The stability of local strong solutions for a shallow water equation

Abstract

We establish the $L^1$ stability of local strong solutions for a shallow water equation which includes the Degasperis-Procesi equation provided that its initial value lies in the Sobolev space $H^s(R)$ with $s>\frac{3}{2}$. The key element in our analysis is that the $L^{\mathrm{\infty }}$ norm of the solutions keeps finite for all finite time t.

MSC:35G25, 35L05.

1 Introduction

From the propagation of shallow water waves over a flat bed, Constantin and Lannes [1] derived the equation

$$g_t+g_x+\frac{3}{2}\rho g{g}_x+\mu (\alpha g_{xxx}+\beta g_{txx})=\rho \mu (\gamma g_x{g}_{xx}+\delta g{g}_{xxx}),$$

(1)

where the constants α, β, γ, δ, ρ and μ satisfy certain restrictions. As illustrated in [1], using suitable mathematical transformations turns Eq. (1) into the form

$$g_t-g_{txx}+k{g}_x+mg{g}_x=a{g}_x{g}_{xx}+bg{g}_{xxx},$$

(2)

where a, b, k and m are constants. We know that the Camassa-Holm and Degasperis-Procesi models are special cases of Eq. (2). Lai and Wu [2] established the well-posedness of local strong solutions and obtained the existence of local weak solutions for Eq. (2).

The aim of this paper is to investigate a special case of Eq. (2). Namely, we study the shallow water equation

$$g_t-g_{txx}+k{g}_x+mg{g}_x=3g_x{g}_{xx}+g{g}_{xxx},$$

(3)

where $k\ge 0$ and $m>0$ are constants. Letting $y=g-{\partial }_{xx}^{2}g$, $v={(m-{\partial }_{xx}^2)}^{-1}g$ and using Eq. (3), we derive the conservation law

$${\int }_{R}yvdx={\int }_R\frac{1+{\xi }^2}{m+{\xi }^2}{|\hat{g}(t,\xi )|}^{2}d\xi ={\int }_R\frac{1+{\xi }^2}{m+{\xi }^2}{|{\hat{g}}_0(\xi )|}^{2}d\xi \sim {\parallel g_0\parallel }_{L^2(R)},$$

(4)

where $g_0=g(0,x)$ and $\hat{g}(t,\xi )$ is the Fourier transform of $g(t,x)$ with respect to variable x. In fact, the conservation law (4) plays an important role in our further investigations of Eq. (3).

For $m=4$, $k=0$, Eq. (3) reduces to the Degasperis-Procesi equation [3]

$$g_t-g_{txx}+4g{g}_x=3g_x{g}_{xx}+g{g}_{xxx}.$$

(5)

Various dynamic properties for Eq. (5) have been acquired by many scholars. Escher et al. [4] and Yin [5] studied the global weak solutions and blow-up structures for Eq. (5), while the blow-up structure for a generalized periodic Degasperis-Procesi equation was obtained in [6]. Lin and Liu [7] established the stability of peakons for Eq. (5) under certain assumptions on the initial value. For other dynamic properties of the Degasperis-Procesi (5) and other shallow water models, the reader is referred to [8–19] and the references therein.

The objective of this work is to establish the $L^1(R)$ stability of local strong solutions for the generalized Degasperis-Procesi equation (3) under the condition that we let the initial value $g_0$ belong to the space $H^s(R)$ with $s>\frac{3}{2}$. Here we address that the $L^1$ stability of local strong solutions for Eq. (3) has never been established in the literature. Our main approaches come from those presented in [20].

This paper is organized as follows. Section 2 gives several lemmas. The main result and its proof are presented in Section 3.

2 Several lemmas

The Cauchy problem of Eq. (3) is written in the form

$$\{\begin{array}{l}{g}_t-g_{txx}+k{g}_x+mg{g}_x=3g_x{g}_{xx}+g{g}_{xxx},\\ g(0,x)=g_0(x),\end{array}$$

(6)

which is equivalent to

$$\{\begin{array}{l}{g}_t+g{g}_x+k{\mathrm{\Lambda }}^{-2}{g}_x+\frac{m-1}{2}{\mathrm{\Lambda }}^{-2}{(g^2)}_x=0,\\ g(0,x)=g_0(x),\end{array}$$

(7)

where ${\mathrm{\Lambda }}^{-2}f=\frac{1}{2}{\int }_R{e}^{-|x-y|}fdy$ for any $f\in L^2(R)$ or $L^{\mathrm{\infty }}(R)$.

