Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application

1.   Introduction

In this paper, we consider the initial value problem for a high-order nonlinear dispersive wave equation

where $k\in N_{0}$ ($N_{0}$ denotes the set of nonnegative integers) and $u$ stands for the unknown function on the line $\mathbb{R}$. Equation (1.1) was known as gkCH equation ^[18], which admits single peakon and multi-peakon traveling wave solutions, and possesses conserved laws

It is shown in ^[18] that this equation is well-posedness in Sobolev spaces $H^{s}$ with $s > 3/2$ on both the circle and the line in the sense of Hadamard by using a Galerkin-type approximation scheme. That is, the data-to-solution map is continuous. Furthermore, it is proved in ^[18] that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates.

For $k = 1$, we obtain the integrable equation with quadratic nonlinearities

which was derived by Camassa and Holm ^[1] and by Fokas and Fuchssteiner ^[12]. It was called as Camassa-Holm equation. It describes the motion of shallow water waves and possesses a Lax pair, a bi-Hamiltonian structure and infinitely many conserved integrals ^[1], and it can be solved by the inverse scattering method. One of the remarkable features of the CH equation is that it has the single peakon solutions

and the multi-peakon solutions

where $p_{i}(t)$, $q_{i}(t)$ satisfy the Hamilton system ^[1]

with the Hamiltonian $H = \frac{1}{2}\sum_{i, j = 1}^{N}p_{i}p_{j}e^{|q_{i}|}$. It is shown that those peaked solitons were orbitally stable in the energy space ^[6]. Another remarkable feature of the CH equation is the so-called wave breaking phenomena, that is, the wave profile remains bounded while its slope becomes unbounded in finite time ^[7,8,9]. Hence, Eq (1.4) has attracted the attention of lots of mathematicians. The dynamic properties related to the equation can be found in ^{[3,4,5,10,11,13,14,15,16,17,22,23,24,25,26,34,35,36]} and the references therein.

For $k = 2$, we obtain the integrable equation with cubic nonlinearities

which was derived by Vladimir Novikov in a symmetry classification of nonlocal PDEs ^[29] and was known as the Novikov equation. It is shown in ^[29] that Eq (1.5) possesses soliton solutions, infinitely many conserved quantities, a Lax pair in matrix form and a bi-Hamiltonian structure. Equation (1.5) can be thought as a generalization of the Camassa-Holm equation. The conserved quantities

and

play an important role in the study of the dynamic properties related to the Eq (1.5). More information about the Novikov equation can be found in Himonas and Holliman ^[19], F. Tiglay ^[30], Ni and Zhou ^[28], Wu and Yin ^[31,32], Yan, Li and Zhang ^[33], Mi and Mu ^[27] and the references therein.

Inspired by the reference cited above, the objective of this paper is to investigate the dynamic properties for the Problems (1.1) and (1.2). More precisely, we firstly establish two blow up criteria and derive a lower bound of the maximal existence time. Then for $k = 2^{p}+1$, $p$ is nonnegative integer, we derive two blow-up phenomena under different initial data. Motivated by the idea from Chen etal's work^[2], we apply the characteristic dynamics of $P = \sqrt{2}u-u_{x}$ and $Q = \sqrt{2}u+u_{x}$ to deduce the first blow-up phenomenon. For the Problems (1.1) and (1.2), in fact, the estimates of $P$ and $Q$ can be closed in the form of

where $\alpha(u)\geq0$ and the nonlocal term $\Theta_{i}$ $(i = 1, 2)$ can be bounded by the estimates from Problems (1.1) and (1.2). From (1.6) the monotonicity of $P$ and $Q$ can be established, and hence the finite-time blow-up follows. In blow-up analysis, one problematic issue is that we have to deal with high order nonlinear term $u^{k-2}u^{3}_{x}$ to obtain accurate estimate. Luckily, we overcome the problem by finding a new $L^{2k}$ estimate $\parallel u_{x}\parallel_{L^{2k}}\leq e^{c_{0}t}\parallel u_{0x}\parallel_{L^{2k}}+c_{0}t$ (see Lemma 4.1). It is shown in ^[18] that Eq (1.1) has peakon travelling solution $u(t, x) = c^{\frac{1}{k}}e^{-|x-ct|}$. Follow the Definition 2.1, we show that the peakon solutions are global weak solutions.

The rest of this paper is organized as follows. For the convenience, Section 2 give some preliminaries. Two blow-up criteria are established in Section 3. Section 4 give two blow-up phenomena. In Section 5, we prove that the peakon solutions are global weak solutions.

