A Razumikhin approach to stability and synchronization criteria for fractional order time delayed gene regulatory networks

1.   Introduction

In this paper, we study the following quasi-linear bi-hyperbolic equation:

under the following dynamic boundary conditions,

and initial conditions

where $a \geq 0$, $b \geq 0$, $t \geq 0$, $x \in \Omega$, $\Omega$ is an open bounded connected region in $\mathbb{R}^n$ ($n \geq 1$) with a smooth boundary $\Gamma: = \partial\Omega$, and $\eta(x)$ represents an outer unit normal vector to the boundary $\Gamma$.

The Eq (1.1) represents a mathematical model of a wave process in a physical domain $\Omega$ over a time interval $[0, T)$. This wave equation involves $w_{tt}$ as the second-time derivative representing acceleration over time, $\Delta^2 w$ as the second-order spatial Laplacian indicating wave irregularities, $\Delta w$ as the first-order Laplacian reflecting propagation speed within the wave, and $bf(-\Delta w)$ as a nonlinear term depicting wave interaction with its negative Laplacian. Overall, these equations and conditions provide a framework for understanding wave behavior, interactions at boundaries, and the evolution of waves from specified initial conditions within a physical space. Applications of such equations extend to various areas, including acoustics, electromagnetics, and mechanics, where wave phenomena play a crucial role in modeling and analysis.

First, we mention some known results of higher-order differential equations under dynamic boundary conditions related to the problems (1.1)–(1.3). Dynamic boundary conditions introduce dependencies on both time and space variables, influencing the behavior and evolution of solutions within specific domains. Recent research has highlighted the significance of dynamic boundary conditions in various mathematical contexts, particularly in studying wave propagation, heat transfer, fluid dynamics, and other physical phenomena. Notably, works by Vasconcellos and Teixeira ^[1] have explored the implications of dynamic boundary conditions on well-posedness. They considered, for $n \leq 3$, the following problem:

where $\phi$ is a non-negative continuous real differentiable function, and $g$ is a continuous non-decreasing real function. They proved the existence and uniqueness of global solutions. Guedda and Labani ^[2] studied the problem

where $p\geq 0$ is a smooth function defined on the boundary of $\Omega$. They studied the global nonexistence of solutions under certain conditions on $f$ and $\phi$. Later, Wu and Tsai ^[3] considered the initial boundary value problem for a Kirchhoff-type plate equation with a source term in a bounded domain. They established the existence of a global solution using an argument similar to that in ^[4]. Vitillaro conducted a study in 2017 focusing on dynamic boundary conditions. In this work ^[5], a wave equation with hyperbolic dynamic boundary conditions, interior and boundary damping effects, and supercritical sources was investigated.

Several authors have extensively studied blow-up phenomena and global nonexistence (see ^[4,6,7,8,9]). Levine ^[8] introduced the concavity method and investigated the nonexistence of global solutions with negative initial energy. Subsequently, Georgiev and Todorova ^[4] expanded upon Levine's work. In 2002, Vitillaro ^[10] further refined the results of Georgiev and Todorova for systems with positive initial energy. Vitillaro also explored blow-up phenomena for wave equations with dynamic boundary conditions in ^[9]. Additionally, Can et al. ^[6,7] investigated the blow-up properties of (1.1) under various boundary conditions, assuming non-positive initial energy. While their result is achieved by applying the Ladyzhenskaya and Kalantarov lemma ^[11], along with a generalized concavity method, our approach is based on the blow-up lemma by Korpusov ^[12], which is another application of the concavity method. In our study of problems (1.1)–(1.3), we obtained both a local existence result and a blow-up result under positive initial energy.

The paper is structured as follows. Section 2 provides essential definitions, theorems, and inequalities. In Section 3, we initially employ the Galerkin approximation method to investigate the existence of the corresponding linear problems (3.1)–(3.3). Subsequently, utilizing the contraction mapping principle, we establish the local existence and uniqueness of regular solutions for problems (1.1)–(1.3). Finally, in the last section, we deduce the blow-up solutions for problems (1.1)–(1.3) under the condition of positive initial energy.

2.   Preliminaries and notations

The Sobolev space is defined by $W^{k, p}(\Omega): = \{u\in L_p(\Omega):D^\alpha u\in L_p(\Omega), \forall\; 0\leq|\alpha|\leq k\}$ for $1\leq p < \infty$, equipped with the following norm:

We denote by $H^k(\Omega) = W^{k, 2}(\Omega)$ the Hilbert-Sobolev space. Throughout this paper, we denote $\|.\|_{L^2(\Omega)} = \|.\|_2$.

