The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects

1.   Introduction

Recently, highly nonlinear models have attracted the focus of many scientists due to their ability to provide more meaningful insights into physical phenomena with memory effects ^[1]. The Burgers-Korteweg-de Vries (Burgers-KdV) equation has gained attention for the purpose of modeling various natural phenomena, such as the propagation of undular bores in shallow water, the flow of liquids containing gas bubbles and the propagation of waves in elastic tubes filled with viscous fluids ^[2,3,4,5]. This equation has attracted physicists, engineers and applied mathematicians from different disciplines who are interested in studying these phenomena. In the field of weak nonlinearity, the Burgers-KdV equation (or KdV equation) is commonly used to describe dispersion waves of finite but small magnitude, and it is not limited to waves in bubble streams. On the other hand, the nonlinear Schrödinger equation (or Ginzburg-Landau equation) is a widely used nonlinear wave equation (or nonlinear evolution equation) for cases involving strong dispersion.

Yatabe et al. ^[6] employed a multi-scale method to derive two KdV-Burgers equations that incorporate a drag force correction term. They verified that the time evolution of wave dissipation, caused by the drag force, differs from that caused by acoustic radiation. From a mathematical perspective, there exists a close relationship between the Burgers equation $(s = 0)$ ^[7] and the KdV equation $(\beta = 0)$ ^[8]. The standard Burgers-KdV equation is given by

where the real constants $\alpha$, $\beta$ and $s$ satisfy that $\alpha \beta s \neq 0$. The nonexistence of a spectral solution for this equation presents a challenge in studying its integrability. Currently, there is no effective analytical method available to solve this type of equation.

Indeed, there have been numerous studies on the Burgers-KdV equation, addressing various aspects such as the existence, uniqueness, well-posedness, stability and solution properties ^{[9,10,11,12,13,14,15]}. Many physical processes can be perturbed by external factors, and the nature of these perturbations can vary across different problems. In recent years, investigating the limit problem with viscous dissipation has received significant attention.

Significant progress has been made in understanding the Burgers-KdV equation. Luc and Francis ^[16] showed that the Burgers-KdV equations are globally well-posed. They established the low regularity of solutions through the use of an algebraic inequality and an a priori estimate. Dlotko ^[17] proved the local and global solvability in $H^2(\mathbb{R})$ of the Cauchy problem for the generalized KdV-Burgers equation by using the parabolic regularization technique. Wang et al. ^[18] obtained an approximate solution to the KdV-Burgers equation with boundary conditions by employing the Adomian decomposition method. Feng and Knobel ^[19] obtained traveling wave solutions from a KdV-Burgers-type equation with higher-order nonlinearities. Zhao and colleagues ^[20,21,22] introduced some localized wave solutions of the high-dimensional integrable systems for the nonlinear mathematical physics. These are just a few examples of the extensive research findings in this area.

Recently, there has been significant interest in the asymptotic connection between ionic dynamical systems and hydrodynamic models. The Euler-Poisson system has been used to derive various nonlinear dispersive equations through the use of reduced perturbation methods, including the KdV equation ^[23], Kadomtsev-Petviashvili equation ^[24], Zakharov-Kuznetsov equation ^[25,26], Burgers equation ^[27] and Schrödinger equation ^[28]. However, directly applying this method to nonlinear systems involving both dissipative and dispersive effects can be challenging. Based on the literature mentioned, this paper focuses on the question of whether the Burgers-KdV system can converge to a similar solution. The reduced perturbation method is not directly applicable when both dispersion and dissipation are present.

When regardless of the magnetic field, we consider the one-dimensional plasma with viscous dissipation ^[29,30], the two-fluid system describing ionic sound waves is reduced to

where $\phi$ is the perturbed potential, $\mu$ is interpreted as the equivalent viscosity coefficient and $n_i$ and $v_i$ represent the density and velocity, respectively.

Standardize the physical quantities in (1.1) as follows:

where $L$ is the characteristic scale of fluctuation, $c_s = \sqrt{\frac{K T_e}{m_i}}$ denotes the ion-acoustic velocity, $K$ is the Boltzmann constant and $n_0$ is the undisturbed density.

When ignoring the influence of the ion pressure term, the dimensionless equations can be simplified to

where $\nu = \frac{\mu}{m_i n_0}$ represents the equivalent kinematic viscosity coefficient and $\lambda_D = \sqrt{KT_e/4\pi n_0 e^2}$ is the Debye length.