2.   Preliminaries

We rewrite Problems (1.1) and (1.2) as follows

which is also equivalent to

Recall that

and $\ast$ denotes the convolution product on $\mathbb{R}$, defined by

Lemma 2.1. Given initial data $u_{0}\in H^{s}$, $s > \frac{3}{2}$, the function $u$ is said to be a weak solution to the initial-value Problem (1.1) and (1.2) if it satisfies the following identity

for any smooth test function $\varphi(t, x)\in C^{\infty}_{c}([0, T)\times \mathbb{R})$. If $u$ is a weak solution on $[0, T)$ for every $T > 0$, then it is called a global weak solution.

The characteristics $q(t, x)$ relating to (2.3) is governed by

Applying the classical results in the theory of ordinary differential equations, one can obtain that the characteristics $q(t, x)\in C^{1}([0, T)\times \mathbb{R})$ with $q_{x}(t, x) > 0$ for all $(t, x)\in [0, T)\times \mathbb{R}$. Furthermore, it is shown in ^[21] that the potential $y = u-u_{xx}$ satisfies

3.   Blow-up criteria and a lower bound of the maximal existence time

In this section, we investigate blow-up criteria and a lower bound of the maximal existence time. Now, we firstly give the first blow-up criterion.

Theorem 3.1. Let $u_{0}\in H^{s}(\mathbb{R})$ with $s > \frac{3}{2}$. Let $T > 0$ be the maximum existence time of the solution $u$ to the Problem (1.1) and (1.2) with the initial data $u_{0}$. Then the correspondingsolution $u$ blows up in finite time if and only if

Proof. Applying a simple density argument, it suffices to consider the case $s = 3$. Let $T > 0$ be the maximal time of existence of solution $u$ to the Problem (1.1) and (1.2) with initial data $u_{0}\in H^{3}(\mathbb{R})$. Due to $y = u-u_{xx}$, by direct computation, one has

So,

Multiplying Eq (2.3) by $2y$ and integrating by parts, we obtain

If there is a $M > 0$ such that $u^{k-1}u_{x} > -M$, from (3.3) we deduce

By virtue of Gronwall's inequality, one has

This completes the proof of Theorem 3.1.

Now we give the second blow-up criterion.

Theorem 3.2. Let $u_{0}\in H^{s}(R)$ with $s > \frac{3}{2}$. Let $T > 0$ be the maximum existence time of the solution $u$ to the Problem (1.1) with the initial data $u_{0}$. Then the correspondingsolution $u$ blows up in finite time if and only if

To complete the proof of Theorem 3.2, the following two lemmas is essential.

Lemma 3.1. (^[11]) The following estimates hold

$(i)$ For $s\geq0$,

$(ii)$ For $s > 0$,

Lemma 3.2. (^[20]) Let $r > 0$. If $u\in H^{r}\cap W^{1, \infty}$ and $v\in H^{r-1}\cup L^{\infty}$, then

where $\Lambda = (1-\partial^{2}_{x})^{\frac{1}{2}}$.

Proof. Applying $\Lambda^{r}$ with $r\geq1$ to two sides of Eq (2.1) and multiplying by $\Lambda^{r}u$ and integrating on $\mathbb{R}$

where $f(u) = \partial_{x}(1-\partial^{2}_{x})^{-1}\bigg[u^{k+1}+\frac{2k-1}{2}u^{k-1}u^{2}_{x}\bigg] +(1-\partial^{2}_{x})^{-1}\bigg[\frac{k-1}{2}u^{k-2}u^{3}_{x}\bigg]$.

Notice that

where Lemmas 3.1, 3.2 and the inequality $\parallel u^{k}\parallel_{H^{r}}\leq k\parallel u\parallel^{k-1}_{L^{\infty}}\parallel u\parallel_{H^{r}}$ were used.

In similar way, from Lemmas 3.1 and 3.2, we have

Notice that

and

Thus, we obtain

It follows from (3.9), (3.10) and (3.15) that

Therefore, if there exists a positive number $M$ such that $\parallel u_{x}\parallel_{L^{\infty}}\leq M$. The Gronwall' inequality gives rise to

This completes the proof of Theorem 3.2.

Theorem 3.3. Assume that $u_{0}\in H^{s}$ with $s > \frac{3}{2}$, $N = max\{3, \frac{8k-3}{6}\}$. Let $\parallel u_{x}(0)\parallel_{L^{\infty}} < \infty$, $T > 0$ be the maximum existence time of the solution$u$ to (2.1) and (2.2) with the initial data $u_{0}$. Then $T$ satisfies

Proof. Notice that the Eq (2.1) is equivalent to the following equation

where $G(x) = \frac{1}{2}e^{-|x|}$ is the Green function of $(1-\partial^{2}_{x})^{-1}$. Multiplying the above equation by $u^{2n-1}$ and integrating the resultant with respect to $x$, in view of Hölder's inequality, we obtain

and

Combining (3.19)–(3.21), integrating over $[0, t]$, it follows that

Letting $n$ tend to infinity in the above inequality, we have

Differentiating (3.18) with respect to $x$, we obtain

Multiplying the above equation by $u^{2n-1}_{x}$ and integrating the resultant with respect to $x$ over $\mathbb{R}$, still in view of Hölder's inequality, we get following estimates

and

Combining (3.25)–(3.31), it gives rise to

Integrating (3.32) over $[0, t]$ and letting $n$ tend to infinity, it follows that

