Ultrasound carotid plaque segmentation via image reconstruction-based self-supervised learning with limited training labels

1.   Introduction

We consider the initial boundary value problem of the following porous elastic system with nonlinear or linear weak damping terms and nonlinear source terms

where $u(x, t)$ and $\phi(x, t)$ are the displacement of the solid elastic material and the volume fraction, respectively, $\mu, \ b, \ \delta$ and $\xi$ are coefficients with physical meaning satisfying

$u_0, \ u_1, \phi_0$ and $\phi_1$ are given initial data, and the assumptions of weak damping terms $g_1, \ g_2$ and nonlinear source terms $f_1, \ f_2$ will be given in Section 2 by Assumption 2.1 and Assumption 2.2, respectively.

In the physical view, elastic solid with voids is an important extension of the classical elasticity theory. It allows the processing of porous solids in which the matrix material is elastic and the interstices are void of material (see ^[8,20] and references therein). Porous media reflects the properties of many materials in the real world, including rocks, soil, wood, ceramics, pressed powder, bones, natural gas hydrates and so on. Due to the diversity of porous media and its special physical properties, such models were widely applied in the past few decades in the petroleum industry, engineering, etc (see ^{[1,12,13,16,17,19]}).

As mathematical efforts, Goodman and Cowin ^[2,8] established the continuum theory and the variational principle of granular materials. Then Nunziato and Cowin ^[3,18] developed the linear and nonlinear theories of porous elastic materials. In recent years, the study of the porous elastic system also attracted a lot of attention ^{[5,6,7,21,22]}. We particularly mention that Freitas et.al. in ^[5] studied the problem (1.1) and proved the global existence and finite time blowup of solutions. Especially, they built up the continuous dependence on initial data of the local solution in the following version

which can also be extended to the global solution with the same form. By denoting $z = (u, \phi)$ and $\tilde{z} = (\tilde{u}, \tilde{\phi})$ the global solutions to problem (1.1) corresponding to the initial data $z_{0}$, $z_{1}$ and $\tilde{z}_{0}$, $\tilde{z}_{1}$, respectively, $\widehat{E}(0)$ is the distance of two sets of different initial data

and

that is

and $\widehat{E}(t)$ is the distance of solutions induced by these two sets of different initial data

The growth estimate (1.2) indicates that the growth of the distance of solutions $\widehat{E}(t)$ is bounded by an exponential growth bound with time $t$. In other words, as the time $t$ goes to infinity, the distance of solutions $\widehat{E}(t)$ of the system is bounded by a very large bound, by which it is hard to explain the solutions $z$ and $\tilde{z}$ of such a dissipative system with the initial data $z_0, z_1$ and $\tilde{z}_0, \tilde{z}_1$, respectively, as both of them are expected to decay to zero as the time $t$ goes to infinity. Hence, the estimate on the growth of the distance of solutions $\widehat{E}(t)$ is proposed to be improved to reflect the decay properties with time $t$ to be consistent with the dissipative behavior of the system. To achieve this, the efforts in the present paper are illustrated by two new continuous dependence results on the initial data for the global-in-time solution. Especially, it is found that the system with the linear damping term behaves differently from that with the nonlinear damping term. Hence in the present paper, we adopt two different estimate strategies to deal with the problem and derive two different conclusions:

$({\rm{i}})$ For the linear damping case, i.e., $g_1(u_t)$ and $g_2(\phi_t)$ take the linear form and satisfy Assumption 2.1, we have

where the positive constants $C_1, C_2, C_3, a, \rho$ are independent of initial data.

$({\rm{ii}})$ For the nonlinear damping case, i.e., $g_1(u_t)$ and $g_2(\phi_t)$ take the nonlinear form and satisfy Assumption 2.1, we have

where $0 < \kappa < 1$, and the positive constants $C_5, C_6, C_7, b_0$ are dependent of initial data.

