see that

which gives the globalness and boundedness of the solution of problem (1.1).

4.   General decay

We denote by $\xi(t) = \min\{\xi_1(t), \xi_2(t)\}, \; h(t) = \max\{h_1(t), h_2(t)\}$ and $G(t) = \min\{G_1(t), G_2(t)\}$.

Theorem 4.1. Let $(u_0, u_1), (v_0, v_1)\in V\times W$. Suppose that (A1)–(A3) hold. Thus, the energy $E(t)$ satisfies

for some positive constants $\beta_1$ and $\beta_2$.

Remark 4.1. (^[16])

1. We recall the Jensen's inequality: Assume $F$ is a concave function on $[a, b], \; f:\Omega \to [a, b]$ and $g$ are in $L^1(\Omega)$, with $g(x)\geq 0$ and $\int_\Omega g(x)\; dx = m > 0$, then

2. From (A2), one has $\lim_{t\rightarrow +\infty}h_i(t) = 0$. Hence, $\exists\; t_1\geq0$ is large enough, verifying

One can easily check, for $i = 1, 2$, that

for some constants $a_i > 0$ and $b_i > 0$. This implies that

Proof of Theorem (4.1): The proof is divided into three steps.

Step 1: In this step, we give estimates for the derivatives (with respect to $t$) of the functionals $\varphi(t)$ and $\psi(t)$ defined below by:

with

and

with

Lemma 4.1. If (A1)–(A3) hold. The functional $\varphi(t)$ defined in (4.4) verifies, along the solution of (1.1),

Proof. We have $\varphi'(t) = \varphi^{'}_1(t)+\varphi^{'}_2(t)$. By using (1.1), we obtain

Since $\int_0^t h_1(s)ds\leq \int_0^{+\infty} h_1(s)ds = 1-l_1$, then, by the use of Cauchy-Schwarz's inequality and Young's inequality, we derive

Inserting (4.9) in (4.8), we get that

Similarly, we infer that

Summing the last two inequalities, we get the desired inequality (4.7).

Lemma 4.2. If (A1)–(A3) hold. The functional defined in (4.6) verifies, for any $0 < \delta < 1$ and for all $t\geq t_1$, along the solution of (1.1),

Here $h_0 = \min\Big\{ \int_0^t h_1(s) ds, \int_0^t h_2(s) ds\Big\}$.

Proof. Differentiating $\psi_1(t)$ with respect to $t$ and using $(1.1)_1$, we get

Now, we estimate the terms in the right-hand side of (4.11) as follows:

$\bullet$ Estimation of the term $\int_0^t h_1(t-s) b\left(y, y(t)-y(s)\right)ds.$

Cauchy Schwarz's inequality and Young's inequality are used to get, for any $\delta > 0$,

$\bullet$ Estimation of the term $- \int_0^t h_1(t-\zeta) \int_0^t h_1(t-s)b\left(y(s), y(t)-y(\zeta)\right)ds\; d\zeta.$

We have

$\bullet$ Estimation of the term $- \int_{\Omega} {{\boldsymbol{\nabla y_t}}} \int_0^t h'_1(t-s)(y(t)-y(s))ds dx.$

One has

$\bullet$ Estimation of the term $\int_\Omega f_1(y, z) \int_0^t h_1(t-s)(y(t)-y(s))ds\; dx.$

By using (2.7) and (3.27), we derive that

$\bullet$ Estimation of the term $-\frac{1}{\rho+1} \int_{\Omega} \vert y_t\vert^{\rho}y_t \int_0^t h'_1(t-s)(y(t)-y(s))ds\; dx.$

Using (3.27) again, we infer that

By combining (4.12)–(4.16), using the fact that $- \left(\int_0^t h_1(s)ds\right)\leq -h_0$ for all $t\geq t_1$ and choosing $\delta_1$ small enough, we derive the estimate (4.10).

Repeating the calculations above with $\psi_2(t)$ yields

Combining (4.10) and (4.17), we obtain the following result.

Corollary 4.1. Assume that (A1)–(A3) hold. Then, the functional $\psi$ satisfies, along the solution, the estimate

for any $0 < \delta < 1$.

Step 2: The aim of this step is to establish the inequality (4.26).

Let's define the functional

where $N > 0$. For $N$ sufficiently large, one has that $F\sim E$, i. e.

for some $c_1, c_2 > 0.$

Let $l = \min\left\{l_1, l_2\right\}$. By using (2.10), (4.7), (4.18), and taking $\delta = \frac{lh_0}{16c}$, we get for any $t\geq t_1$

Taking $N$, such that

to obtain that

By the virtue of (2.10) and (4.3), we infer that for any $t\geq t_1$,

and similarly

Hence, (4.21) becomes

where $\alpha > 0$. Define $\mathcal{H}(t) = F(t)+cE(t)$. It is easy to see that $\mathcal{H}(t)\sim E(t)$. Using (4.22), we get

The following two situations are then distinguished.

First Case: $G_1(t)$ and $G_2(t)$ are linear.

By multiplying (4.23) by $\xi(t)$ and using (A2) and (2.10) to obtain

Since $\xi$ is non-increasing, then by using (4.24), the functional $\mathcal {F}(t) = \xi(t)\mathcal{H}(t)+cE(t)$ satisfies for any $t\geq t_1$,

It is obvious that $\mathcal{F} \sim E$, and then we get the existence of some positive constant $m_1$, such that

By applying Gronwall's Lemma, there exists a constant $m_2 > 0$, such that

and then we have

where $m_3 > 0$.