Combining (3.23) and (3.33), we deduce that

Define $h(t) = \parallel u\parallel_{L^{\infty}}+\parallel u_{x}\parallel_{L^{\infty}}$ and $N = max\{3, \frac{8k-3}{6}\}$. Then $h(0) = \parallel u(0)\parallel_{L^{\infty}}+\parallel u_{x}(0)\parallel_{L^{\infty}}$. One can obtain

Solving (3.35), we get

Therefore, let $T = \frac{1}{2N\parallel u_{0}\parallel^{k-2}_{H^{1}}(\parallel u(0)\parallel_{L^{\infty}}+\parallel u_{x}(0)\parallel_{L^{\infty}})^{2}}$, for all $t\leq T$, $\parallel u_{x}\parallel_{L^{\infty}}\leq h(t)$ holds.

4.   Blow-up phenomena

For the convenience, we firstly give several Lemmas.

Lemma 4.1. ($L^{2k}$ estimate)Let $u_{0}\in H^{s}$, $s\geq3/2$ and $\parallel u_{0x}\parallel_{L^{2k}} < \infty$, $k = 2^{p}+1$, $p$ is nonnegative integer. Let $T$ be the lifespan of the solution to problem (1.1). The estimate

holds for $t\in [0, T)$.

Proof. Differentiating the first equation of Problem (2.1) respect with to $x$, we have

Multiplying the above equation by $2ku^{2k-1}_{x}$ and integrating the resultant over $\mathbb{R}$, we obtain

In view of Hölder's inequality, we derive the following estimates

and

$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$

On the other hand,

and

$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$

Therefore, if $2k-2 = 2$ $(k = 2)$, combining (4.3) and (4.7), we have

If $2k-4 = 2$ $(k = 3)$, combining (4.4) and (4.8), we have

If $2k-8 = 2$ $(k = 5)$, combining (4.5) and (4.9), we have

and if $2k-16 = 2$ $(k = 9)$, combining (4.6) and (4.10), we have

$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$

Therefore, if and only if $k = 2^{p}+1$, $p$ is nonnegative integer, the following inequality follows

where $c_{0} = c_{0}(\parallel u_{0}\parallel_{H^{1}(\mathbb{R})})$.

In view of Gronwall's inequality, we have for $t\in [0, T)$

This completes the proof of Lemma 4.1.

Remark. In case of $p = 0$, then $k = 2$, and we obtain

which is exactly the same as the Lemma 3.2 in ^[23].

Lemma 4.2. Given that $u_{0}\in H^{s}$, $s\geq3/2$. Let $\parallel u_{0x}\parallel_{L^{2k}} < \infty$, $k = 2^{p}+1$, $p$ is nonnegative integer and $T$ be the lifespan of the solution to Problem (1.1) and (1.2). The estimate

holds.

Proof. Applying Hölder'inequality, Lemma 4.1 and (1.3), we have

Lemma 4.3. Given that $u_{0}\in H^{s}$, $s\geq3$. For $k = 2^{p}+1$ and $p$ is nonnegative integer. Then

Proof. From the Young's inequality, Lemma 4.2 and (1.3), it follows that

Therefore,

Integrating over the time interval $[0, t]$ yields

So,

Theorem 4.1. Let $\parallel u_{0x}\parallel_{L^{2k}} < \infty$, $k = 2^{p}+1$, $p$ is nonnegative integer and $u_{0}\in H^{s}(\mathbb{R})$ for $s > \frac{3}{2}$. Suppose that there exist some $0 < \lambda < 1$ and $x_{1}\in \mathbb{R}$ such that $\frac{1}{2}u^{k-1}_{0}(2u^{2}_{0}-u^{2}_{0x})+C^{2}\leq 0$, $\sqrt{2}u_{0} < -u_{0x}$, $u_{0}(x_{1}) > 0$ and

Then the corresponding solution $u(t, x)$ blows up in finite time $T^{\ast}$ with

where $C = \sqrt{c\alpha+c_{0}}$.

