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The stability of solutions for the Fornberg–Whitham equation in \(L^{1}(\mathbb{R})\) space

Boundary Value Problems20182018:142

https://doi.org/10.1186/s13661-018-1065-0

  • Received: 26 April 2018
  • Accepted: 13 September 2018
  • Published:

Abstract

The \(L^{2}(\mathbb{R})\) conservation law of solutions for the nonlinear Fornberg–Whitham equation is derived. Making use of the Kruzkov’s device of doubling the space variables, the stability of the solutions in \(L^{1}(\mathbb{R})\) space is established under certain assumptions on the initial value.

Keywords

  • \(L^{1}\) stability
  • Local strong solutions
  • The Fornberg–Whitham equation

MSC

  • 35G25
  • 35L05

1 Introduction

In this article, we investigate the Fornberg–Whitham(FW) equation
$$ V_{t}-V_{txx}-V_{x}+\frac{3}{2}VV_{x}= \frac{9}{2}V_{x}V_{xx}+\frac {3}{2}VV_{xxx}, $$
(1)
which was first written down in Whitham [1]. The numerical and theoretical analysis of solutions for Eq. (1) are made in Fornberg and Whitham [2] in which the peakon solution
$$ V(t,x)=\frac{8}{9}e^{-\frac{1}{2}|x-\frac{4}{3}t|} $$
(2)
is found.

Recently, Holmes and Thompson [3] have established the existence and uniqueness of the FW equation in the Besov space in both non-periodic and periodic cases and discussed the sharpness of continuity on the data-to-solution map. A Cauchy–Kowalevski type result, which guarantees the existence and uniqueness of real analytic solutions for Eq. (1), is given and the blow-up criterion for solutions is obtained in [3]. Haziot [4] employs the estimates derived from the FW equation itself and some conclusions in [5] to derive sufficient conditions on the initial value which lead to wave breaking of solutions. For the detailed discussion about the discovery of wave breaking, we refer the reader to [2, 58].

We know that the dynamic properties of the Fornberg–Whitham equation are related to those of the Camassa–Holm (CH) [9], Degasperis–Procesi (DP) [10], and Novikov equations [11]. The four types of equations possess the peakon solutions. Here, we recall several works on the study of the CH, DP, and Novikov equations. The well-posedness of the Cauchy problem for a generalized CH equation is established in Himonas and Holliman [12]. The nonuniform dependence of the periodic CH equation and the well-posedness of the DP equation are discussed in [13] and [14], respectively. The continuity properties of the data-to-solution map for the periodic b-family equation including the CH and DP equations are obtained in [15]. Coclite and Karlsen [16] discuss the existence and stability of the entropy solution for the DP equation. The existence and uniqueness of global solutions for the DP equation are studied in Liu and Yin [17] in the case that the initial data satisfy the sign condition. Escher et al. [18] investigate the global weak solutions and blow-up structure for the DP model under certain assumptions. Matsuno [19] finds out the multisoliton solutions of the DP equation and analyzes their peakon limits. The uniform stability of peakons for the Camassa–Holm model is established in Constantin and Strauss [7]. Using the conservation law and assuming that the initial data satisfy the sign condition, Lin and Liu [20] obtain the stability of peakons for the Degasperis–Procesi equation. The Cauchy problem for the Novikov equation is considered in [21]. A generalized Novikov model with peakon solutions is studied in [22]. For other studies of the CH, DP, and Novikov equations, the reader is referred to [2129] and the references therein.

Motivated by the works made in Coclite and Karlsen [16], the aim of this article is to investigate the stability of local strong solutions for the Fornberg–Whitham equation (1). We find out the \(L^{2}\) conservation law to the FW model. Assuming that the initial data belong to the space \(L^{1}(\mathbb{R})\cap H^{s}(\mathbb{R})\) with \(s> \frac{3}{2}\), we obtain the stability of local strong solution in the space \(L^{1}(\mathbb{R})\). We state that the \(L^{1}\) stability for Eq. (1) has never been established in the previous literature works. The main technique used in this work is the device of doubling the space variables presented in [30].

The structure of this paper is that several lemmas are given in Sect. 2 and the proof of our main result is presented in Sect. 3.

2 Several lemmas

Consider the Cauchy problem of Eq. (1)
$$ \textstyle\begin{cases} V_{t}-V_{txx}-V_{x}+\frac{3}{2}VV_{x}=\frac{9}{2}V_{x}V_{xx}+\frac {3}{2}VV_{xxx}, \\ V(0,x)=V_{0}(x). \end{cases} $$
(3)
Letting \(\Lambda^{2}=1-\partial_{x}^{2}\) and noting the expression \(VV_{xxx}=\frac{1}{2}(V^{2})_{xxx}-3V_{x}V_{xx}\), multiplying both sides of the first equation of problem (3) by \(\Lambda^{-2}\), we obtain the nonlocal form of problem (3) in the form
$$ \textstyle\begin{cases} V_{t}+\frac{3}{2}VV_{x}-(1-\partial_{x}^{2})^{-2}V_{x}=0, \\ V(0,x)=V_{0}(x), \end{cases} $$
(4)
where \(\Lambda^{-2}g=\frac{1}{2}\int_{R}e^{-|x-y|}g\, dy\) for any \(g\in L^{\infty}\) or \(g\in L^{p}(\mathbb{R})\) with \(1\leq p\leq\infty\).