Second Case: $G_1(t)$ or $G_2(t)$ is nonlinear. Defining $J_1$ and $J_2$ by

and

Since $b(y(t), y(t))+b(y(t-s), y(t-s))\leq \frac{2}{l}\left(E(t)+E(t-s)\right)\leq \frac{4}{l}E(0)$, for all $0 < s < t$, we infer that

and similarly

By taking $0 < \lambda < 1$ sufficiently small, we get, for all $t > 0$,

Now, defining $K_1(t)$ and $K_2(t)$ by

and

One can easily check that $K_i(t)\leq-c E'(t)$, for $i = 1, 2$.

Given that $G_1(0) = 0$ and the strict convexity of $G_1$ on $(0, r]$, one has then $G_1(\kappa x)\leq \kappa G_1(x), \; \forall\; 0\leq\kappa\leq 1$ and $x\in(0, r]$. Now, using (A1), (4.25) and Jensen's inequality, we obtain

Note that $\overline{G}_1$ is an extension of $G_1$, satisfying $\overline{G}_1$ as strictly convex and strictly increasing on $(0, +\infty)$. Thus, we have

Similarly, we have

where $\overline{G}_2$ is an extension of $G_2$.

We infer from (4.23) that

Step 3: Here, we shall prove the desired inequality (4.1).

We set $G = \min\{\overline{G}_1, \overline{G}_2\}$. For $\varepsilon_0 < r$, using (4.26) and since $E'\leq0$, $\overline{G}_i^{'} > 0$, $\overline{G}_i^{''} > 0$, $i = 1, 2$, we claim that the functional ${\mathcal{G}}$, defined by

is equivalent to $E(t)$ and satisfies

The convex conjugate of $G$ in the Young's sense (see ^[25]) is denoted by $G^{*}$ and satisfies

The following inequality holds true:

with $A = G'\left(\varepsilon_0\frac{E(t)}{E(0)}\right)$ and $B_i = \overline{G}_{i}^{-1}\left(\frac{\lambda K_i(t)}{\xi_i(t)}\right), \; i = 1, 2.$

Using (4.27), (4.28) and (4.29), we obtain

Since $K_i(t)\leq -cE'(t)$ (for $i = 1, 2)$, we infer that

Consequently, letting $\mathcal{G}_1 = \xi \mathcal{G}+cE$, we have: $\alpha_1\mathcal{G}_1(t)\leq E(t)\leq\alpha_2\mathcal{G}_1(t),$ for some $\alpha_1, \alpha_2 > 0$.

Thus, we get

where $\beta_1 > 0$ and $\mathcal{G}_2(t) = tG'(\varepsilon_0t)$. Since $\mathcal{G}_2'(t) = G'(\varepsilon_0t)+\varepsilon_0tG''(\varepsilon_0t)$, then using the strict convexity of $G_{i} (i = 1, 2)$ on $(0, r]$, we have $\mathcal{G}_2'(t), \mathcal{G}_2(t) > 0$ on $(0, 1]$. Since $\mathcal{G}_1\sim E$ and using (4.31), one derives that

and

with $\beta_2 > 0$. Integrating the last inequality over $(t_1, t)$ yields

Now, the function $G_0$ defined by $G_0(t) = \int^r_t\frac{1}{sG'(s)}ds,$ is strictly decreasing on $(0, r]$ and satisfies $\lim\limits_{t\rightarrow0}G_0(t) = +\infty$. Thus, we deduce that

This inequality together with (4.32) yields to (4.1). This ends the proof of Theorem (4.1). □

5.   Examples

In this section, we give two examples that illustrate explicit formulas for the decay rates of the energy.

1. Let $h_1(t) = h_2(t) = pe^{-k(1+t)^q}, \; t\geq0$, where $p > 0$, $0 < q\leq1$ and $p > 0$ is chosen so that $h_i$ satisfies (2.4). We can see, for $i = 1, 2$, that

where $\xi_i(t) = qk(1+t)^{q-1}$ and $G_i(t) = t$. From (4.1), it holds that

2. Let $h_i(t) = \frac{p_i}{(1+t)^{q_i}}, \; i = 1, 2$, where $q_i > 0$ and $p_i > 0$ is chosen such that, (2.4) holds true. One has, for $i = 1, 2$,

where $\xi_i(t) = \frac{q_i}{p_i^{\frac{1}{q_i}}}$ and $G_i(t) = t^{\frac{q_i+1}{q_i}}.$

Putting $q_3 = \min\{q_1, q_2\}$. Therefore, it follows from (4.1) that

6.   Conclusions

This paper focuses on the existence and the asymptotic stability of solutions for a system of two coupled quasi-linear Kirchhoff plate equations in a bounded domain of $\mathbb{R}^2$, subject only to viscoelasticity dissipative terms and with the presence of rotational forces and source terms. Each one of these two equations describes the motion of a plate, which is clamped along one portion of its boundary and has free vibrations on the other portion of the boundary. This work is motivated by previous results concerning coupled quasi-linear wave equations ^[14,15,16] and single quasi-linear plate equation ^[12,13].

As future works, we can change the type of damping by considering, for example, weak damping (of the form $y_t$), Balakrishnan-Taylor damping (of the form $(\nabla y, \nabla y_t) \Delta y$) or strong damping (of the form $\Delta^2 y_t)$.

Acknowledgments

This work is supported by Researchers Supporting Project number (RSPD2023R736), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The author declares that there are no conflicts of interest.