Proof. We track the dynamics of $P(t) = \big(\sqrt{2}u-u_{x}\big)(t, q(t, x_{1}))$ and $Q(t) = \big(\sqrt{2}u+u_{x}\big)(t, q(t, x_{1}))$ along the characteristics

and

Then we obtain

The expected monotonicity conditions on $P$ and $Q$ indicate that we would like to have

It is shown from assumptions of Theorem 4.1 that the initial data satisfies

Therefore, along the characteristics emanating from $x_{1}$, the following inequalities hold

and

Therefore over the time of existence the following inequalities always hold

Letting $h(t) = \sqrt{-PQ(t)}$ and using the estimate $\frac{Q-P}{2}\geq h(t)$, we have

We focus on the time interval $0\leq t\leq T_{2}: = \frac{(1-\lambda)u_{0}(x_{1})}{c_{0}+c\alpha}$, it implies that

Solving for $0\leq t\leq T_{2}$

we obtain

It is observed from assumption of Theorem 4.1 that $T_{0} < T_{2}$, (4.32) implies that $h\rightarrow +\infty$ as $t\rightarrow T^{\ast}$

Theorem 4.2. Let $\parallel u_{0x}\parallel_{L^{2k}} < \infty$, $k = 2^{p}+1$, $p$ is nonnegative integer and $u_{0}\in H^{s}(\mathbb{R})$ for $s > \frac{3}{2}$. Suppose that there exist some $0 < \lambda < 1$ and $x_{2}\in \mathbb{R}$ such that $u_{0}(x_{2}) > 0$, $u_{0x}(x_{2}) < -\sqrt{\frac{b}{a}}$ and

where $a = \frac{k}{2}\lambda^{k-1} u^{k-1}_{0}(x_{2})$ and $b = c_{0}+c\alpha$.Then the corresponding solution $u(t, x)$ blows up in finite time $T^{**}$ with

Proof. Now, we prove the blow-up phenomenon along the characteristics $q(t, x_{2})$. From (2.1), it follows that

and

Setting $M(t) = u(t, q(t, x_{1}))$ and using the Young's inequality, Lemmas 4.2 and 4.3 and (2.8) again, from (4.37), we get

where $b = c_{0}+c\alpha$. Now, we focus on the time interval $0\leq t\leq T_{3}: = \frac{(1-\lambda)u_{0}(x_{2})}{c_{0}+c\alpha}$, it implying that

Therefore, for $0\leq t\leq T_{3}$, we deduce from (4.38) that

where $a = \frac{1}{2}\lambda^{k-1} u^{k-1}_{0}(x_{2})$.

It is observed from assumption of Theorem 4.2 that $u_{0x}(x_{2}) < -\sqrt{\frac{b}{a}}$ and $T_{4} < T_{3}$. Solving (4.40) results in

where $T^{\ast\ast}\leq T_{4} = \frac{1}{2\sqrt{ab}}\ln\bigg(\frac{u_{0x}(x_{2})-\sqrt{\frac{b}{a}}}{u_{0x}(x_{2})+\sqrt{\frac{b}{a}}}\bigg)$.

5.   Peakon solutions

In this section, we will turn our attention to peakon solution for the Problem (1.1).

Theorem 5.1. The peakon function of the form

is a global weak solution to (1.1) and (1.2) in the sense of Definition 2.1.

Proof. Let $u = ae^{-|x-ct|}$ be peakon solution for the Problems (1.1) and (1.2), where $a\neq0$ is an undetermined constant. We firstly claim that

Hence, using (2.6), (5.2) and integration by parts, we derive that

On the other hand,

Directly calculate

Therefore, we obtain

Note that $G_{x} = -\frac{1}{2}sign(x)e^{-|x|}$. For $x > ct$,

We directly compute $I_{1}$ as follows

In a similar procedure, we obtain

and

Substituting (5.8)–(5.10) into (5.7), we deduce that for $x > ct$

where $\Omega = -\frac{1}{2}\frac{k(k+2)}{k+1}a^{k+1}$.

For $x < ct$,

We directly compute $\Delta_{1}$ as follows

In a similar procedure, one has

and

Therefore, from (5.13)–(5.15), we deduce that for $x < ct$

where $\Omega = \frac{1}{2}\frac{k(k+2)}{k+1}a^{k+1}$.

Due to $u = ae^{-|x-ct|}$,

To ensure that $u = ae^{-|x-ct|}$ is a global weak solution of (2.1) in the sense of Definition 2.1, we let

Solving (5.17), we get

From (5.18), we derive that

which along with (5.11), (5.16) and (5.17) gives rise to

Therefore, we conclude that

for every test function $\varphi(t, x)\in C^{\infty}_{c}([0, +\infty)\times \mathbb{R})$, which completes the proof of Theorem 5.1.

Acknowledgements

This work is supported by the GuiZhou Province Science and Technology Basic Project [Grant number QianKeHe Basic [2020]1Y011], Department of Guizhou Province Education project [grant number QianJiaoHe KY Zi [2019]124] and the GuiZhou Province Science and Technology Plan Project [Grant number QianKeHe Platform Talents [2018]5784-09].

Conflict of interest

There are no conflict of interest.