By observing (1.3) and (1.4), we find that these two continuous dependence results can reasonably reflect the decay property of the dissipative system (1.1). The difference between (1.3) and (1.4) is that the parameters in (1.3) do not depend on the initial data, while the parameters in (1.4) depend on the initial data. Hence although (1.3) and (1.4) are in the similar form, we present and prove them separately.

Additionally, to develop the finite time blowup of the solution to problem (1.1) at the arbitrary positive initial energy level derived in ^[22], we estimate the lower bound of the blowup time in the present paper for the nonlinear weak damping case by noticing that the linear weak damping case was discussed in ^[22]. For more relative works on the blowup of solutions to the hyperbolic equations at high initial energy, please refer to ^{[10,11,14,15,25]}. We can also refer to ^[9,23,24] for the works about the blowup of solutions to parabolic equations.

The rest of the present paper is organized as follows. In Section 2, we give some notations, assumptions about damping terms and source terms, and functionals and manifolds for the potential well theory. In Section 3, we deal with the continuous dependence on initial data of the global solution for the linear weak damping case. In Section 4, we establish the continuous dependence on initial data of the global solution for the nonlinear weak damping case. In Section 5, we estimate the lower bound of blowup time at the arbitrarily positive initial energy level for the nonlinear weak damping case.

2.   Preliminaries

2.1.   Notations and assumptions

We denote the $L^{2}$-inner product by

and the norm of $L^{p}(0, L)$ by

As we are dealing with the system of two equations, for $z = (u, \phi)$ and $\hat{z} = (\hat{u}, \hat{\phi})$, we introduce

and

Further, we consider the Hilbert space

with inner products given by

for $z = (u, \phi)$, $\hat{z} = (\hat{u}, \hat{\phi})$, where $\mu$, $\delta$, $\xi$, $b$ are the coefficients of the terms in the equations in problem (1.1). Therefore, we have

The norm $\|z\|_{V}$ is equivalent to the corresponding usual norm on $V$, i.e., ${H_{0}^1(0, L)}\times {H_{0}^1(0, L)}$, introduced in ^[20]. For $1 < q < +\infty$, we define

which means

for $z\in V$. Here, due to $H^{1}_{0}(0, L)\hookrightarrow L^{q}(0, L)$ for $1 < q < +\infty$, we see $0 < R_{q} < +\infty$. And we denote

and

where $f_{j}(u, \phi)$, $j = 1, 2$, are the source terms, and $g_{1}(u_{t})$ and $g_{2}(\phi_{t})$ are the damping terms in the equations in problem (1.1).

We give the following assumptions about damping terms, i.e., $g_{1}(u_{t})$ and $g_{2}(\phi_{t})$, and source terms, i.e., $f_{j}(u, \phi)$, $j = 1, 2$, in the equations in problem (1.1).

Assumption 2.1. (Damping terms) Let $g_{1}, g_{2}: \mathbb{R}\rightarrow \mathbb{R}$ be continuous, monotone increasing functions with $g_{1}(0) = g_{2}(0) = 0$. In addition, there exist positive constants $\alpha > 0$ and $\beta > 0$ such that

${\rm{(i)}}$ for $|s|\geq1$

and

${\rm{(ii)}}$ for $|s| < 1$

and

Assumption 2.2. (Source terms) For the functions $f_{j}\in C^{1}(\mathbb{R}^{2})$, $j = 1, 2$, there exists a positive constant $C$ such that

where $\eta = (\eta_{1}, \eta_{2})\in\mathbb{R}^{2}$, $f_{j}(\eta) = f_{j}(\eta_{1}, \eta_{2})$, $j = 1, 2$, and

There exists a nonnegative function $F\in C^{2}(\mathbb{R}^{2})$ satisfying

and

for all $\lambda > 0$, where $F(\eta) = F(\eta_{1}, \eta_{2})$ and

According to ^[5], Assumption 2.2 implies that there exists a constant $M > 0$ such that

2.2.   Potential well

Next, we recall some functionals and manifolds for the potential well theory. We recall the potential energy functional

and the Nehari functional

The energy functional is defined as

And the Nehari manifold is defined as

Then we can define the depth of the potential well

By above, we introduce the stable manifold

Next, since we need to apply the decay rate of the energy in investigating continuous dependence on the initial data of the solution, we recall the following notations used in the investigation of the decay rate of the energy in ^[5]

where

and $\mathcal{M}(s)$ attains the maximum value at

Here, Proposition 2.11 in ^[5] shows the fact $\hat{d}\leq d$.