Lemma 1

If \(V_{0}(x)\in H^{s}(\mathbb{R})\), \(s>\frac{3}{2}\) and \(V(t,x)\) is the solution of problem (4), then
$$ \int_{R}V^{2}(t,x)\,dx= \int_{R}V_{0}^{2}(x)\,dx. $$
(5)

Proof

Setting \((1-\partial_{x}^{2})^{-2}V=W\), we get \(W-W_{xx}=V\) and
$$ \int_{R}V\bigl(1-\partial_{x}^{2} \bigr)^{-2}V_{x}\, dx= \int_{R}VW_{x}\, dx= \int _{R}(W-W_{xx})W_{x}\, dx=0, $$
(6)
from which we have
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{R}V^{2}\,dx =& \int_{R} VV_{t}\,dx \\ =& \int_{R} \biggl[-\frac{3}{2}V^{2}V_{x}+V \bigl(1-\partial_{x}^{2}\bigr)^{-2}V_{x}) \biggr]\,dx \\ =&0+ \int_{R} \bigl[V\bigl(1-\partial_{x}^{2} \bigr)^{-2}V_{x}) \bigr]\,dx \\ =&0, \end{aligned}$$
which completes the proof. □

Lemma 2

([3, 4, 23])

Assume \(V(0,x)=V_{0}(x)\in H^{s}(\mathbb{R})\), \(s>\frac{3}{2}\). Then problem (3) or (4) has a unique strong solution V satisfying
$$ V\in C\bigl([0,T);H^{s}(\mathbb{R})\bigr)\cap C^{1} \bigl([0,T);H^{s-1}(\mathbb {R})\bigr), $$
where \(T=T(V_{0})>0\) is the maximal existence time.
Consider the ordinary differential equation
$$ \textstyle\begin{cases} p_{t}=\frac{3}{2}V(t,p),\quad t\in[0, T), \\ p(0,x)=x. \end{cases} $$
(7)

Lemma 3

Assume that \(V_{0}\in H^{s}\), \(s\geq3\), and \(T>0\) is the maximal existence time of the solution for problem (7). Then there exists a unique solution \(p\in C^{1}([0, T)\times\mathbb{R})\) to problem (7) and the map \(p(t, \cdot)\) is an increasing diffeomorphism of R with \(p_{x}(t,x)>0\) for \((t,x)\in[0, T)\times\mathbb{R}\).

Proof

Using Lemma 2, we have \(V\in C^{1}([0, T); H^{s-1}(\mathbb{R}))\) and \(H^{s}\in C^{1}(\mathbb{R})\). Therefore, we know that functions \(V(t,x)\) and \(V_{x}(t,x)\) are bounded, Lipschitz in space, and \(C^{1}\) in time. Making use of the existence and uniqueness theorem of ordinary differential equations, we conclude that problem (7) has a unique solution \(p\in C^{1}([0, T)\times\mathbb{R})\).

We differentiate (7) about the variable x and get
$$ \textstyle\begin{cases} \frac{d}{dt}p_{x}=\frac{3}{2}V_{x}(t,p)p_{x},\quad t\in[0, T), \\ p_{x}(0,x)=1, \end{cases} $$
(8)
which results in
$$ p_{x}(t,x)=e^{\int_{0}^{t} \frac{3}{2}V_{x}(\tau, p(\tau,x))\,d\tau}. $$
(9)
For every \(T'< T\), applying the Sobolev imbedding theorem gives rise to
$$ \sup_{(\tau,x)\in[0,T')\times R} \bigl\vert V_{x}(\tau, x) \bigr\vert < \infty, $$
(10)
from which we know that there exists a constant \(K_{0}>0\) to satisfy \(p_{x}(t,x)\geq e^{-K_{0}t}>0\) for \((t,x)\in[0, T)\times\mathbb{R}\). The proof is finished. □

Lemma 4

Suppose that T is the maximal existence time of the solution V to problem (4) and \(V_{0}\in H^{s}(\mathbb{R})\), \(s>\frac{3}{2}\). Then
$$\begin{aligned}& \bigl\| V(t,x)\bigr\| _{L^{\infty}}\leq t\| V_{0}\| _{L^{2}}+\| V_{0}\|_{L^{\infty}},\quad \forall t\in[0, T], \end{aligned}$$
(11)
$$\begin{aligned}& \bigl|\Lambda^{-2}V_{x}\bigr|\leq\| V_{0}\|_{L^{2}}, \quad \forall t\in[0, T]. \end{aligned}$$
(12)