Let $Q_g(t,x)=\frac{m-1}{2}{\mathrm{\Lambda }}^{-2}(g^2)+k{\mathrm{\Lambda }}^{-2}g$ and $J_g={\partial }_x(\frac{m-1}{2}{\mathrm{\Lambda }}^{-2}(g^2)+k{\mathrm{\Lambda }}^{-2}g)$, we have

$$g_t+\frac{1}{2}{\left(g^2\right)}_x+J_g=0.$$

(8)

Lemma 2.1 For problem (6) with $m>0$, it holds that

$${\int }_{R}yvdx={\int }_R\frac{1+{\xi }^2}{m+{\xi }^2}{|\hat{g}(t,\xi )|}^{2}d\xi ={\int }_R\frac{1+{\xi }^2}{m+{\xi }^2}{|{\hat{g}}_0(\xi )|}^{2}d\xi \sim {\parallel g_0\parallel }_{L^2(R)}.$$

(9)

In addition, there exist two positive constants $c_1$ and $c_2$ depending only on m such that

$$c_1{\parallel g_0\parallel }_{L^2(R)}\le c_1{\parallel g\parallel }_{L^2(R)}\le c_2{\parallel g_0\parallel }_{L^2(R)}.$$

Proof Letting $y=g-{\partial }_{xx}^{2}g$ and $v={(m-{\partial }_{xx}^2)}^{-1}g$ and using Eq. (3), we have $g=mv-{\partial }_{xx}^{2}v$ and

$$\begin{array}{rcl}\frac{d}{dt}{\int }_{R}yvdx& =& {\int }_R{y}_{t}vdx+{\int }_{R}y{v}_{t}dx=2{\int }_{R}v{y}_{t}dx\\ =& 2{\int }_R[{(-\frac{m}{2}{g}^2)}_x+k{g}_x+\frac{1}{2}{\partial }_{xxx}^3{g}^2]vdx\\ =& 2{\int }_R[{(-\frac{m}{2}{g}^2)}_{x}v+k{g}_{x}v+\frac{1}{2}{\partial }_x{g}^2{\partial }_{xx}^{2}v]dx\\ =& {\int }_R[{(-m{g}^2)}_{x}v+k{g}_{x}v+{\left(g^2\right)}_x(mv-g)]dx\\ =& -{\int }_{R}g{\left(g^2\right)}_{x}dx+k{\int }_R(m{v}_x-v_{xxx})vdx\\ =& k{\int }_R{v}_{xx}{v}_{x}dx\\ =& 0,\end{array}$$

from which we complete the proof. □

Lemma 2.2 ([2])

If $g_0\in H^s(R)$ with $s>\frac{3}{2}$, there exist maximal $T=T(u_0)>0$ and a unique local strong solution $g(t,x)$ to problem (6) such that

$$g(t,x)\in C([0,T);H^s(R))C^1([0,T);H^{s-1}(R)).$$

Firstly, we study the differential equation

$$\{\begin{array}{l}{p}_t=g(t,p),t\in [0,T),\\ p(0,x)=x.\end{array}$$

(10)

Lemma 2.3 Let $g_0\in H^s(R)$, $s>3$ and let $T>0$ be the maximal existence time of the solution to problem (10). Then problem (10) has a unique solution $p\in C^1([0,T)\times R,R)$. Moreover, the map $p(t,\cdot )$ is an increasing diffeomorphism of R with $p_x(t,x)>0$ for $(t,x)\in [0,T)\times R$.

Proof From Lemma 2.2, we have $g\in C^1([0,T);H^{s-1}(R))$ and $H^{s-1}(R)\in C^1(R)$. Thus we conclude that both functions $g(t,x)$ and $g_x(t,x)$ are bounded, Lipschitz in space and $C^1$ in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (10) has a unique solution $p\in C^1([0,T)\times R,R)$.

Differentiating (10) with respect to x yields

$$\{\begin{array}{l}\frac{d}{dt}{p}_x=g_x(t,p)p_x,t\in [0,T),\\ p_x(0,x)=1,\end{array}$$

(11)

which leads to

$$p_x(t,x)=exp\left({\int }_0^t{g}_x(\tau ,p(\tau ,x))d\tau \right).$$

(12)

For every $T^{\prime }<T$, using the Sobolev embedding theorem yields

$$\underset{(\tau ,x)\in [0,T^{\prime })\times R}{sup}|g_x(\tau ,x)|<\mathrm{\infty }.$$

It is inferred that there exists a constant $K_0>0$ such that $p_x(t,x)\ge e^{-K_{0}t}$ for $(t,x)\in [0,T)\times R$. This completes the proof. □

Lemma 2.4 Assume $g_0\in H^s(R)$ with $s>\frac{3}{2}$. Let T be the maximal existence time of the solution g to Eq. (3). Then we have

$${\parallel g(t,x)\parallel }_{L^{\mathrm{\infty }}}\le c_0{\parallel g_0\parallel }_{L^2}^{2}t+{\parallel g_0\parallel }_{L^{\mathrm{\infty }}}\mathrm{\forall }t\in [0,T],$$

(13)

where constant $c_0$ depends on m, k.