Definition 2.1. Let $w(t)$ be a weak solution of the problem defined by Eqs (1.1)–(1.3). We define the maximal existence time $T_\infty$ as follows:

(ⅰ) If $w(t)$ exists for $0\leq T < \infty$, then $T_\infty = +\infty$.

(ⅱ) If there exists a $T_0\in(0, \infty)$ such that $w(t)$ exists for $0\leq T < T_0$, but does not exist at $T = T_0$, then $T_\infty = T_0$.

In order to prove the blow-up result, we will utilize the following lemma due to Korpusov.

Lemma 2.2. ^[12] Let $\psi(t)\in C^2(0, T)$ and consider the differential inequality

Assume that the following conditions

hold with $\psi(t)\geq 0$, and $\psi(0) > 0$. Then the time $T > 0$ can not be arbitrarily large. That is,

where $T_\infty$ is the maximal existence time interval for $\psi(t)$ and

such that $\lim_{t\uparrow T_\infty}\psi(t) = +\infty.$

Now, we state the assumptions on the function $f$:

(A1) $f:H_0^2(\Omega)\to L^2(\Omega)$ is locally Lipschitz with the Lipschitz constant $L_f$, that is, for every $x\in H_0^2(\Omega)$, there exists a neighborhood $V$ of $x$ and a positive constant $L_f$ depending on $V$ such that

for each $y, z\in V$.

(A2) The function $f$ with its primitive $F(u) = \displaystyle{\displaystyle{\int}}_0^u f(s)ds$ has the property:

for all $u \in\mathbb{R}$ and for some positive real number $\gamma$.

Example 2.3. Consider the function $f(u) = u^2$. This function satisfies property (2.1) based on the conditions $f(0) = 0$ and the behavior of its primitive, $F(u) = \displaystyle{\int}_0^u f(s) ds = \displaystyle{\int}_0^u s^2 ds = \frac{u^3}{3}$. Specifically, we can establish the inequality $uf(u) = u^3 \geq 2(2\gamma + 1) \frac{u^3}{3}$, which holds true for some $\gamma \leq \frac{1}{4}$.

3.   Local existence

In this section, we delve into the local existence of solutions for the wave Eqs (1.1)–(1.3) employing the contraction mapping principle. Initially, we examine the following linear initial boundary value problem:

Lemma 3.1. Suppose that $w_0\in U$, $w_1\in H$, and $h\in W^{1, 2}(0, T;L^2(\Omega))$. Then, the problems (3.1)–(3.3) admit a unique solution $w$ such that

where $U = \{w\in H_0^2(\Omega):\frac{\partial \Delta w}{\partial \eta}|_\Gamma = -a \Delta w_t\}$, and $H = H^1_0(\Omega)\cap H^2(\Omega)$.

Proof. We initially employed the Galerkin approximation method to investigate the existence of solutions to this linear problem. Let $(\phi_n)_{n\in\mathbb{N}}$ be a basis in $U$, and $V_n$ denote the subspace generated by $\phi_1, ..., \phi_n$ ($n = 1, 2, ...$). Consider $w_n(t) = \sum_{i = 1}^n r_{in}(t)\phi_i$ as the solution of the approximation problem corresponding to (3.1)–(3.3) for $\phi\in V_n$. Then, we have:

with initial conditions satisfying

First, we verify the existence of solutions to (3.4)–(3.6) on some interval $[0, t_n)$, $0 < t_n < T$, and then use standard differential equations techniques ^[13] to extend the solution across the entire interval $[0, T]$. To achieve this, we need to establish the following a priori estimates.

Setting $\phi = 2w'_n(t)$ in (3.4), integrating over $(0, t)$, and utilizing boundary conditions yield:

From this, we obtain:

where $C_0 = \|w'_n(0)\|_2^2+\| \Delta w_n(0)\|_2^2+\|\nabla w_n(0)\|_2^2+\displaystyle{\int}_0^T\|h\|_2^2 dt$, and utilizing the estimate:

The conditions (3.5) and (3.6), and the property of $h$ imply that $C_0$ is bounded. Now, for all $0\leq t\leq T$, applying Gronwall's inequality in (3.7), we obtain

where $M_1$ is a positive constant.