Proof

Using the density argument presented in [17], we only need to consider the case \(s=3\) to prove Lemma 4. If the initial value \(V_{0}\in H^{3}(\mathbb{R})\), we obtain \(V\in C([0,T),H^{3}(\mathbb{R}))\cap C^{1}([0,T), H^{2}(\mathbb{R}))\). From (4), we have
$$\begin{aligned} V_{t}+\frac{3}{2}VV_{x} =&\frac{1}{2} \int_{-\infty}^{\infty}e^{-|x-y|}\frac {\partial}{\partial y}V(t,y) \,dy \\ =&\frac{1}{2} \int_{-\infty}^{x}e^{-x+y}\frac{\partial}{\partial y}V(t,y) \,dy+\frac{1}{2} \int_{x}^{\infty}e^{-y+x}\frac{\partial}{\partial y}V(t,y) \,dy \\ =&-\frac{1}{2} \int_{-\infty}^{x} e^{-x+y}V(t,y)\,dy+ \frac{1}{2} \int _{x}^{\infty}e^{-y+x}V(t,y) \,dy \end{aligned}$$
(13)
and
$$\begin{aligned} \frac{dV(t,p(t,x))}{dt} =&V_{t}\bigl(t,p(t,x)\bigr)+V_{x} \bigl(t,p(t,x)\bigr)\frac {dp(t,x)}{dt} \\ =&\biggl(V_{t}+\frac{3}{2}{}VV_{x}\biggr) \bigl(t,p(t,x)\bigr). \end{aligned}$$
(14)
Using the identity \(\int_{-\infty}^{\infty}e^{-2|x-y|}\,dy=1\) and \(\| V\|_{L^{2}}=\| V_{0}\|_{L^{2}}\) (see Lemma 1), we have
$$\begin{aligned}& \biggl\vert -\frac{1}{2} \int_{-\infty}^{x} e^{-x+y}V(t,y)\,dy+ \frac{1}{2} \int _{x}^{\infty}e^{-y+x}V(t,y)\,dy \biggr\vert \\& \quad \leq\frac{1}{2} \int_{-\infty}^{x} e^{-x+y} \bigl\vert V(t,y) \bigr\vert \,dy+\frac{1}{2} \int _{x}^{\infty}e^{-y+x} \bigl\vert V(t,y) \bigr\vert \,dy \\& \quad \leq \biggl( \int_{-\infty}^{\infty}e^{-2 \vert x-y \vert }\,dy \biggr)^{\frac{1}{2}} \biggl( \int_{-\infty}^{\infty}V^{2}(t,y)\,dy \biggr)^{\frac{1}{2}} \\& \quad \leq\| V\|_{L^{2}(\mathbb{R})} \\& \quad =\| V_{0}\|_{L^{2}(\mathbb{R})}, \end{aligned}$$
(15)
from which together with (13) we derive that (12) holds.
From (13)–(15), we derive that
$$\begin{aligned} \biggl\vert \int_{0}^{t}\frac{dV(t,p(t,x))}{dt}\,dt \biggr\vert \leq& \frac{1}{2} \int_{0}^{t} \biggl\vert \int _{-\infty}^{\infty}e^{-| p(t,x)-y|}\frac{\partial}{\partial y}V(t,y) \,dy \biggr\vert \,dt \\ \leq& t\| V_{0}\|_{L^{2}(\mathbb{R})}, \end{aligned}$$
(16)
from which we obtain
$$ \bigl\vert V\bigl(t,p(t,x)\bigr) \bigr\vert \leq \bigl\Vert V \bigl(t,p(t,x)\bigr) \bigr\Vert _{L^{\infty}}\leq t\| V_{0} \|_{L^{2}(\mathbb {R})}+\| V_{0}\|_{L^{\infty}}. $$
(17)
Using Lemma 3, for every \(t\in[0,T')\), \(T'< T\), we get that there exists a function \(K(t)>0\) such that
$$ e^{-K(t)}\leq p_{x}(t,x)\leq e^{K(t)},\quad x\in \mathbb{R}. $$
(18)
We deduce from (18) that the function \(p(t,\cdot)\) is strictly increasing on \(\mathbb{R}\) with \(\lim_{x\rightarrow\pm\infty}p(t, x)=\pm\infty\) as long as \(t\in[0,T')\). Applying (17) produces
$$ \bigl\Vert V(t,x) \bigr\Vert _{L^{\infty}}= \bigl\Vert V\bigl(t,p(t,x) \bigr) \bigr\Vert _{L^{\infty}}\leq t \Vert V_{0} \Vert _{L^{2}(\mathbb {R})}+ \Vert V_{0} \Vert _{L^{\infty}}. $$
The proof is finished. □

Lemma 5

Suppose that \(V_{1}(t,x)\) and \(V_{2}(t,x)\) are two solutions of problem (4) with initial data \(V_{1,0}(x), V_{2,0}(x)\in H^{s}(\mathbb{R})\) (\(s>\frac{3}{2}\)), respectively. Assume \(f(t,x)\in C_{0}^{\infty}([0,\infty)\times(-\infty,\infty)\). Then
$$\begin{aligned}& \int_{-\infty}^{\infty} \biggl\vert \Lambda^{-2} \frac{\partial}{\partial x}V_{1}(t,x)-\Lambda^{-2}\frac{\partial}{\partial x}V_{2}(t,x) \biggr\vert \bigl\vert f(t,x) \bigr\vert \,dx \\& \quad \leq c_{0} \int_{-\infty}^{\infty} \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \,dx, \end{aligned}$$
(19)
where \(c_{0}>0\) depends on f.