Proof Let $Z(x)=\frac{1}{2}{e}^{-|x|}$, we have ${(1-{\partial }_x^2)}^{-1}f=Z\star f$ for all $f\in L^2(R)$ and $g=Z\star y(t,x)$. Using a simple density argument presented in [6], it suffices to consider $s=3$ to prove this lemma. If T is the maximal existence time of the solution g to Eq. (3) with the initial value $g_0\in H^3(R)$ such that $g\in C([0,T),H^3(R))\cap C^1([0,T),H^2(R))$. From (7), we obtain

$$g_t+g{g}_x=-(m-1)Z\star (g{g}_x)-kZ\star g_x.$$

(14)

Since

$$\begin{array}{rcl}-Z\star (g{g}_x)& =& -\frac{1}{2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-|x-y|}g{g}_{y}dy\\ =& -\frac{1}{2}{\int }_{-\mathrm{\infty }}^x{e}^{-x+y}g{g}_{y}dy-\frac{1}{2}{\int }_x^{+\mathrm{\infty }}{e}^{x-y}g{g}_{y}dy\\ =& \frac{1}{4}{\int }_{\mathrm{\infty }}^x{e}^{-|x-y|}{g}^{2}dy-\frac{1}{4}{\int }_x^{\mathrm{\infty }}{e}^{-|x-y|}{g}^{2}dy\end{array}$$

(15)

and

$$\begin{array}{rcl}\frac{dg(t,p(t,x))}{dt}& =& g_t(t,p(t,x))+g_x(t,p(t,x))\frac{dp(t,x)}{dt}\\ =& (g_t+g{g}_x)(t,p(t,x)),\end{array}$$

(16)

from (16), we have

$$\frac{dg}{dt}=\frac{m-1}{4}{\int }_{-\mathrm{\infty }}^{p(t,x)}{e}^{-|p(t,x)-y|}{g}^{2}dy-\frac{m-1}{4}{\int }_{p(t,x)}^{\mathrm{\infty }}{e}^{-|p(t,x)-y|}{g}^{2}dy-kZ\star g_x,$$

(17)

from which we get

$$\begin{array}{rcl}\left|\frac{dg(t,p(t,x))}{dt}\right|& \le & \frac{|m-1|}{4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-|p(t,x)-y|}{g}^{2}dy+|kZ\star g_x|\\ \le & \frac{|m-1|}{4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}^{2}dy+k\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{2}{e}^{-|x-y|}{g}_{y}dy\right|\\ \le & \frac{|m-1|}{4}{\parallel g\parallel }_{L^2}^2+k{\parallel g\parallel }_{L^2}\\ \le & c{\parallel g\parallel }_{L^2(R)}\\ \le & c{\parallel g_0\parallel }_{L^2(R)},\end{array}$$

(18)

where c is a positive constant independent of t. Using (18) results in

$$-ct{\parallel g_0\parallel }_{L^2(R)}+g_0\le g(t,p(t,x))\le ct{\parallel g_0\parallel }_{L^2(R)}+g_0.$$

(19)

Therefore,

$$\left|g(t,p(t,x))\right|\le {\parallel g(t,p(t,x))\parallel }_{L^{\mathrm{\infty }}}\le ct{\parallel g_0\parallel }_{L^2(R)}+{\parallel g_0\parallel }_{L^{\mathrm{\infty }}}.$$

(20)

Using the Sobolev embedding theorem to ensure the uniform boundedness of $g_x(s,\eta )$ for $(s,\eta )\in [0,t]\times R$ with $t\in [0,T^{\prime })$, from Lemma 2.3, for every $t\in [0,T^{\prime })$, we get a constant $C(t)$ such that

$$e^{-C(t)}\le p_x(t,x)\le e^{C(t)},x\in R.$$

We deduce from the above equation that the function $p(t,\cdot )$ is strictly increasing on R with ${lim}_{x\to \pm \mathrm{\infty }}p(t,x)=\pm \mathrm{\infty }$ as long as $t\in [0,T^{\prime })$. It follows from (20) that

$${\parallel g(t,x)\parallel }_{L^{\mathrm{\infty }}}={\parallel g(t,p(t,x))\parallel }_{L^{\mathrm{\infty }}}\le ct{\parallel g_0\parallel }_{L^2(R)}+{\parallel g_0\parallel }_{L^{\mathrm{\infty }}}.$$

(21)

 □

Lemma 2.5 Assume $g_0\in L^2(R)$. Then

$${\parallel Q_g\parallel }_{L^{\mathrm{\infty }}(R_{+}\times R)},{\parallel J_g\parallel }_{L^{\mathrm{\infty }}(R_{+}\times R)}\le c_0{\parallel g_0\parallel }_{L^2}^2,$$

(22)

where $c_0$ is a constant independent of t.