3.   Continuous dependence on initial data of the global solution for linear weak damping case

In this section, we consider the model equations in (1.1) with the linear weak damping terms, i.e., $r = m = \hat{r} = \hat{m} = 1$. First, we need the following decay result of the energy.

Lemma 3.1. (Decay of the energy) Let Assumption 2.1 and Assumption 2.2 hold with $r = m = \hat{r} = \hat{m} = 1$. For any $0 < \sigma < 1$, if $\mathcal{E}(z_{0}, z_{1}) < \sigma \hat{d}$ and $z_{0}\in W$, then one has

for $t > 0$, where $\lambda_{0}$ and $K_{0}$ will be defined in the proof.

Proof. We define

where $\varepsilon > 0$. Here, according to Cauchy-Schwartz inequality, Young inequality, and (2.5), we have

and

According to Theorem 2.12(ⅳ) in ^[5], we know

which means that (3.2) and (3.3) turn to

and

According to (3.5) and (3.6), we know

where

and

We calculate the derivative of the auxiliary functional $H(t)$ with respect to time $t$ as

In (3.8), we have

Here, the notation $\nabla F$ is defined by (2.13). Thus, according to (2.11), we know $\nabla F(z) = \mathcal{F}(z)$, which means (3.9) turns to

Testing the both sides of the first equation in (1.1) by $u_{t}$ and integrating both sides over $[0, L]$, we have

And testing the both sides of the second equation in (1.1) by $\phi_{t}$ and integrating both sides over $[0, L]$, we have

By substituting (3.11) and (3.12) into (3.10), we have

Next, we use Assumption 2.1 to deal with (3.13). In Assumption 2.1, for $|s|\geq1$, according to (2.6) with $m = 1$ and (2.7) with $r = 1$, we know that

Then taking the absolute value of (3.14) gives

For $|s| < 1$, according to (2.8) with $\hat{m} = 1$ and (2.9) with $\hat{r} = 1$, we know that (3.15) also holds. Meanwhile, since $g_{1}(0) = g_{2}(0) = 0$ and $g_{j}(s)$, $j = 1, 2$, are assumed to be the increasing functions, for $j = 1, 2$, we know $g_{j}(s) > 0$ for $s > 0$ and $g_{j}(s) < 0$ for $s < 0$, which gives $g_{j}(s)s\geq0$, $j = 1, 2$, for $s\in \mathbb{R}$. Thus, we have

which makes (3.13) turn to

We deal with the term $\varepsilon\left(z_{tt}, z\right)$ in (3.8). Testing the both sides of the first equation in problem (1.1) by $u$ and integrating both sides over $[0, L]$, we have

And testing the both sides of the second equation in problem (1.1) by $\phi$ and integrating both sides over $[0, L]$, we have

By (3.17) plus (3.18), we have

According to (3.16) and (3.19), we know that (3.8) turns to

Next, we deal with the term $\varepsilon\left|\left(\mathcal{G}(z_{t}), z\right)\right|$ in (3.20). By using (3.15) and Hölder inequality, we know

Then, We deal with $\varepsilon\left(\mathcal{F}(z), z\right)$ in (3.20). Here, we first need to give

For all $\lambda > 0$, taking the derivative of both sides of (2.12) with respect to $\lambda$, we know

where $\nabla F$ is defined by (2.13). By taking $\lambda = 1$ in (3.23) and using (2.11), we obtain (3.22). According to (3.22) and (2.14), we have