Proof

We have
$$\begin{aligned}& \int_{-\infty}^{\infty} \biggl\vert \Lambda^{-2} \frac{\partial}{\partial x}V_{1}(t,x)-\Lambda^{-2}\frac{\partial}{\partial x}V_{2}(t,x) \biggr\vert \bigl\vert f(t,x) \bigr\vert \,dx \\& \quad \leq \int_{-\infty}^{\infty} \bigl\vert \partial_{x} \Lambda^{-2}(V_{1}-V_{2}) \bigr\vert \bigl\vert f(t,x) \bigr\vert \,dx \\& \quad \leq \int_{-\infty}^{\infty} \biggl\vert \int_{-\infty}^{\infty }e^{- \vert x-y \vert }\operatorname{sign}(x-y) \bigl(V_{1}(t,y)-V_{2}(t,y)\bigr)\,dy \biggr\vert \bigl\vert f(t,x) \bigr\vert \,dx \\& \quad \leq \int_{-\infty}^{\infty} \int_{-\infty}^{\infty }e^{- \vert x-y \vert } \bigl\vert V_{1}(t,y)-V_{2}(t,y) \bigr\vert \bigl\vert f(t,x) \bigr\vert \,dy\,dx \\& \quad \leq c_{0} \int_{-\infty}^{\infty} \vert V_{1}-V_{2} \vert \,dy, \end{aligned}$$
in which we have applied the Tonelli theorem. The proof is completed. □
Assume that \(\delta(\sigma)\) is a function which is infinitely differentiable on \((-\infty, +\infty)\) such that \(\delta(\sigma)\geq 0\), \(\delta(\sigma)=0\) for \(|\sigma|\geq1\) and \(\int_{-\infty}^{\infty}\delta(\sigma)\,d\sigma=1\). For an arbitrary \(h>0\), set \(\delta_{h}(\sigma )=\frac{\delta(h^{-1}\sigma)}{h}\). We conclude that \(\delta_{h}(\sigma)\) is a function in \(C^{\infty}(-\infty, \infty)\) and
$$ \textstyle\begin{cases} \delta_{h}(\sigma)\geq0,\qquad \delta_{h}(\sigma)=0 \quad \text{if } |\sigma|\geq h, \\ |\delta_{h}(\sigma)|\leq\frac{c}{h}, \qquad \int_{-\infty}^{\infty}\delta _{h}(\sigma)\,d\sigma=1. \end{cases} $$
(20)
Suppose that the function \(W_{1}(x)\) is locally integrable in \((-\infty, \infty)\). The approximation function of \(W_{1}\) is defined by
$$ W_{1}^{h}(x)=\frac{1}{h} \int_{-\infty}^{\infty}\delta\biggl(\frac {x-y}{h} \biggr)W_{1}(y)\,dy,\quad h>0. $$
(21)
We call \(x_{0}\) a Lebesgue point of function \(W_{1}(x)\) if
$$ \lim_{h\rightarrow0}\frac{1}{h} \int_{|x-x_{0}|\leq h} \bigl\vert W_{1}(x)-W_{1}(x_{0}) \bigr\vert \,dx=0. $$

We introduce notation about the concept of a characteristic cone. For any \(M>0\), we define \(M>N=\max_{t\in[0, T]}\| V\| _{L^{\infty}}<\infty\). Let designate the cone \(\{(t,x): |x|< M-Nt, 0\leq t\leq T_{0}=\min(T, MN^{-1}) \}\). We let \(S_{\tau}\) designate the cross section of the cone by the plane \(t=\tau, \tau\in[0, T_{0}]\).

Let \(K_{r+2\rho}=\{x: |x|\leq r+2\rho\}\) where \(r>0\), \(\rho>0\) and \(\zeta _{T}=[0,T]\times\mathbb{R}\). The space of all infinitely differentiable functions \(f(t,x)\) with compact support in \([0,T]\times\mathbb{R}\) is denoted by \(C_{0}^{\infty}(\zeta_{T})\).

Lemma 6

([30])

Let the function \(U(t,x)\) be a bounded and measurable function in some cylinder \(\Omega_{T}=[0, T]\times K_{r}\). If for some \(\rho\in(0, \min[r, T])\) and any number \(h\in(0,\rho )\), then the following function
$$ U_{h}=\frac{1}{h^{2}} \iiiint_{|\frac{t-\tau}{2}|\leq h, \rho\leq\frac {t+\tau}{2}\leq T-\rho, |\frac{x-y}{2}|\leq h, |\frac{x+y}{2}|\leq r-\rho} \bigl\vert U(t,x)-U(\tau, y) \bigr\vert \, dx \, dt\, dy\, d\tau $$
satisfies \(\lim_{h\rightarrow0}U_{h}=0\).

Lemma 7

([30])

If the function \(G(U)\) satisfies a Lipschitz condition on the interval \([-N, N]\), then the function
$$G_{1}(U_{1},U_{2})=\operatorname{sign}(U_{1}-U_{2}) \bigl(G(U_{1})-G(U_{2})\bigr) $$
satisfies the Lipschitz condition in \(U_{1}\) and \(U_{2}\), respectively.

Lemma 8

Suppose that V is the strong solution of problem (4), \(f(t,x)\in C_{0}^{\infty}(\zeta_{T})\) and \(f(0,x)=0\). Then
$$ \iint_{\zeta_{T}} \biggl\{ |V-k|f_{t}+\frac{3}{4} \operatorname {sign}(V-k)\bigl[V^{2}-k^{2} \bigr]f_{x}+\operatorname{sign}(V-k)\Lambda^{-2}V_{x}f \biggr\} \, dx\, dt=0, $$
(22)
where k is an arbitrary constant.