Proof Using (7), we get

$$Q_g(t,x)=\frac{m-1}{4}{\int }_R{e}^{-|x-y|}{g}^2(t,y)dy+\frac{k}{2}{\int }_R{e}^{-|x-y|}gdy,$$

(23)

$$\begin{array}{rl}{J}_g(t,x)=& \frac{m-1}{4}{\int }_R{e}^{-|x-y|}sign(y-x)g^2(t,y)dy\\ +\frac{k}{2}{\int }_R{e}^{-|x-y|}sign(y-x)g(t,y)dy.\end{array}$$

(24)

It follows from (23)-(24) and Lemma 2.1 that (22) holds. □

Lemma 2.6 Assume that $g_1(t,x)$ and $g_2(t,x)$ are two local strong solutions of equation (3) with initial data $g_{10},g_{20}\in H^s(R)$, $s>\frac{3}{2}$, respectively. Then, for any $f(t,x)\in C_0^{\mathrm{\infty }}([0,\mathrm{\infty })\times R)$, it holds that

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|J_{g_1}(t,x)-J_{g_2}(t,x)||f(t,x)|dx\le c_0(1+t){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1-g_2|dx,$$

(25)

where $c_0>0$ depends on t, f, ${\parallel g_{10}\parallel }_{L^2(R)}$, ${\parallel g_{20}\parallel }_{L^2(R)}$, ${\parallel g_{10}\parallel }_{L^{\mathrm{\infty }}(R)}$ and ${\parallel g_{20}\parallel }_{L^{\mathrm{\infty }}(R)}$.

Proof We have

$$\begin{array}{r}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|J_{g_1}(t,x)-J_{g_2}(t,x)||f(t,x)|dx\\ \le \frac{|m-1|}{2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|{\partial }_x{\mathrm{\Lambda }}^{-2}(g_1^2-g_2^2)\right||f(t,x)|dx\\ +\frac{k}{2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-|x-y|}|sign(x-y)||g_1-g_2||f(t,x)|dydx\\ =\frac{|m-1|}{4}\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-|x-y|}|sign(x-y)||g_1^2-g_2^2|dy|f(t,x)|dx\right|\\ +c_0{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1-g_2|dy{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-|x-y|}|f(t,x)|dx\\ \le \frac{|m-1|}{4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|(g_1-g_2)(g_1+g_2)|dy\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-|x-y|}|f(t,x)|dx\right|\\ +c_0{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1-g_2|dy\\ \le c_0(1+t){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1-g_2|dy,\end{array}$$

in which we have used the Tonelli theorem and Lemma 2.4. The proof is completed. □

We define $\delta (\sigma )$ to be a function which is infinitely differentiable on $(-\mathrm{\infty },+\mathrm{\infty })$ such that $\delta (\sigma )\ge 0$, $\delta (\sigma )=0$ for $|\sigma |\ge 1$ and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\sigma )d\sigma =1$. For any number $h>0$, we let ${\delta }_h(\sigma )=\frac{\delta (h^{-1}\sigma )}{h}$. Then we know that ${\delta }_h(\sigma )$ is a function in $C^{\mathrm{\infty }}(-\mathrm{\infty },\mathrm{\infty })$ and

$$\{\begin{array}{l}{\delta }_h(\sigma )\ge 0,{\delta }_h(\sigma )=0\text{if }|\sigma |\ge h,\\ |{\delta }_h(\sigma )|\le \frac{c}{h},{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\delta }_h(\sigma )=1.\end{array}$$

(26)

Assume that the function $u(x)$ is locally integrable in $(-\mathrm{\infty },\mathrm{\infty })$. We define an approximation function of u as

$$u^h(x)=\frac{1}{h}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(\frac{x-y}{h}\right)u(y)dy,h>0.$$

(27)

We call $x_0$ a Lebesgue point of the function $u(x)$ if

$$\underset{h\to 0}{lim}\frac{1}{h}{\int }_{|x-x_0|\le h}|u(x)-u(x_0)|dx=0.$$

At any Lebesgue points $x_0$ of the function $u(x)$, we have ${lim}_{h\to 0}{u}^h(x_0)=u(x_0)$. Since the set of points which are not Lebesgue points of $u(x)$ has measure zero, we get $u^h(x)\to u(x)$ as $h\to 0$ almost everywhere.

We introduce notation connected with the concept of a characteristic cone. For any $R_0>0$, we define $N>{max}_{t\in [0,T]}{\parallel g\parallel }_{L^{\mathrm{\infty }}}<\mathrm{\infty }$. Let ℧ designate the cone $\{(t,x):|x|<R_0-Nt,0\le t\le T_0=min(T,R_0{N}^{-1})\}$. We let $S_{\tau }$ designate the cross section of the cone ℧ by the plane $t=\tau$, $\tau \in [0,T_0]$.