By using (2.5), (3.24) turns to

Then, we estimate the term $\|z\|^{p-1}_{V}$ in (3.25). According to Theorem 2.12 (ⅱ) in ^[5], we know $z(t)\in W$ for $t > 0$. By using $I(z(t)) > 0$, i.e., $z(t)\in W$, we have

which means

Meanwhile, according to (3.16), i.e.,

we have $\mathcal{E}(z(t), z_{t}(t))\leq\mathcal{E}(z_{0}, z_{1})$. Thus, we know

i.e.,

for $t > 0$, which implies that (3.25) turns to

Due to $\mathcal{E}(z_{0}, z_{1}) < \sigma \hat{d}$ being assumed, we know that (3.28) turns to

where $\hat{d}$ is defined by (2.17). Substituting (3.21) and (3.29) into (3.20), we have

By using Young inequality for $\delta_{0} > 0$ and inequality (2.5) for $q = 2$, we know that (3.30) turns to

In (3.31), we choose $\delta_{0} > 0$ to make $1-\sigma^{\frac{p-1}{2}}-\beta\delta_{0} R_{2} > 0$ hold, where $1-\sigma^{\frac{p-1}{2}} > 0$ due to $\sigma\in(0, 1)$. Then, we select $\varepsilon > 0$ such that $\alpha-\varepsilon-\frac{\varepsilon\beta}{\delta_{0}} > 0$ and

To deal with (3.31), we first have

According to Theorem 2.12 (ⅳ) in ^[5], (3.32) turns to

Due to (3.7), i.e., $H(t)\leq \alpha_{2}\mathcal{E}(z(t), z_{t}(t))$, (3.33) turns to

Thus, we know that (3.31) implies

where

By using Gronwall's inequality, (3.35) gives

According to (3.7), (3.37), and the assumptions $\mathcal{E}(z_{0}, z_{1}) < \sigma\hat{d}$ and $0 < \sigma < 1$, we have

where

Theorem 3.2. (Continuous dependence on initial data for linear weak damping case) Let Assumption 2.1 and Assumption 2.2 hold with $r = m = \hat{r} = \hat{m} = 1$. For any $0 < \sigma < 1$, suppose $\mathcal{E}(z_{0}, z_{1}) < \sigma \hat{d}$, $z_{0}\in W$, $\mathcal{E}(\tilde{z}_{0}, \tilde{z}_{1}) < \sigma \hat{d}$ and $\tilde{z}_{0}\in W$. Let $z = (u, \phi)$ and $\tilde{z} = (\tilde{u}, \tilde{\phi})$ be the global solutions to problem (1.1) with the initial data $z_{0}$, $z_{1}$, and $\tilde{z}_{0}$, $\tilde{z}_{1}$, respectively. Then one has

where

$\lambda_{0}$ and $K_{0}$ are defined by (3.36) and (3.39), respectively, $R_{4(p-1)}$ is the best embedding constant defined in (2.4) taking $q = 4(p-1)$,

and

Proof. We denote $w: = z-\tilde{z}$. According to the proof of Theorem 2.5 in ^[5], we notice that

holds by Assumption 2.1 and Assumption 2.2. In the following, we shall finish this proof by considering the following two steps. In Step $\rm{I}$, we shall derive a similar estimate of the growth of $\widehat{E}(t)$ to (135) in ^[5]. As we build this estimate for the global solution instead of the local solution treated in ^[5], we have to rebuild all the necessary estimates based on the conditions for global existence theory.

Step Ⅰ: Global estimate of $\widehat{E}(t)$ for global solution.