Assuming a finite small quantity of density perturbation, let $\Delta n = n_i - n_0$; we have that $n-1 = \frac{\Delta n}{n_0} \ll 1$; taking $\frac{\Delta n}{n_0} = \varepsilon \ll 1$, let $\lambda_D / L^2 = \alpha \varepsilon$ and $\nu / L c_s = \beta \varepsilon$; this is what the weak dispersion and viscosity require. The quantity $\varepsilon$ is a quantity that describes the strength of the nonlinearity. When both the weak dispersion and the weak dissipative effect are equivalent to $\varepsilon$, we can obtain the Burgers-KdV equation via the perturbation method. The system (1.1a) can be rewritten as follows:

1.1.   Derivation of Burgers-KdV equation

By applying the following Gardner-Morikawa transformation ^[29] to (1.3),

we obtain the following parameterized system:

where the small quantity $\varepsilon$ is also the amplitude of initial disturbance and $\lambda_0$ is the velocity parameter.

Assume that the variables have the following expansions:

we then incorporate (1.6) into system (1.5), terminating each expansion at different orders of magnitude $\varepsilon$.

At the order of $O(\varepsilon)$, we have

a simpler calculation gives

At the order of $O(\varepsilon^2)$, we have

Taking $\partial_x$ of (1.9c), multiplying (1.9a) by $\lambda_0$ and then putting them into (1.9b), we obtain the homogeneous Burgers-KdV equation for $n_e^{(1)}$:

where we applied (1.8). Note that system (1.10) and (1.9) are self contained for $(n^{(1)}, v^{(1)}, \phi^{(1)})$.

From (1.9), we can express $(n^{(2)}, v^{(2)})$ in the following form:

where we have used (1.8b) and the function $f(1)$ only depends on $n_e^{(1)}$ and its derivatives.

At the order of $O(\varepsilon^3)$, we have

Taking $\partial_x$ of (1.12c), multiplying (1.12a) by $\lambda_0$ and then putting them into (1.12b), we obtain the inhomogeneous Burgers-KdV equation for $n_e^{(2)}$:

where we have applied (1.11) and $G(1)$ depends only on $n_e^{(1)}$ and its derivatives.

At the order of $O(\varepsilon^{k+1})$, we can get the linearized inhomogeneous Burgers-KdV equation for $n_e^{(k)}$:

where $G(k-1)$ only depends on $n_e^{(1)}, \ldots, n_e^{(k-1)}$, which can all be determined from the preceding $k$-$1$ steps. The system (1.14) is also self-contained and only depends on $n_e^{(k)}$.

Equations (1.10), (1.13) and (1.14) contain both third derivative terms and second derivative terms which arise from the interplay of dispersion and dissipation in the system. Finding a general Gardner-Morikawa transformation that keeps the equation unchanged is not possible in this case. This implies that the general reduced perturbation method fails to preserve the original form of the equation. Therefore we assumed that the magnitudes of the dispersion and dissipation terms are at the same level.

Remark 1.1. If the magnitudes of dispersion and dissipation terms are not on the same order, the approximate evolution equation may be reduced to either the KdV equation ($\lambda^2_D / L^2 \gg \nu/L_{c_s}$) or the Burgers equation ($\lambda^2_D / L^2 \ll \nu/L_{c_s}$) respectively. These reduced equations capture the dominant behavior of the system when either dispersion or dissipation is significantly stronger than the other.

Remark 1.2. Through qualitative analysis, (1.10), (1.13) and (1.14) can be likened to a nonlinear oscillator equation with a damping term, which can be solved in the form of an oscillating shock wave by applying qualitative analysis for sufficiently large damping. On the contrary, when the damping is small enough, the soliton-mode solution with slow attenuation of amplitude can be obtained.

Theorem 1.1. Let $s_1 \geq 2$ be a sufficiently large integer. Then for any given initial data $n_0^{(1)}\in H^{ s_1 }(\mathbb{R})$, there exists $\tau_* > 0$ such that the initial value problem (1.8) and (1.10) has the unique solution

with initial data $(n_{e0}^{(1)}, \lambda_0 n_{e0}^{(1)}, n_{e0}^{(1)})$. Moreover, we can extend the solution to any time interval $[-\tau, \tau]$ by the conservation of the Burgers-KdV equation ^[30,31].