Let $K_{r+2\rho }=\{x:|x|\le r+2\rho \}$, where $r>0$, $\rho >0$ and ${\pi }_T=[0,T]\times R$ for an arbitrary $T>0$. The space of all infinitely differentiable functions $f(t,x)$ with compact support in $[0,T]\times R$ is denoted by $C_0^{\mathrm{\infty }}({\pi }_T)$.

Lemma 2.7 ([20])

Let the function $u(t,x)$ be bounded and measurable in cylinder ${\mathrm{\Omega }}_T=[0,T]\times K_r$. If for $\rho \in (0,min[r,T])$ and any number $h\in (0,\rho )$, then the function

$$V_h=\frac{1}{h^2}{⨌}_{|\frac{t-\tau }{2}|\le h,\rho \le \frac{t+\tau }{2}\le T-\rho ,|\frac{x-y}{2}|\le h,|\frac{x+y}{2}|\le r-\rho }|u(t,x)-u(\tau ,y)|dxdtdyd\tau$$

satisfies ${lim}_{h\to 0}{V}_h=0$.

Lemma 2.8 ([20])

Let $|\frac{\partial G(u)}{\partial u}|$ be bounded. Then the function

$$H(u,v)=sign(u-v)(G(u)-G(v))$$

satisfies the Lipschitz condition in u and v, respectively.

Lemma 2.9 Let g be the strong solution of problem (7), $f(t,x)\in C_0^{\mathrm{\infty }}({\pi }_T)$ and $f(0,x)=0$. Then

$${\iint }_{{\pi }_T}\{|g-k|f_t+sign(g-k)\frac{1}{2}[g^2-k^2]f_x-sign(g-k)J_g(t,x)f\}dxdt=0,$$

(28)

where k is an arbitrary constant.

Proof Let $\mathrm{\Phi }(g)$ be an arbitrary twice smooth function on the line $-\mathrm{\infty }<g<\mathrm{\infty }$. We multiply the first equation of problem (7) by the function ${\mathrm{\Phi }}^{\prime }(g)f(t,x)$, where $f(t,x)\in C_0^{\mathrm{\infty }}({\pi }_T)$. Integrating over ${\pi }_T$ and transferring the derivatives with respect to t and x to the test function f, for any constant k, we obtain

$${\iint }_{{\pi }_T}\{\mathrm{\Phi }(g)f_t+[{\int }_k^g{\mathrm{\Phi }}^{\prime }(z)zdz]f_x-{\mathrm{\Phi }}^{\prime }(g)J_g(t,x)f\}dxdt=0,$$

(29)

in which we have used ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[{\int }_k^g{\mathrm{\Phi }}^{\prime }(z)zdz]f_{x}dx=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[f{\mathrm{\Phi }}^{\prime }(g)g{g}_x]dx$.

Integration by parts yields

$$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[{\int }_k^g{\mathrm{\Phi }}^{\prime }(z)zdz]f_{x}dx& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[\frac{1}{2}(g^2-k^2){\mathrm{\Phi }}^{\prime }(g)\\ -\frac{1}{2}{\int }_k^g(z^2-k^2){\mathrm{\Phi }}^{″}(z)dz]f_{x}dx.\end{array}$$

(30)

Let ${\mathrm{\Phi }}^h(g)$ be an approximation of the function $|g-k|$ and set $\mathrm{\Phi }(g)={\mathrm{\Phi }}^h(g)$. Using the properties of $sign(g-k)$, (29), (30) and sending $h\to 0$, we have

$${\iint }_{{\pi }_T}\{|g-k|f_t+sign(g-k)\frac{1}{2}[g^2-k^2]f_x-sign(g-k)J_g(t,x)f\}dxdt=0,$$

(31)

which completes the proof. □

In fact, the proof of (28) can also be found in [20].

For $g_{10}\in H^s(R)$ and $g_{20}\in H^s(R)$ with $s>\frac{3}{2}$, using Lemma 2.2, we know that there exists $T>0$ such that two local strong solutions $g_1(t,x)$ and $g_2(t,x)$ of Eq. (3) satisfy

$$g_1(t,x),g_2(t,x)\in C([0,T);H^s(R))C^1([0,T);H^{s-1}(R)),t\in [0,T).$$

(32)

3 Main result

Now, we give the main result of this work.

Theorem 3.1 Assume that $g_1$ and $g_2$ are two local strong solutions of Eq. (3) with initial data $g_{10},g_{20}\in L^1(R)\cap H^s(R)$, $s>\frac{3}{2}$. For $T>0$ in (32), it holds that

$${\parallel g_1(t,\cdot )-g_2(t,\cdot )\parallel }_{L^1(R)}\le c{e}^{ct}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_{10}(x)-g_{20}(x)|dx,t\in [0,T],$$

(33)

where c depends on ${\parallel g_{10}\parallel }_{L^{\mathrm{\infty }}(R)}$, ${\parallel g_{20}\parallel }_{L^{\mathrm{\infty }}(R)}$, ${\parallel g_{10}\parallel }_{L^2(R)}$, ${\parallel g_{20}\parallel }_{L^2(R)}$ and T.