To estimate $w''_n(0)$ in $L^2$-norm, we set $t = 0$ in (3.4) and $\phi = 2w''_n(0)$:

By employing (3.5) and (3.6), we find a positive constant $M_2$ such that:

Next, we aim to establish an upper bound for $\|w''_n(t)\|_2$. Replacing $\phi = 2w''_n(t)$ in (3.4) after differentiating it with respect to $t$ gives

Hence, by integrating (3.12) over $(0, t)$ and using the inequalities (3.8), (3.9) and (3.11), we obtain

Using Gronwall's inequality for the inequality

and (3.5) and (3.6), we can derive

for any $t\in [0, T]$ with a positive $M_3$, which is independent of $n\in\mathbb{N}$. Using (3.9) and (3.14), we may conclude that

Thus, by taking the limit in (3.4) and utilizing the above convergences, we obtain:

for all $\sigma\in D(0, T)$ and for all $u\in U$. From the above identity, we have

since $w'', \Delta w$ and $h\in L^\infty(0, T;L^2(\Omega))$ and we deduce $\Delta^2 w\in L^\infty(0, T;L^2(\Omega))$, so $w\in L^\infty(0, T;U)$.

To prove the uniqueness of the solution, let $w_1$ and $w_2$ be two solutions of (3.1)–(3.3). Then $v = w_1-w_2$ satisfies

for $\phi\in U$. Also, we have

Now, if we set $\phi = 2v'(t)$ in (3.19), then we have

By Gronwall's inequality, we conclude that

Therefore, we have uniqueness. Now, we establish the local existence of the problems (1.1)–(1.3).

Theorem 3.2. Suppose that $f:H_0^2(\Omega)\to L^2(\Omega)$, and that $w_0\in U$, and $w_1\in H$, then there exists a unique solution $w$ with $w\in L^\infty(0, T; U)$ and $w_t\in L^\infty(0, T; H)$.

Proof. Define the following space for $T > 0$ and $R_0 > 0$:

Then $X_{T, R_0}$ is a complete metric space with the distance

where $x, y\in X_{T, R_0}$.

By Lemma 3.1, for any $u\in X_{T, R_0}$, the problem

has a unique solution $w$ of (3.21). We define the nonlinear mapping $Bu = w$, and then, we shall show that there exists $T > 0$ and $R_0 > 0$ such that

(ⅰ) $B:X_{T, R_0}\to X_{T, R_0},$

(ⅱ) In the space $X_{T, R_0}$, the mapping $B$ is a contraction according to the metric given in (3.20).

After multiplication by $2w_t$ in Eq (3.21), and integration over $\Omega$, we find

Taking into account the assumption (A1) on $f$, we obtain

Then, by integrating (3.22) over $(0, t)$ and using the above inequality, we deduce

Thus, by Gronwall's inequality, we have

Therefore, if the parameters $T$ and $R_0$ satisfy $e_1(w_0)e^{\displaystyle{\int}_0^t 4b^2L_f^2+1}\leq R_0^2$, we obtain

Hence, it implies that $B$ maps $X_{T, R_0}$ into itself.

Let us now prove (ⅱ). To demonstrate that $B$ is a contraction mapping with respect to the metric $d(., .)$ given above, we consider $u_i\in X_{T, R_0}$ and $w_i\in X_{T, R_0}$, where $i = 1, 2$ are the corresponding solutions to (3.21). Let $v(t) = (w_1-w_2)(t)$, then $v$ satisfies the following system:

with initial conditions

and boundary conditions

Multiplying (3.25) by $2v_t$, and integrating it over $\Omega$, we find

where

and

To proceed the estimates of $I_i$, $i = 2, 3$, we observe that

and

Thus, by using (3.27) and (3.28) in (3.26), we get

So, from Gronwall's inequality, it follows that

By (3.20), we have

where $C(T, R_0) = 4b^2L_f^2T^2e^{R_0^2T}$. Hence, under inequality (3.24), $B$ is a contraction mapping if $C(T, R_0) < 1$. Indeed, we choose $R_0$ to be sufficiently large and $T$ to be sufficiently small so that (3.24) and (3.29) are simultaneously satisfied. By applying the contraction mapping theorem, we obtain the local existence result.

Remark 3.3. The application of the contraction mapping theorem in Theorem 3.2 guarantees the existence of a unique local solution $w(t)$ defined in the ball $B(0, R_0)\subset H_0^2(\Omega)$. Since $U\times (H_0^1(\Omega)\cap H^2(\Omega))$ is dense in $H_0^2(\Omega)\times L^2(\Omega)$, we can obtain the similar priori estimates in Theorem 3.2 for $\|w(t)\|_{H_0^2(\Omega)}$ and this norm remains bounded as $t\to T_\infty$. So, we can conclude that the solution can be extended to the whole space $H_0^2(\Omega)$.