Theorem 1.2. Let $k\geq 2$ and $s_k \leq s_1 - 3(k-1)$ be sufficiently large integers. Then, for any $\tau > 0$ and any given initial data $(n_0^{(k)}, v_0^{(k)}, n_{e0}^{(k)})\in H^{ s_k }(\mathbb{R})$, the initial value problem of (1.14) with initial data $(n_0^{(k)}, v_0^{(k)}, n_{e0}^{(k)})$ has the unique solution

1.2.   Remainder system

Next, we will give a strict proof to show that $n_e^{(1)}$ converges to a solution of the Burgers-KdV equation as $\varepsilon \rightarrow 0$. Suppose that $(n^{(k)}, v^{(k)}, n_{e}^{(k)})$, $1\leq k \leq 4$ are sufficiently smooth. Let $(n, v, n_e)$ be a solution of (1.5) and have the following expansions:

where $(n^{(k)}, v^{(k)}, n_{e}^{(k)})$ for $1\leq k \leq 4$ satisfies (1.8), (1.10) and (1.13) respectively, and $(N, V, N_e)$ is the remainder.

Just for the sake of calculation, we denote

Putting (1.15) into the scaled system (1.5), a simper calculation gives the remainder system, as follows

where

Next, we give some basic estimates for the remainder term $\mathcal{R}_3$.

Lemma 1.1. For the $s = 1, 2, \ldots$ integers, there exists some constant $C = C(\|n_e^{(i)}\|_{H^{\delta}})$ and $C = C(\|n_e^{(i)}\|_{H^{\delta}}, \sqrt{\varepsilon}\|N_e\|_{H^s})$ such that $\|\mathcal{R}_1, \mathcal{R}_2\|_{H^s} \leq C (\|n_e^{(i)}\|_{H^{\delta}})$ and

where $\delta = \max\{2, s-1\}$; due to the fact that $H^s$ is an algebra, the proof of Lemma 1.1 is obvious; also, the constant $C$ is nondecreasing.

1.3.   Main results

The main results are given as follows:

Theorem 1.3. Let $s_i \geq 2$ in Theorems 1.1 and 1.2 be sufficiently large and $(n^{(1)}, v^{(1)}, n_e^{(1)}) \in H^{s_1}$ be a solution constructed in Theorem 1.1 for the Burgers-KdV equation with initial data $(n_0^{(1)}, v_0^{(1)}, n_{e0}^{(1)})$ in $H^{s_1}$ satisfying (1.8). Let $(n^{(i)}, v^{(i)}, n_e^{(i)}) \in H^{s_i} (i = 2, 3, 4)$ be a solution of (1.14), as constructed in Theorem 1.2 with initial data $(n_0^{(i)}, v_0^{(i)}, n_{e0}^{(i)})$ in $H^{s_i}$. Let $(N^\varepsilon_0, V^\varepsilon_0, N^\varepsilon_{e0}) \in H^s$ and assume the following:

Then, for any $\tau > 0$, there exists $\varepsilon_0 > 0$ such that, if $0 < \varepsilon < \varepsilon_0$, the solution of the system (1.5) with initial data $(n_0, v_0, n_{e0})$ can be expressed as follows:

such that, for all $0 < \varepsilon < \varepsilon_0$,

From (1.22), we obtain the uniform $H^2$-norm of the remainder $(N, V, N_e)$ on $\varepsilon$, and it satisfies

where $C$ is independent of $\varepsilon$ and BKdV denotes the Burgers-KdV equation.

In order to give the proof of Theorem1.3, we define the following weighted norm:

2.   The uniform estimate

In this section, we will give the uniform estimates of the system (1.16) and show that (1.16) has smooth solutions for significantly small $\tau_\varepsilon > 0$, as dependent on $\varepsilon > 0$. Let $\tilde{C}$ be a constant that is independent of $\varepsilon$. By the classical theorem, we know that there exists $\tau_\varepsilon > 0$ such that on $[0, \tau_\varepsilon]$,

Note that $n$ is bounded, i.e., $1/2 < n < 3/2$, and $|v| < 1/2$ for $\varepsilon < \varepsilon_1$. There exists some constant $C_1 = C_1(\varepsilon \tilde{C})$ for any $\alpha, \beta \geq 0$ such that

where $C_1$ is chosen to be nondecreasing in its argument.

Next, we will give some lemmas to show the relation between $N$ and $N_e$.