Proof For arbitrary $T>0$ and $f(t,x)\in C_0^{\mathrm{\infty }}({\pi }_T)$, we assume that $f(t,x)=0$ outside the cylinder

$$\uplus =\{(t,x)\}=[\rho ,T-2\rho ]\times K_{r-2\rho },0<2\rho \le min(T,r).$$

(34)

We set

$$\eta =f(\frac{t+\tau }{2},\frac{x+y}{2}){\delta }_h\left(\frac{t-\tau }{2}\right){\delta }_h\left(\frac{x-y}{2}\right)=f(\cdots ){\lambda }_h(\ast ),$$

(35)

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 (26). Note that

$${\eta }_t+{\eta }_{\tau }=f_t(\cdots ){\lambda }_h(\ast ),{\eta }_x+{\eta }_y=f_x(\cdots ){\lambda }_h(\ast ).$$

(36)

Using the Kruzkov device of doubling the variables [20] and Lemma 2.9, we have

$$\begin{array}{r}{⨌}_{{\pi }_T\times {\pi }_T}\{|g_1(t,x)-g_2(\tau ,y)|{\eta }_t\\ +sign(g_1(t,x)-g_2(\tau ,y))(\frac{g_1^2(t,x)}{2}-\frac{g_2^2(\tau ,y)}{2}){\eta }_x\\ -sign(g_1(t,x)-g_2(\tau ,y))J_{g_1}(t,x)\eta \}dxdtdyd\tau =0.\end{array}$$

(37)

Similarly, we have

$$\begin{array}{r}{⨌}_{{\pi }_T\times {\pi }_T}\{|g_2(\tau ,y)-g_1(t,x)|{\eta }_{\tau }\\ +sign(g_2(\tau ,y)-g_1(t,x))(\frac{g_2^2(\tau ,y)}{2}-\frac{g_1^2(t,x)}{2}){\eta }_y\\ -sign(g_2(\tau ,y)-g_1(t,x))J_{g_2}(\tau ,y)\eta \}dxdtdyd\tau =0,\end{array}$$

(38)

from which we obtain

$$\begin{array}{rcl}0& \le & {⨌}_{{\pi }_T\times {\pi }_T}\{|g_1(t,x)-g_2(\tau ,y)|({\eta }_t+{\eta }_{\tau })\\ +sign(g_1(t,x)-g_2(\tau ,y))(\frac{g_1^2(t,x)}{2}-\frac{g_2^2(\tau ,y)}{2})({\eta }_x+{\eta }_y)\}dxdtdyd\tau \\ +|{⨌}_{{\pi }_T\times {\pi }_T}sign(g_1(t,x)-g_2(t,x))(J_{g_1}(t,x)-J_{g_2}(\tau ,y))\eta dxdtdyd\tau |\\ =& I_1+I_2+\left|{⨌}_{{\pi }_T\times {\pi }_T}{I}_{3}dxdtdyd\tau \right|.\end{array}$$

(39)

We will show that

$$\begin{array}{rcl}0& \le & {\iint }_{{\pi }_T}\{|g_1(t,x)-g_2(t,x)|f_t\\ +sign(g_1(t,x)-g_2(t,x))(\frac{g_1^2(t,x)}{2}-\frac{g_2^2(t,x)}{2})f_x\}dxdt\\ +|{\iint }_{{\pi }_T}sign(g_1(t,x)-g_2(t,x))[J_{g_1}(t,x)-J_{g_2}(t,x)]fdxdt|.\end{array}$$

(40)

In fact, the first two terms in the integrand of (39) can be represented in the form

$$A_h=A(t,x,\tau ,y,g_1(t,x),g_2(\tau ,y)){\lambda }_h(\ast ).$$

From Lemma 2.4 and the assumptions on solutions $g_1$, $g_2$, we have ${\parallel g_1\parallel }_{L^{\mathrm{\infty }}}<C_T$ and ${\parallel g_2\parallel }_{L^{\mathrm{\infty }}}<C_T$. From Lemma 2.8, we know that $A_h$ satisfies the Lipschitz condition in $g_1$ and $g_2$, respectively. By the choice of η, we have $A_h=0$ outside the region

$$\{(t,x;\tau ,y)\}=\{\rho \le \frac{t+\tau }{2}\le T-2\rho ,\frac{|t-\tau |}{2}\le h,\frac{|x+y|}{2}\le r-2\rho ,\frac{|x-y|}{2}\le h\}$$

