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

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.

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, 5–8].

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 [21–29] 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. □