We estimate the term $\int_{0}^{t}\int_{0}^{L}\left(\mathcal{F}(z(\tau))-\mathcal{F}(\tilde{z}(\tau))\right) w_{t}(\tau){\rm d}x{\rm d}\tau$ in (3.43) as follows

Here, according to the proof of Lemma 3.2 in ^[5], we notice that (2.10) in Assumption 2.2 gives

which means (3.44) turns to

Next, we deal with $A_{1}$ and $A_{2}$ separately. For $A_{1}$, by Hölder inequality and Young inequality, we have

By the similar process, we can deal with $A_{2}$ as

According to (3.47), (3.48) and Hölder inequality, we know that (3.46) turns to

In (3.49), by noticing $z = (u, \phi)$, $\tilde{z} = (\tilde{u}, \tilde{\phi})$, we see that

By using Hölder inequality and Young inequality, we know (3.50) turns to

Next, we deal with $\int_{0}^{L}(|u|^{p-1}+|\tilde{u}|^{p-1}+|\phi|^{p-1}+|\tilde{\phi}|^{p-1}+1)^{4}{\rm d}x$ in (3.49). For $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$, $k_{5}\geq0$, we have

From above observation, we have

According to (3.51) and (3.52), (3.49) turns to

By using (2.5), we know that (3.53) turns to

According to (3.27) and the assumptions $\mathcal{E}(z_{0}, z_{1}) < \sigma \hat{d}$ and $\mathcal{E}(\tilde{z}_{0}, \tilde{z}_{1}) < \sigma \hat{d}$, we have

and

Substituting (3.55) and (3.56) into (3.54), we have

Due to (3.57), we know

Substituting (3.58) into (3.43), we have

Then, by a variation of Gronwall's inequality (see Appendix), we have

As the growth estimate (3.60) we derived in Step $\rm{I}$ does not reflect the decay of the solution, we shall deal with the decay terms and the non-decay terms separately in Step $\rm{II}$ to upgrade the results obtained in Step $\rm{I}$, i.e., (3.60), by giving an improved estimate to reflect the dissipative property of the system (1.1).

Step Ⅱ: Decay estimate of $\widehat{E}(t)$.

According to (3.46), we have

By the similar process dealing with $A_{1}$ and $A_{2}$, we can treat $A_{3}$ and $A_{4}$ as

and

By the similar process of obtaining (3.54), i.e.,

we can use (3.62) and (3.63) to give

Due to (3.26), (2.16) and Lemma $3.1$, we know

and

According to (3.65) and (3.66), we have

and

which mean

By substituting (3.67) into (3.64), we obtain

Substituting (3.68) into (3.61) and using Hölder inequality and (2.5), we have

According to (3.60), we know

i.e.,

for $0 < a < 1$. Meanwhile, combining (3.65) and (3.66), we also have

i.e.,

and

Combining (3.70), (3.71) and (3.72), we have

We choose $0 < a < \min\left\{\frac{2\lambda_{0}}{\bar{M}C+\lambda_{0}}, 1\right\}$ such that

in (3.74). Meanwhile, according to (3.70), (3.72) and (3.73), we notice that

Due to (3.74) and (3.76), we see that (3.69) turns to

By substituting (3.77) into (3.43), we obtain

where

Here, according to (3.79), we notice that $D\leq\frac{N}{\lambda_{1}}$, which means that (3.78) turns to

i.e.,

We define

Thus, we can rewrite (3.81) as

By applying Gronwall's inequality, (3.83) gives

which means (3.80) turns to

For $0 < \rho < 1$, according to (3.84), we have

Here, by using Young inequality, we know

According to (3.65) and (3.66), we know

and

By substituting (3.87) and (3.88) into (3.86), we have

which means that (3.85) turns to (3.40).