Proof

Here we mention that the method to prove this lemma comes from [30]. We assume that \(\Phi(V)\) is an arbitrary twice differentiable function on the line \(-\infty< V<\infty\). We multiply the first equation of Eq. (4) by the function \(\Phi'(V)f(t,x)\), where \(f(t,x)\in C_{0}^{\infty}(\zeta_{T})\). Integrating over \(\zeta_{T}\) and integrating by parts (transferring the derivatives with respect to t and x to function f), for any constant k, we have
$$\int_{-\infty}^{\infty} \biggl[ \int_{k}^{V}\Phi'(z)z\, dz \biggr]f_{x}\, dx=- \int_{-\infty }^{\infty} \bigl[f\Phi'(V)VV_{x} \bigr]\,dx $$
and
$$ \iint_{\zeta_{T}} \biggl\{ \Phi(V)f_{t}+\frac{3}{2} \biggl[ \int_{k}^{V}\Phi '(z)z\, dz \biggr] f_{x}-\Phi'(V)\Lambda^{-2}V_{x}f \biggr\} \, dx\, dt=0. $$
(23)
Integration by parts yields
$$\begin{aligned} \int_{-\infty}^{\infty} \biggl[ \int_{k}^{V}\Phi'(z)z\, dz \biggr]f_{x}\, dx =& \int _{-\infty}^{\infty} \biggl[\frac{1}{2} \Phi'(V)V^{2} -\frac{1}{2}\Phi'(k)k^{2} \\ &{} -\frac{1}{2} \int_{k}^{V} \bigl(z^{2}-k^{2} \bigr)\Phi''(z)\, dz \biggr]f_{x}\, dx. \end{aligned}$$
(24)
Choosing that \(\Phi^{h}(V)\) is an approximation of the function \(|V-k|\), setting \(\Phi(V)=\Phi^{h}(V)\), and making use of the properties of the \(\operatorname{sign}(V-k)\), (23), (24) and sending \(h\rightarrow0\), we notice that the last term in (24) becomes zero. Thus, we have
$$ \iint_{\zeta_{T}} \biggl\{ |V-k|f_{t}+\frac{3}{4} \operatorname {sign}(V-k)\bigl[V^{2}-k^{2} \bigr]f_{x}+\operatorname{sign}(V-k)\Lambda^{-2}V_{x} f \biggr\} \, dx\, dt=0. $$
(25)
The proof is finished. □

3 Main result

Now, we give the main result of this work.

Theorem 1

Assume that \(V_{1}\) and \(V_{2}\) are two local strong solutions of Eq. (1) with initial data \(V_{1,0}(x),V_{2,0}(x)\in L^{1}(\mathbb{R})\cap H^{s}(\mathbb{R})\), \(s>\frac {3}{2}\). Let T be the maximal existence time of the solutions. Then
$$ \bigl\Vert V_{1}(t,\cdot)-V_{2}(t,\cdot) \bigr\Vert _{L^{1}(\mathbb{R})}\leq c_{0}e^{c_{0}t} \int_{-\infty}^{\infty} \bigl\vert V_{10}(x)-V_{20}(x) \bigr\vert \,dx,\quad t\in[0, T), $$
(26)
where \(c_{0}>0\) is a constant.