(41)

and

$$\begin{array}{rcl}{⨌}_{{\pi }_T\times {\pi }_T}{A}_{h}dxdtdyd\tau & =& {⨌}_{{\pi }_T\times {\pi }_T}[A(t,x,\tau ,y,g_1(t,x),g_2(\tau ,y))\\ -A(t,x,t,x,g_1(t,x),g_2(t,x))]{\lambda }_h(\ast )dxdtdyd\tau \\ +{⨌}_{{\pi }_T\times {\pi }_T}A(t,x,t,x,g_1(t,x),g_2(t,x)){\lambda }_h(\ast )dxdtdyd\tau \\ =K_{11}(h)+K_{12}.\end{array}$$

(42)

Considering the estimate $|\lambda (\ast )|\le \frac{c}{h^2}$ and the expression of function $A_h$, we have

$$\begin{array}{rl}|K_{11}(h)|\le & c[h+\frac{1}{h^2}\\ \times {⨌}_{|\frac{t-\tau }{2}|\le h,\rho \le \frac{t+\tau }{2}\le T-\rho ,|\frac{x-y}{2}|\le h,|\frac{x+y}{2}|\le r-\rho }|g_2(t,x)-g_2(\tau ,y)|dxdtdyd\tau ],\end{array}$$

(43)

where the constant c does not depend on h. Using Lemma 2.7, we obtain $K_{11}(h)\to 0$ as $h\to 0$. The integral $K_{12}$ does not depend on h. In fact, substituting $t=\alpha$, $\frac{t-\tau }{2}=\beta$, $x=\zeta$, $\frac{x-y}{2}=\xi$ and noting that

$${\int }_{-h}^h{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\lambda }_h(\beta ,\xi )d\xi d\beta =1,$$

(44)

we have

$$\begin{array}{rcl}{K}_{12}& =& 2^2{\iint }_{{\pi }_T}{A}_h(\alpha ,\zeta ,\alpha ,\zeta ,g_1(\alpha ,\zeta ),g_2(\alpha ,\zeta ))\{{\int }_{-h}^h{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\lambda }_h(\beta ,\xi )d\xi d\beta \}d\zeta d\alpha \\ =& 4{\iint }_{{\pi }_T}A(t,x,t,x,g_1(t,x),g_2(t,x))dxdt.\end{array}$$

(45)

Hence

$$\underset{h\to 0}{lim}{⨌}_{{\pi }_T\times {\pi }_T}{A}_{h}dxdtdyd\tau =4{\iint }_{{\pi }_T}A(t,x,t,x,g_1(t,x),g_2(t,x))dxdt.$$

(46)

Since

$$I_3=sign(g_1(t,x)-g_2(\tau ,y))(J_{g_1}(t,x)-J_{g_2}(\tau ,y))f{\lambda }_h(\ast )={\overline{I}}_3(t,x,\tau ,y){\lambda }_h(\ast )$$

(47)

and

$$\begin{array}{r}{⨌}_{{\pi }_T\times {\pi }_T}{I}_{3}dxdtdyd\tau \\ ={⨌}_{{\pi }_T\times {\pi }_T}[{\overline{I}}_3(t,x,\tau ,y)-{\overline{I}}_3(t,x,t,x)]{\lambda }_h(\ast )dxdtdyd\tau \\ +{⨌}_{{\pi }_T\times {\pi }_T}{\overline{I}}_3(t,x,t,x){\lambda }_h(\ast )dxdtdyd\tau =K_{21}(h)+K_{22},\end{array}$$

(48)

we obtain

$$\begin{array}{rl}|K_{21}(h)|\le & c(h+\frac{1}{h^2}\\ \times {⨌}_{|\frac{t-\tau }{2}|\le h,\rho \le \frac{t+\tau }{2}\le T-\rho ,|\frac{x-y}{2}|\le h,|\frac{x+y}{2}|\le r-\rho }|J_{g_2}(t,x)-J_{g_2}(\tau ,y)|dxdtdyd\tau ).\end{array}$$

(49)

Using Lemma 2.7, we have $K_{21}(h)\to 0$ as $h\to 0$. Using (44), we have

$$\begin{array}{rcl}{K}_{22}& =& 2^2{\iint }_{{\pi }_T}{\overline{I}}_3(\alpha ,\zeta ,\alpha ,\zeta ,g_1(\alpha ,\zeta ),g_2(\alpha ,\zeta ))\{{\int }_{-h}^h{\lambda }_h(\beta ,\xi )d\xi d\beta \}d\zeta d\alpha \\ =& 4{\iint }_{{\pi }_T}{\overline{I}}_3(t,x,t,x,g_1(t,x),g_2(t,x))dxdt\\ =& 4{\iint }_{{\pi }_T}sign(g_1(t,x)-g_2(t,x))(J_{g_1}(t,x)-J_{g_2}(t,x))f(t,x)dxdt.\end{array}$$

(50)

From (42), (46), (48), (49) and (50), we prove that inequality (40) holds.