Lemma 2.1. Let $(N, V, N_e)$ be a solution to (1.16) and $\alpha \geq 0$ be an integer. There exist some constants $0 < \varepsilon_1 < 1$ and $C_1 = C_1(\varepsilon \tilde{C})$ such that for every $0 < \varepsilon < \varepsilon_1$,

Proof. When $\alpha = 0$, taking the inner product of (1.16c) with $N_e$, we have

where $\alpha$ is a constant and $\frac{1}{2} < n_e < \frac{3}{2}$; therefore, there exists a constant C such that

so the left of (2.5) is larger than $\|N_e \|^2 + \varepsilon \|\partial_x N_e \|^2$.

Due to the fact that

combined with Young's inequality, we get

Due to the fact that $\tilde{n}_e$ is bounded in $L^\infty$, by the Hölder inequality, we have

From Lemma 1.1, it holds that $\varepsilon < \varepsilon_1$, which is sufficiently small such that

Therefore, by the Hölder inequality, we have

Taking the inner product of (1.16c) with $\varepsilon \partial^2_x N_e$, we have

by the Hölder inequality, the result is as follows

Then, applying the $L^2$-norm of (1.16c), we obtain some $C$ such that

Putting (2.10), (2.12) and (2.13) together, we obtain (2.3) for $\alpha = 0$.

For the higher-order cases, we can differentiate (1.16c) with respect to $\partial^\alpha$, take the inner product with $\partial^\alpha_x N_e$ and $\varepsilon \partial^{\alpha+2}_x N_e$ and then perform the same procedure as for the case $\alpha = 0$; we can complete this lemma.

Lemma 2.2. Let $(N, V, N_e)$ be a solution to (1.16) and $\alpha \geq 0$ be an integer. There exist some constants $C$ and $C_1 = C_1(\varepsilon \tilde{C})$ such that

and

Proof. Due to the fact that $1/2 < n < 3/2$ and $|v| < 1/2$, taking the $L^2$-norm of (1.16a) gives

By using Lemma 2.1 with $\alpha = 1$, we have (2.14).

Then, taking $\partial_x$ of (1.16a), we obtain

By the Sobolev embedding inequality, we have

then by Lemma 2.1, we complete this lemma.

Lemma 2.3. Let $(N, V, N_e)$ be a solution to (1.16) and $\alpha \geq 0$ be an integer. There exist some constants $C_1 = C_1(\varepsilon \tilde{C})$ and $\varepsilon_1 > 0$ such that for any $0 < \varepsilon < \varepsilon_1$,

Proof. For the case of $\alpha = 0$, taking $\partial_t$ of (1.16c) and then taking the inner product with $\partial_t N_e$, we have

where $\alpha$ is a constant and $1/2 < n_e < 3/2$, so there exists a constant C such that $\varepsilon \frac{\alpha }{n_e}\int |\partial_{tx} N_e|^2\geq C \varepsilon \int |\partial_{tx} N_e|^2$; therefore, the left of (2.20) is greater than or equal to $C (\|\partial_t N_e\|^2+\varepsilon \int |\partial_{tx} N_e|^2)$. Now, we estimate the right of (2.20). For $A_1$, for any small $\gamma > 0$, by Young's inequality, we have

Due to the fact that

by the Hölder inequality and Sobolev embedding inequality, we obtain

and

Similar to $A_2$, by the Hölder inequality, we obtain

Therefore, we have

When $\alpha = 1$, we take $\partial_{xt}$ of (1.16c); then, taking the inner product with $\varepsilon\partial_{xt} N_e$, we have

Due to the fact that

for arbitrary $\gamma > 0$, by Young's inequality, we obtain

Using (2.1), by the Hölder inequality and Sobolev embedding inequality, we have

and

Similar to the process for $B_{2}$, by the Sobolev embedding inequality, we have

therefore, we have

We can give by the similar process for the case of $\alpha\geq2$.