Proof

From Lemma 2, we know the existence of local strong solutions for Eq. (1). Let \(f(t,x)\in C_{0}^{\infty}(\zeta_{T})\). Assume \(f(t,x)=0\) outside the cylinder
$$ \uplus=\bigl\{ (t,x)\bigr\} =[\rho, T-2\rho]\times K_{r-2\rho},\quad 0< 2\rho \leq\min(T, r). $$
(27)
We let
$$ \xi=f\biggl(\frac{t+\tau}{2},\frac{x+y}{2}\biggr)\delta_{h} \biggl(\frac{t-\tau}{2}\biggr)\delta _{h}\biggl(\frac{x-y}{2} \biggr)=f(\cdots)\lambda_{h}(\ast), $$
(28)
where \((\cdots)=(\frac{t+\tau}{2}, \frac{x+y}{2})\) and \((\ast)=(\frac {t-\tau}{2}, \frac{x-y}{2})\). The function \(\delta_{h}(\sigma)\) is defined in (20). We obtain
$$ \xi_{t}+\xi_{\tau}=f_{t}(\cdots) \lambda_{h}(\ast), \qquad \xi_{x}+\xi_{y}=f_{x}( \cdots)\lambda _{h}(\ast). $$
(29)
We apply the technique of Kruzkov’s device of doubling the space variables [30]. In (22), we set \(k=V_{1}(\tau,y)\) and \(f=\xi (t,x,\tau,y)\) for a fixed point \((\tau,y)\). We note that \(V_{1}(\tau, y)\) is defined almost everywhere in \(\zeta_{T}=[0,T]\times\mathbb{R}\). We integrate (22) over \(\zeta_{T}\) for variable \((\tau, y)\) and then get
$$\begin{aligned}& \iiiint_{\zeta_{T}\times\zeta_{T}} \biggl\{ \bigl\vert V_{1}(t,x)-V_{2}( \tau,y) \bigr\vert \xi _{t} \\& \quad {}+\frac{3}{4}\operatorname{sign}\bigl(V_{1}(t,x)-V_{2}( \tau,y)\bigr) \biggl(\frac {V_{1}^{2}(t,x)}{2} -\frac{V_{2}^{2}(\tau,y)}{2} \biggr) \xi_{x} \\& \quad {}+\operatorname{sign}\bigl(V_{1}(t,x)-V_{2}(\tau,y) \bigr)\Lambda^{-2}\partial _{x}\bigl(V_{1}(t,x) \bigr)\xi \biggr\} \, dt\, dx\, dy\, d\tau=0. \end{aligned}$$
(30)
Similarly, it has
$$\begin{aligned}& \iiiint_{\zeta_{T}\times\zeta_{T}} \biggl\{ \bigl\vert V_{2}( \tau,y)-V_{1}(t,x) \bigr\vert \xi_{\tau } \\& \quad {}+\frac{3}{4}\operatorname{sign}\bigl(V_{2}( \tau,y)-V_{1}(t,x)\bigr) \biggl(\frac {V_{2}^{2}(\tau,y)}{2} -\frac{V_{1}^{2}(t,x)}{2} \biggr)\xi_{y} \\& \quad {}+\operatorname{sign}\bigl(V_{2}(\tau,y)-V_{1}(t,x) \bigr)\Lambda^{-2}\partial _{y}\bigl(V_{2}(\tau,y) \bigr)\xi \biggr\} \, dx\, dt\, dy\, d\tau= 0. \end{aligned}$$
(31)
Using (30) and (31), we acquire the inequality
$$\begin{aligned} 0 \leq& \iiiint_{\zeta_{T}\times\zeta_{T}}\biggl\{ \bigl\vert V_{1}(t,x)-V_{2}( \tau,y) \bigr\vert (\xi _{t}+\xi_{\tau}) \\ &{}+\frac{3}{4}\operatorname{sign}\bigl(V_{1}(t,x)-V_{2}( \tau,y)\bigr) \biggl(\frac {V_{1}^{2}(t,x)}{2} -\frac{V_{2}^{2}(\tau,y)}{2}\biggr) ( \xi_{x}+\xi_{y})\biggr\} \, dx\, dt\, dy\, d\tau \\ &{}+ \biggl\vert \iiiint_{\zeta_{T}\times\zeta_{T}}\operatorname {sign}\bigl(V_{1}(t,x)-V_{2}(t,x) \bigr) \\ &{}\times\bigl(\Lambda^{-2}\partial_{x}V_{1}(t,x)- \Lambda ^{-2}\partial_{y}V_{2}(\tau,y)\bigr)\xi\, dx\, dt\, dy\, d\tau \biggr\vert . \\ =& L_{1}+L_{2}+ \biggl\vert \iiiint_{\zeta_{T}\times\zeta_{T}}L_{3}\, dx\, dt\, dy\, d\tau \biggr\vert . \end{aligned}$$
(32)
We claim that the following inequality
$$\begin{aligned} 0 \leq& \iint_{\zeta_{T}}\biggl\{ \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert f_{t} \\ &{} +\frac{3}{4}\operatorname{sign}\bigl(V_{1}(t,x)-V_{2}(t,x) \bigr) \biggl(\frac {V_{1}^{2}(t,x)}{2} -\frac{V_{2}^{2}(t,x)}{2}\biggr)f_{x}\biggr\} \, dx\, dt \\ &{} + \biggl\vert \iint_{\zeta_{T}} \operatorname{sign}\bigl(V_{1}(t,x)-V_{2}(t,x) \bigr)\Lambda ^{-2}\partial_{x}\bigl[V_{1}(t,x)-V_{2}(t,x) \bigr]f \, dx\, dt \biggr\vert \end{aligned}$$
(33)
holds.
In fact, for the choice of ξ, the first two terms in the integrand of (32) can be represented in the form
$$ D_{h}=D\bigl(t,x,\tau,y,V_{1}(t,x),V_{2}(\tau,y) \bigr)\lambda_{h}(\ast). $$
From Lemma 4, we know \(\| V_{1}\|_{L^{\infty}}< C_{T}\) and \(\| V_{2}\|_{L^{\infty}}< C_{T}\); from Lemma 7, we know \(D_{h}\) satisfies the Lipschitz condition in \(V_{1}\) and \(V_{2}\), respectively. By the choice of ξ, we derive that \(D_{h}=0\) outside the region
$$\begin{aligned} \bigl\{ (t,x; \tau,y)\bigr\} =& \biggl\{ \rho\leq\frac{t+\tau}{2}\leq T-2\rho, \frac {|t-\tau|}{2}\leq h, \\ & \frac{|x+y|}{2}\leq r-2\rho, \frac{|x-y|}{2}\leq h \biggr\} . \end{aligned}$$