Let

$$\mu (t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1(t,x)-g_2(t,x)|dx.$$

(51)

We define

$${\theta }_h={\int }_{-\mathrm{\infty }}^{\sigma }{\delta }_h(\sigma )d\sigma ({\theta }_h^{\prime }(\sigma )={\delta }_h(\sigma )\ge 0)$$

(52)

and choose two numbers ρ and $\tau \in (0,T_0)$, $\rho <\tau$. In (40), we choose

$$f=[{\theta }_h(t-\rho )-{\theta }_h(t-\tau )]\chi (t,x),h<min(\rho ,T_0-\tau ),$$

(53)

where

$$\chi (t,x)={\chi }_{\epsilon }(t,x)=1-{\theta }_{\epsilon }(|x|+Nt-R+\epsilon ),\epsilon >0.$$

(54)

We note that the function $\chi (t,x)=0$ outside the cone ℧ and $f(t,x)=0$ outside the set ⊎. For $(t,x)\in \mathrm{\mho }$, we have the relations

$$0={\chi }_t+N|{\chi }_x|\ge {\chi }_t+N{\chi }_x.$$

(55)

Applying (53)-(55) and (40), we have the inequality

$$\begin{array}{rcl}0& \le & {\iint }_{{\pi }_{T_0}}\left\{[{\delta }_h(t-\rho )-{\delta }_h(t-\tau )]{\chi }_{\epsilon }|g_1(t,x)-g_2(t,x)|\right\}dxdt\\ +{\int }_0^{T_0}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[{\theta }_h(t-\rho )-{\theta }_h(t-\tau )]|[J_{g_1}(t,x)-J_{g_2}(t,x)]\chi (t,x)|dxdt.\end{array}$$

(56)

Using Lemma 2.6 and letting $\epsilon \to 0$ and $R_0\to \mathrm{\infty }$, we obtain

$$\begin{array}{rcl}0& \le & {\int }_0^{T_0}\left\{[{\delta }_h(t-\rho )-{\delta }_h(t-\tau )]{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1(t,x)-g_2(t,x)|dx\right\}dt\\ +c_0(1+T_0){\int }_0^{T_0}[{\theta }_h(t-\rho )-{\theta }_h(t-\tau )]{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1(t,x)-g_2(t,x)|dxdt.\end{array}$$

(57)

By the properties of the function ${\delta }_h(\sigma )$ for $h\le min(\rho ,T_0-\rho )$, we have

$$\begin{array}{rcl}|{\int }_0^{T_0}{\delta }_h(t-\rho )\mu (t)dt-\mu (\rho )|& =& |{\int }_0^{T_0}{\delta }_h(t-\rho )|\mu (t)-\mu (\rho )|dt|\\ \le & c\frac{1}{h}{\int }_{\rho -h}^{\rho +h}|\mu (t)-\mu (\rho )|dt\to 0\text{as }h\to 0,\end{array}$$

(58)

where c is independent of h. Letting

$$L(\rho )={\int }_0^{T_0}{\theta }_h(t-\rho )\mu (t)dt={\int }_0^{T_0}{\int }_{-\mathrm{\infty }}^{t-\rho }{\delta }_h(\sigma )d\sigma \mu (t)dt,$$

(59)

we get

$$L^{\prime }(\rho )=-{\int }_0^{T_0}{\delta }_h(t-\rho )\mu (t)dt\to -\mu (\rho ),\text{as }h\to 0,$$

(60)

from which we obtain

$$L(\rho )\to L(0)-{\int }_0^{\rho }\mu (\sigma )d\sigma \text{as }h\to 0.$$

(61)

Similarly, we have

$$L(\tau )\to L(0)-{\int }_0^{\tau }\mu (\sigma )d\sigma \text{as }h\to 0.$$

(62)

It follows from (61) and (62) that

$$L(\rho )-L(\tau )\to {\int }_{\rho }^{\tau }\mu (\sigma )d\sigma \text{as }h\to 0.$$

(63)

Send $\rho \to 0$, $\tau \to t$, and note that

$$\begin{array}{rcl}|g_1(\rho ,x)-g_2(\rho ,x)|& \le & |g_1(\rho ,x)-g_{10}(x)|\\ +|g_2(\rho ,x)-g_{20}(x)|+|g_{10}(x)-g_{20}(x)|.\end{array}$$

(64)

Thus, from (57), (58), (63)-(64), we have

$$\begin{array}{rcl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1(t,x)-g_2(t,x)|dx& \le & {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_{10}-g_{20}|dx\\ +c_0(1+T_0){\int }_0^t{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|g_1(t,x)-g_2(t,x)|dxdt,\end{array}$$

(65)

from which we complete the proof by using the Gronwall inequality. □