Next, we define a weak solution for the initial and boundary value problem, as follows:

Definition 3.4. A weak solution to the problems (1.1)–(1.3) on $(0, T)$ is any function $w\in C(0, T;H_0^2(\Omega))\cap C(0, T;L^2(\Omega))$, with $w_0\in H_0^2(\Omega)$ and $w_1\in L^2(\Omega)$ verifying

for all test functions $\phi$ in $C(0, T;U)\cap C(0, T;L^2(\Omega))$.

4.   Blow-up

In this section, we study the existence of blow-up solutions for the initial and boundary value problems (1.1)–(1.3). We recall the definition for blow-up of the solutions to the problems (1.1)–(1.3).

Definition 4.1. Suppose $w$ is a solution to (1.1)–(1.3) in the maximal existence time interval $[0, T_\infty)$, $0 < T_\infty\leq\infty$. Then $w$ blows up at $T_\infty$ if ${\lim\sup}_{t\to T_\infty, t < T_\infty}\|w\|_2 = +\infty$.

We introduce the energy functional $E(t)$ as:

Furthermore, we define the function $\psi(t)$ as follows:

The subsequent lemma demonstrates that our energy functional $E(t)$ defined in (4.1) is a non-increasing function.

Lemma 4.2. Under the assumption (2.1) for the energy function $E(t), t > 0$, the inequality $E(t)\leq\; E(0)$ holds.

Proof. Multiplying Eq (1.1) by $-2 \Delta w_t$ in $L^2(\Omega)$ yields the equality:

By using Green's Formula and the boundary conditions (1.2), we obtain

Then, we have,

It is obvious from (4.4) that $E(t)\leq E(0)$ for all $t\geq 0$.

Theorem 4.3. Under the assumptions on the parameter of our problem, the functional $\psi(t)$ given by (4.2) satisfies the following inequality:

where

Proof. Differentiating the function $\psi$ defined in Eq (4.2) for $t$, we obtain

Taking one more derivative with respect to $t$ and utilizing Green's formula gives:

Since

we obtain,

By using the inequality (2.1) we have,

Thus, we obtain from the inequalities (4.6) and (4.4) that

Multiplying both sides of the following inequality by $\psi(t)$:

we get

From (4.5), we obtain:

Applying Schwartz's and Hölder's inequalities, we obtain:

Now, we introduce the following notations:

Hence, from (4.9), we have

By Cauchy's inequality, we obtain

On the other hand,

and we also have

so, we get

Consequently, by subtracting (4.10) from (4.7), we obtain,

as desired.

Theorem 4.4. For each fixed $w_0\in W_0^{1, p}(\Omega)$, there exists $w_1\in L^2(\Omega)$ satisfying the conditions

Hence, by Lemma 2.2 we have the following upper bound for the existence time $T_0 = T_0(u_0, u_1) > 0$ of the solution:

where

and

with

Proof. It is sufficient to prove the resulting conditions in (4.11) are compatible. Firstly, we choose a non-trivial initial function $w_0(x)\in W_0^{1, p}(\Omega)$ in such a way that

Fix $w_0(x)$ and put $w_1(x) = \lambda w_0(x)$ with $\lambda > 0$ so large that the initial energy is guaranteed to be positive:

Note that $\psi'(0) = 2\lambda\|\nabla w_0\|_2^2+\displaystyle{\int}_\Gamma a (\Delta w_0)^2 > 0$. Then the condition (4.11) takes the form,

Write $\lambda = 1/a^{1/2}$, where $a > 0$. Then a series of the transformations in (4.14) yields the inequality that coincides with (4.13). This proves that the conditions (4.11) are compatible for sufficiently small $a > 0$.

Remark 4.5. Consider the function $f$ from Assumption (A2) and the functions $w_0$ and $w_1$ that satisfy the following conditions:

(i) By Theorem 4.4, the bounded function $\psi$ defined in Eq (4.2) and its derivative $\psi'$ satisfy Lemma 2.2.

(ii) Additionally, the initial energy functional $E(0)$ defined in Eq (4.12) is positive.

Therefore, a positive number exists $T > 0$ such as $T < T_\infty$, where $\psi(t) \to +\infty$ as $t \to T_\infty$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions.

Conflict of interest

The authors declare there are no conflicts of interest.