(34)
Furthermore, we get
$$\begin{aligned}& \iiiint_{\zeta_{T}\times\zeta_{T}}D_{h}\, dx\, dt\, dy\, d\tau \\& \quad = \iiiint_{\zeta _{T}\times\zeta_{T}} \bigl[D\bigl(t,x,\tau,y,V_{1}(t,x),V_{2}( \tau,y)\bigr) \\& \qquad {} -D\bigl(t,x,t,x,V_{1}(t,x),V_{2}(t,x)\bigr) \bigr] \lambda_{h}(\ast)\, dx\, dt\, dy\, d\tau \\& \qquad {} + \iiiint_{\zeta_{T}\times\zeta_{T}} D\bigl(t,x,t,x,V_{1}(t,x),V_{2}(t,x) \bigr)\lambda _{h}(\ast)\, dx\, dt\, dy\, d\tau \\& \quad = B_{11}(h)+B_{12}. \end{aligned}$$
(35)
Noticing \(|\lambda(\ast)|\leq\frac{c}{h^{2}}\) and the definition of \(D_{h}\) gives rise to
$$\begin{aligned}& \bigl\vert B_{11}(h) \bigr\vert \\& \quad \leq c \biggl[h+ \frac{1}{h^{2}} \\& \qquad {}\times \iiiint_{|\frac{t-\tau}{2}|\leq h, \rho\leq\frac{t+\tau }{2}\leq T-\rho, |\frac{x-y}{2}|\leq h, |\frac{x+y}{2}|\leq r-\rho} \bigl\vert V_{1}(t,x)-V_{2}(\tau, y) \bigr|\, dx\, dt\, dy\, d\tau \biggr], \end{aligned}$$
(36)
where the constant c does not depend on h. Using Lemma 6, we get \(B_{11}(h)\rightarrow0\) as \(h\rightarrow0\). The integral \(B_{12}\) does not depend on h. Substituting \(t=\alpha\), \(\frac{t-\tau}{2}=\beta\), \(x=\eta\), \(\frac{x-y}{2}=\mu\) and noting the identity
$$ \int_{-h}^{h} \int_{-\infty}^{\infty}\lambda_{h}(\beta,\mu) \,d\mu \,d\beta =1, $$
(37)
we derive that
$$\begin{aligned} B_{12} =&2^{2} \iint_{\zeta_{T}} D_{h}\bigl(\alpha,\eta,\alpha, \eta,V_{1}(\alpha,\eta),V_{2}(\alpha,\eta)\bigr) \biggl\{ \int _{-h}^{h} \int_{-\infty}^{\infty}\lambda_{h}(\beta,\mu)\,d\mu \,d\beta \biggr\} \, d\eta\, d\alpha \\ =& 4 \iint_{\zeta_{T}}D_{h}\bigl(t,x,t,x, V_{1}(t,x),V_{2}(t,x) \bigr)\, dx\, dt. \end{aligned}$$
(38)
Thus, we have
$$ \lim_{h\rightarrow0} \iiiint_{\zeta_{T}\times\zeta_{T}}D_{h}\, dx\, dt\, dy\, d\tau =4 \iint_{\zeta_{T}}D\bigl(t,x,t,x, V_{1}(t,x),V_{2}(t,x) \bigr)\, dx\, dt. $$
(39)
We write
$$\begin{aligned} L_{3} =&\operatorname{sign}\bigl(u(t,x)-v(\tau,y)\bigr) \bigl( \Lambda^{-2}\partial _{x}V_{1}(t,x)- \Lambda^{-2}\partial_{y}V_{2}(\tau,y) \bigr)f( \cdots)\lambda_{h}(\ast ) \\ =&\overline{L_{3}}(t.x,\tau,y)\lambda_{h}( \ast) \end{aligned}$$
(40)
and
$$\begin{aligned} \iiiint_{\zeta_{T}\times\zeta_{T}} L_{3}\, dx\, dt\, dy\, d\tau =& \iiiint_{\zeta _{T}\times\zeta_{T}} \bigl[\overline{L_{3}}(t.x,\tau,y)- \overline {L_{3}}(t.x,t,x) \bigr]\lambda_{h}(\ast)\, dx\, dt\, dy\, d\tau \\ &{} + \iiiint_{\zeta_{T}\times\zeta_{T}}\overline{L_{3}}(t.x,t,x)\lambda _{h}(\ast)\, dx\, dt\, dy\, d\tau \\ =&B_{21}(h)+B_{22}, \end{aligned}$$
(41)
from which we have
$$\begin{aligned}& \bigl\vert B_{21}(h) \bigr\vert \\& \quad \leq c \biggl(h+ \frac{1}{h^{2}} \iiiint_{|\frac{t-\tau}{2}|\leq h, \rho\leq\frac{t+\tau }{2}\leq T-\rho, |\frac{x-y}{2}|\leq h, |\frac{x+y}{2}|\leq r-\rho} \bigl\vert \Lambda ^{-2} \partial_{x}V_{1}(t,x) \\& \qquad {}-\Lambda^{-2} \partial_{y}V_{2}(\tau,y) \bigr\vert \, dx\, dt\, dy\, d \tau \biggr). \end{aligned}$$
(42)
Using Lemmas 5 and 6, we have \(B_{21}(h)\rightarrow0\) as \(h\rightarrow 0\). Using (37), we have
$$\begin{aligned} B_{22} =&2^{2} \iint_{\zeta_{T}}\overline{L_{3}}\bigl(\alpha,\eta,\alpha, \eta ,V_{1}(\alpha,\eta),V_{2}(\alpha,\eta)\bigr) \biggl\{ \int_{\mathbb{R}} \int _{-h}^{h}\lambda_{h}(\beta,\mu)\,d\mu \,d\beta \biggr\} \, d\eta \, d\alpha \\ =& 4 \iint_{\zeta_{T}}\overline{L_{3}}\bigl(t,x,t,x, V_{1}(t,x),V_{2}(t,x)\bigr)\, dx\, dt \\ =&4 \iint_{\zeta_{T}}\operatorname{sign}\bigl(V_{1}(t,x)-V_{2}(t,x) \bigr) (\Lambda ^{-2}\partial_{x}\bigl[V_{1}(t,x)-V_{2}(t,x) \bigr]f(t,x)\, dx\, dt. \end{aligned}$$
(43)
From (36), (37), (42), and (43), we prove that inequality (33) holds.
Set
$$ X(t)= \int_{-\infty}^{\infty} \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \,dx. $$
(44)
Let
$$ \gamma_{h}= \int_{-\infty}^{\sigma}\delta_{h}(\tau)\,d\tau \qquad \bigl(\gamma_{h}'(\sigma )=\delta_{h}( \sigma)\geq0 \bigr) $$
(45)