2.1.   Zeroth, first and second order estimates

Proposition 2.1. Let $(N, V, N_e)$ be a solution to (1.16) and $\kappa = 0, 1, 2$; then,

Proof. We take $\partial_x^\kappa$ of (1.16b) and the inner product of $\partial_x^\kappa V$. Integrating by parts yields

The second term vanishes by integration by parts. The third term can be divided into two parts as follows

for the first part, integration by parts gives

where, when $\kappa = 0$, there is no such "summation" term. Regarding the second part, after integration by parts, for $0\leq\kappa\leq2$, we have

For the fourth therm, similar to the first term,

For the fifth therm, similar to the first term, integration by parts gives

By Lemma (1.1), we have

Now, we estimate the right side of (2.34) for $0 < \kappa \leq 2$; taking $\partial_x^{\kappa}$ of (1.16a), we have

Incorporating this into the right side of (2.34), we have

We first estimate the term for $3 \leq i \leq 8$; for the term $I_{3}$, by the Hölder inequality and Sobolev embedding inequality, we have

Similar to $I_{3}$, simple calculation gives

and

By Lemma (1.1), with a similar calculation to that for $I_{3}$, we obtain

Lemma 2.4. Let $(N, V, N_e)$ be a solution to (1.16); we have

where $I_1$ is defined in (2.35).

Proof. For the first term $I_{1}$, by using a simpler calculation, we can decompose it into two parts:

Taking $\partial_x^{\kappa+1}$ of(1.16c), we have

inserting it into $I_{11}$ gives

Regarding estimate of $I_{111}$, by commutator estimation and Sobolev embedding, we obtain

Regarding estimate of $I_{12}$, by performing integration by parts twice, commutator estimation and Sobolev embedding, we have

The estimate of $I_{13}$ is similar to that of $I_{12}$:

By Lemma 1.1, we have

With a similar estimate to $I_{11}$, for $0 < \gamma \leq 2$, we obtain

Combining the estimate of $I_{11}$ and $I_{12}$, we can get (2.36).

Lemma 2.5. Let $(N, V, N_e)$ be a solution to (1.16); then, the following inequality holds

where $I_2$ is defined in (2.35).

Proof. Taking $\partial_t \partial_x^\kappa$ of (1.16c), we have

inserting it into $I_2$ gives

Regarding the estimate of $I_{21}$, through integration by parts, we obtain

where the second term on the right-hand side is bounded as follows

where we have used the fact that

for the third term, by the Hölder inequality and Sobolev embedding inequality, we have

therefore, we have

For the estimate of $I_{22}$, we have the following decomposition:

For the term $I_{221}$, through integration by parts, and by using the Hölder inequality and Sobolev embedding inequality, we have

where we used the fact that

For the term $I_{222}$, by the Hölder inequality and Sobolev embedding inequality, we have

For the term $I_{223}$, by the Hölder inequality and Sobolev embedding inequality, we have

Therefore, we have the estimate

For the estimate of $I_{23}$, by the Hölder inequality and Sobolev embedding inequality, we have

Regarding the estimate of $I_{24}$, by Lemma 1.1, we have

Combining (2.38), (2.39), (2.40) and (2.41), we complete this lemma.

2.2.   Third order estimates

Proposition 2.2. Let $(N, V, N_e)$ be a solution to (1.16); then,

Proof. Taking $\partial_x^3$ of (1.16b) and then taking the inner product with $\varepsilon \partial_x^3 V$, we have

Estimate the left-hand side of (2.43). The second term vanishes by integration by parts. For the third term, by the Hölder inequality and Sobolev embedding inequality, we get

Regarding the estimate for the fourth term, similar to the third term, we have

Regarding the estimate for the fifth term, we have the following decomposition:

Integration by parts twice yields

Through by parts, and by using the Sobolev embedding inequality, we have

Similarly, through integration by parts, and by using the Sobolev embedding inequality, we have

For the last term, by Lemma 1.1, we have

Estimate the right-hand side of (2.43). Taking $\partial_x^3$ of (1.16a) and inserting it into the right-hand side of (2.43) gives

We first give the estimate of $\tilde{I}_{i}$ for $3\leq i \leq 6$. For $\tilde{I}_{3}$, we have the following decomposition:

Regarding the estimate for $\tilde{I}_{31}$, by the Hölder inequality and Sobolev embedding, we have

Regarding the estimate for $\tilde{I}_{32}$, through integration by parts, and by using the Hölder inequality and Sobolev embedding, we have

Via a similar calculation process, we have

By Lemma 1.1, we have

Lemma 2.6. Let $(N, V, N_e)$ be a solution to (1.16); we have

Proof. We have the following decomposition:

Regarding the estimate for $\tilde{I}_{11}$, taking $\partial_x^4$ of (1.16c) and inserting it into $\tilde{I}_{11}$, we have

Regarding the estimate for $\tilde{I}_{111}$, by the Hölder inequality and Sobolev embedding, we have

where we used the fact that

Regarding the estimate for $\tilde{I}_{112}$ and $\tilde{I}_{113}$, through integration by parts, using the Hölder inequality and Sobolev embedding, we have