4.   Continuous dependence on initial data of the global solution for nonlinear weak damping case

In this section, we consider the continuous dependence of the global solution on the initial data for the nonlinear weak damping case of the model equations in problem (1.1) by supposing that $m\geq1$, $r\geq1$, and $\hat{m} = \hat{r} = 1$ in Assumption 2.1, which means that the weak damping terms $g_j(s)$, $j = 1, 2$, take the nonlinear form for $|s|\geq1$ and linear form for $|s| < 1$. These conditions are applied to improve the estimate (1.2) and reflect the decay property of (1.1), which was clearly clarified in Corollary 2.14 in ^[5], that is, the condition $\hat{m} = \hat{r} = 1$ is necessary to obtain the exponential decay of the energy, which helps to get the exponential decay, and the absence of such linear condition can only lead to the polynomial decay of the energy. Hence although we discuss the nonlinear weak damping case here, we still need to assume that the terms $g_j(s)$, $j = 1, 2$, take the linear form for $|s| < 1$.

Theorem 4.1. (Continuous dependence on initial data for nonlinear weak damping case) Let Assumption 2.1 and Assumption 2.2 hold with $\hat{r} = \hat{m} = 1$, $\mathcal{E}(z_{0}, z_{1}) < \mathcal{M}(s_0-\nu)$, $\mathcal{E}(\tilde{z}_{0}, \tilde{z}_{1}) < \mathcal{M}(s_0-\nu)$, $\|z_{0}\|_{V}\leq s_{0}-\nu$, and $\|\tilde{z}_{0}\|_{V}\leq s_{0}-\nu$ for some $\nu > 0$. Let $z = (u, \phi)$ and $\tilde{z} = (\tilde{u}, \tilde{\phi})$ are the global solutions to the problem (1.1) with the initial data $z_{0}$, $z_{1}$, and $\tilde{z}_{0}$, $\tilde{z}_{1}$, respectively, where $\mathcal{M}$ and $s_{0}$ are defined in (2.18) and (2.19), respectively. Then one has

where

and $\theta > 0$, $\tilde{\theta} > 0$, and $T > 0$ satisfy

and

$\bar{M}$ is defined in (3.42),

and

Proof. Due to Corollary 2.14 in ^[5], for any $T > 0$, we know that there exist $\theta$ and $\tilde{\theta}$ to make (4.3) and (4.4) hold, where $\theta$ is dependent on $\mathcal{E}(z_{0}, z_{1})$ and $T$, and $\tilde{\theta}$ is dependent on $\mathcal{E}(\tilde{z}_{0}, \tilde{z}_{1})$ and $T$. According to Proposition 2.11 in ^[5], the assumptions $\mathcal{E}(z_{0}, z_{1}) < \mathcal{M}(s_0-\nu)$, $\|z_{0}\|_{V}\leq s_{0}-\nu$, and $\mathcal{E}(\tilde{z}_{0}, \tilde{z}_{1}) < \mathcal{M}(s_0-\nu)$, $\|\tilde{z}_{0}\|_{V}\leq s_{0}-\nu$ give $z_{0}\in W$ and $\tilde{z}_{0}\in W$, respectively. Here $\mathcal{M}(s_0-\nu) < \hat{d}$ can be observed according to (2.17). Thus, we know (3.26) also holds. According to these facts and (2.16), we have

and

Due to (4.2), we know that (4.6) and (4.7) turn to

and

respectively. According to (4.8) and (4.9), we have

and

which mean

Next, we need to use the estimate (3.64) to continue this proof. More precisely, by substituting (4.10) into (3.64), we obtain

By substituting (4.11) into (3.61) and the similar process of obtaining (3.69), we have

According to (3.60), we know

i.e.,

where $b_0$ is defined by (4.5). Meanwhile, combining (4.8) and (4.9), we know

i.e.,

and

where $b_0 > 0$ and $1-b_0 > 0$ are ensured by (4.5). According to (4.13), (4.14) and (4.15), we have

According to (4.5), we have

i.e.,

in (4.17). Meanwhile, according to (4.13), (4.15) and (4.16), we notice that

Due to (4.17) and (4.19), we know that (4.12) turns to

By substituting (4.20) into (3.43), we obtain

where

Here, according to (4.22), we notice that $D_{1}\leq\frac{N_{1}}{\lambda_{2}}$, which means that (4.21) turns to

i.e.,

By similar process of obtaining (3.84), we have

For $0 < \kappa < 1$, according to (4.25), we know

By the similar process of obtaining (3.86), we have

According to (4.8) and (4.9), we know

and

By substituting (4.28) and (4.29) into (4.27), we have

which means that (4.26) turns to (4.1).