and choose two numbers ρ and \(\tau\in(0, T_{0}), \rho<\tau\). In (33), we choose
$$ f=\bigl[\gamma_{h}(t-\rho)-\gamma_{h}(t-\tau)\bigr] \chi(t,x),\quad h< \min(\rho, T_{0}-\tau ), $$
(46)
where
$$ \chi(t,x)=\chi_{\varepsilon}(t,x)=1-\gamma_{\varepsilon}\bigl( \vert x \vert +Nt-M+\varepsilon \bigr),\quad \varepsilon>0. $$
(47)
We know that the function \(\chi(t,x)=0\) outside the cone and \(f(t,x)=0\) outside the set . If \((t,x)\in\mho\), we get the relations
$$ 0=\chi_{t}+N|\chi_{x}|\geq\chi_{t}+N \chi_{x}. $$
(48)
Applying (46)–(48) and (33), we have
$$\begin{aligned} 0 \leq& \int_{0}^{T_{0}} \int_{-\infty}^{\infty} \bigl\{ \bigl[\delta_{h}(t- \rho )-\delta_{h}(t-\tau)\bigr]\chi_{\varepsilon}\bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \bigr\} \, dx\, dt \\ &{} + \int_{0}^{T_{0}} \int_{-\infty}^{\infty}\bigl[\gamma_{h}(t-\rho)- \gamma _{h}(t-\tau)\bigr] \bigl\vert (\Lambda^{-2} \partial_{x}\bigl[V_{1}(t,x)-V_{2}(t,x)\bigr] \chi(t,x) \bigr\vert \, dx\, dt. \end{aligned}$$
(49)
Using Lemma 5 and letting \(\varepsilon\rightarrow0\) and \(M\rightarrow \infty\), we obtain
$$\begin{aligned} 0 \leq& \int_{0}^{T_{0}} \biggl\{ \bigl[\delta_{h}(t- \rho)-\delta_{h}(t-\tau)\bigr] \int _{-\infty}^{\infty} \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \,dx \biggr\} \,dt \\ &{}+c_{0}(1+T_{0}) \int_{0}^{T_{0}}\bigl[\gamma_{h}(t-\rho)- \gamma_{h}(t-\tau)\bigr] \int _{-\infty}^{\infty} \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \, dx\, dt. \end{aligned}$$
(50)
Using the properties of the function \(\delta_{h}(\sigma)\) for \(h\leq\min (\rho, T_{0}-\rho)\) yields
$$\begin{aligned} \biggl\vert \int_{0}^{T_{0}}\delta_{h}(t-\rho)X(t)\,dt-X( \rho) \biggr\vert =& \biggl\vert \int _{0}^{T_{0}}\delta_{h}(t-\rho) \bigl\vert X(t)-X(\rho) \bigr\vert \,dt \biggr\vert \\ \leq& c\frac{1}{h} \int_{\rho-h}^{\rho+h} \bigl\vert X(t)-X(\rho) \bigr\vert \,dt\rightarrow0\quad \text{as } h\rightarrow0, \end{aligned}$$
(51)
where c is independent of h. Denoting
$$ L(\rho)= \int_{0}^{T_{0}}\gamma_{h}(t-\rho)X(t)\,dt= \int_{0}^{T_{0}} \int_{-\infty }^{t-\rho}\delta_{h}(\sigma) \,d\sigma X(t)\,dt, $$
(52)
we get
$$ L'(\rho)=- \int_{0}^{T_{0}}\delta_{h}(t-\rho)X(t)\,dt \rightarrow-X(\rho)\quad \text{as } h\rightarrow0, $$
(53)
and
$$ L(\rho)\rightarrow L(0)- \int_{0}^{\rho}X(\sigma)\,d\sigma \quad \text{as } h \rightarrow0. $$
(54)
Similarly, we obtain
$$ L(\tau)\rightarrow L(0)- \int_{0}^{\tau}X(\sigma)\,d\sigma \quad \text{as } h \rightarrow0. $$
(55)
It follows from (54) and (55) that
$$ L(\rho)-L(\tau)\rightarrow \int_{\rho}^{\tau}X(\sigma)\,d\sigma \quad \text{as } h \rightarrow0. $$
(56)
Send \(\rho\rightarrow0\), \(\tau\rightarrow t\), and note that
$$\begin{aligned} \bigl\vert V_{1}(\rho,x)-V_{2}(\rho,x) \bigr\vert \leq& \bigl\vert V_{1}(\rho,x)-V_{10}(x) \bigr\vert \\ &{} + \bigl\vert V_{2}(\rho,x)-V_{20}(x) \bigr\vert + \bigl\vert V_{10}(x)-V_{20}(x) \bigr\vert . \end{aligned}$$
(57)
Thus, from (50), (51), (56)–(57), we have
$$\begin{aligned} \int_{-\infty}^{\infty} \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \,dx \leq& \int_{-\infty}^{\infty } \vert V_{10}-V_{20} \vert \,dx \\ &{} +c_{0} \int_{0}^{t} \int_{-\infty}^{\infty} \bigl\vert V_{1}(t,x)-V_{2}(t,x) \bigr\vert \, dx\, dt. \end{aligned}$$
(58)
Using the Gronwall inequality and (58), we complete the proof. □

Declarations

Acknowledgements

The authors are very grateful to the reviewers for their helpful and valuable comments and suggestions, which have led to a meaningful improvement of the paper.

Availability of data and materials

Not applicable.

Funding

This work is supported by the National Natural Science Foundation of China (No. 11471263).

Authors’ contributions

The authors contributed equally to the writing of this paper. They read and approved the final manuscript.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, China

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