By Lemma 1.1, we have

Therefore, we have

Regarding the estimate for $\tilde{I}_{12}$, through integration by parts, and by using the the Hölder inequality and Sobolev embedding, we have

Regarding the estimate for $\tilde{I}_{13}$, through integration by parts, and by using the Hölder inequality and Sobolev embedding, we have

Lemma 2.7. Let $(N, V, N_e)$ be a solution to (1.16); we have

Proof. Taking $\partial_t\partial_x^3$ of (1.16c), and then inserting the result in $\tilde{I}_{2}$, we have

Regarding the estimate for $\tilde{I}_{21}$, through integration by parts, we have

For $\tilde{I}_{211}$, by using the Hölder inequality and Sobolev embedding, we have

For $\tilde{I}_{212}$, through integration by parts, by using the Hölder inequality and Sobolev embedding, we have

For $\tilde{I}_{213}$, it is similar to $\tilde{I}_{212}$:

Therefore, we have

Regarding the estimate for $\tilde{I}_{22}$, we have the following decomposition:

For the term $\tilde{I}_{221}$, through integration by parts, we have

By using the Hölder inequality and Sobolev embedding, we have

Regarding the term $\tilde{I}_{224}$, due to the term $\partial_t\partial^3_x N_e$, even if we raise the Sobolev order or the expansion order, it cannot be controlled in terms of $\||(V, N_e) |\|_\varepsilon$, the essential reason is related to Lemmas 2.2 and 2.3. But, by observing its features, we can use (1.16c) to complete this estimate. Simple calculation gives

by using (1.16c), we have

Inserting it into $\tilde{I}_{2214}$, and using the Hölder inequality and Sobolev embedding, we have

Regarding the terms $\tilde{I}_{2215}$ and $\tilde{I}_{2216}$, by using the Hölder inequality, we obtain

Therefore, we have

Regarding the estimate of $\tilde{I}_{222}$, by using the Hölder inequality and Sobolev embedding, we have

where we note that

The difficulty is that the term $\partial_t\partial^4_x N_e$ cannot be controlled in terms of $\||(V, N_e) |\|_\varepsilon$; we can use (1.16c) to complete this estimate. Simple calculation gives

by using (1.16c), through integration by parts, by using the Hölder inequality and Sobolev embedding, we have

Regarding the estimate of $\tilde{I}_{223}$, through integration by parts, and by using the Hölder inequality and Sobolev embedding, we have

where we note that

by using (1.16c), we have

therefore, we have

Regarding the estimate of $\tilde{I}_{224}$, through integration by parts, and by using the Hölder inequality and Sobolev embedding, we have

where we note that

by using (1.16c) we have

Therefore, we have

Regarding the estimate for $\tilde{I}_{23}$, similar to $\tilde{I}_{22}$, we have

By Lemma 1.1, we have

Combining these estimates, we complete this lemma.

3.   Conclusions

Proof of Theorem 1.3. Combining Propositions 2.1 and 2.2, we have

Integrating the inequality over $(0, t)$ yields

where $C$ is an absolute constant.

Since $C_1$ is nondecreasing and depends on $\||(V, N_e)|\|^2_\varepsilon$ through $\varepsilon\||(V, N_e)|\|^2_\varepsilon$, let $C_1^* = c(1)$ and $C_2 > C sup_{\varepsilon < 1}\||(V, N_e)(0)|\|^2_\varepsilon$. For any arbitrarily given $\tau > 0$, we choose $\tilde{C}$ such that $\tilde{C} > e^{4 C_1^\prime}\tau (1+C_2)(1+C_1^\prime)$. Then there exists $\varepsilon_0 > 0$ such that $\varepsilon \tilde{C} \leq 1$ for all $\varepsilon < \varepsilon_0$; we have

By Lemma 2.1, we have

By the Grönwall inequality, we complete the proof of Theorem 1.3.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

The author (R. Rong) was supported by Xiamen Institute of Technology High-level Talents Research Launch Project (No. YKJ22050R) and Fujian Province Young and Middle-aged Teacher Education Research Project (No. JAT220337). The author (H. Liu) was supported by the National Natural Science Foundation of China (Nos. 12271293, 11901342) and Natural Science Foundation of Shandong Province (No. ZR2023MA002).

Conflict of interest

All authors declare no conflict of interest that may influence the publication of this paper.