5.   Lower bound estimate of blowup time for positive initial energy and nonlinear weak damping case

The finite time blowup at the positive initial energy level was established for the linear weak damping case and nonlinear weak damping case in ^[22], and for the linear weak damping case, the lower and upper bounds of the blowup time were also estimated there. Hence in this section, we shall estimate the lower bound of the blowup time at the positive initial energy level for the nonlinear weak damping case.

Theorem 5.1. (Lower bound of blowup time for positive initial energy and nonlinear weak damping case) Let Assumption 2.1 and Assumption 2.2 hold, and $\mathcal{E}(z_{0}, z_{1})\geq 0$. Suppose $z(x, t)$ is the solution to problem (1.1). If $z(x, t)$ blows up at a finite time $T_{0}$, then we have the estimate of blowup time

where

and

Proof. Let $z = (u, \phi)$ be a weak solution to problem (1.1). We suppose that such solution blows up at a finite time $T_{0}$. Our goal is to obtain an estimate of the lower bound of $T_{0}$.

For $t\in [0, T_{0})$, we define

then, by Hölder inequality and Young inequality, we have

Next task is to estimate the terms in the last line of (5.2). By (2.14) and (2.16), we obtain

According to (3.16), we know

where $\mathcal{E}(z_{0}, z_{1})\geq 0$. We notice that (5.3) and (5.4) give

which means

and

Combining (2.5) and (5.6), we see

By substituting (5.7) and (5.8) into (5.2), we have

We consider the function $h(x): = x^{p}, x > 0, p > 1$. Since $h^{''}(x) = p(p-1)x^{p-2} > 0$, $h(x)$ is a convex function. Thus it gives that

that is to say

Then, due to $\mathcal{E}(z_{0}, z_{1})\geq0$ and $G(t)\geq0$, we can get

which means that (5.9) turns to

i.e.,

Recalling the assumption that the solution of problem (1.1) blows up in finite time $T_{0}$, we have

Then, integrating both sides of (5.11) on $(0, T_{0})$ and combining (5.12), we get

Thus, the proof of Theorem 5.1 is completed.

6.   Appendix: a variation of Gronwall's inequality

In Sept Ⅰ of the proofs of Theorem $3.2$, by the classical form of Gronwall's inequality (integral form) shown in Appendix B.2 of ^[4], we know that (3.59) gives

In (6.1), the growth order of the distance of the solutions, i.e., $\widehat{E}(t)$, is controlled by the product of an exponential function and a polynomial function, which is higher than that in (1.2) established for the local solution. In Sept Ⅱ of the proofs of Theorem $3.2$, in order to build the growth estimate of $\widehat{E}(t)$ in the same form as (1.2) for the global solution, i.e., (3.60), we need the following variation of Gronwall's inequality.

Proposition 6.1. For a nonnegative, summable function $\zeta(t)$ on $[0, \bar{T}]$ with satisfying

for the constants $\bar{C}_{1}, \bar{C}_{2}\geq0$, one has

for a.e. $0\leq t\leq \bar{T}$.

Proof. We use the similar idea of proving the classical form of Gronwall's inequality shown by Appendix B in ^[4] to give the proofs. We first define the auxiliary function

By direct calculation, we have

Substituting (6.2) into (6.5), we have

which means

i.e.,

According to (6.4) and (6.6), we have

Substituting (6.7) into (6.2), we obtain (6.3).

Use of AI tools declaration

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

Acknowledgments

Runzhang Xu was supported by the National Natural Science Foundation of China (12271122) and the Fundamental Research Funds for the Central Universities. Chao Yang was supported by the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072022GIP2403).

Conflict of interest

The authors declare there is no conflict of interest.