On the well-posedness of the exp-Rabelo equation

Abstract

The exp-Rabelo equation describes pseudo-spherical surfaces. It is a nonlinear evolution equation. In this paper, the well-posedness of bounded from above solutions for the initial value problem associated with this equation is studied.

1 Introduction

Bäcklund transformations have been useful in the calculation of soliton solutions of certain nonlinear evolution equations of physical significance [9, 22, 25, 26] restricted to one space variable $x$ and a time coordinate $t$. The classical treatment of the surface transformations, which provide the origin of Bäcklund theory, was developed in [11]. Bäcklund transformations are local geometric transformations, which construct from a given surface of constant Gaussian curvature ${-}1$ a two parameter family of such surfaces. To find such transformations, one needs to solve a system of compatible ordinary differential equations [10].

In [15, 16], the authors used the notion of differential equation for a function $u(t,x)$ that describes a pseudo-spherical surface, and they derived some Bäcklund transformations for nonlinear evolution equations which are the integrability condition $sl(2,R)$-valued linear problems [13, 14, 18, 19, 26].

In [20], the authors had derived some Bäcklund transformations for nonlinear evolution equations of the Ablowitz–Kaup–Newell–Segur (AKNS) class. These transformations explicitly express the new solutions in terms of the known solutions of the nonlinear evolution equations and corresponding wave functions which are solutions of the associated AKNS system [2, 32].

In [17], the authors used Bäcklund transformations derived in [15, 16] in the construction of exact soliton solutions for some nonlinear evolution equations describing pseudo-spherical surfaces which are beyond the AKNS class. In particular, they analyzed the following equation [3]:

$$\begin{aligned} \partial _x \left( \partial _tu +\alpha g(u)\partial _x u +\beta \partial _x u\right) =\gamma g'(u), \quad \alpha ,\,\beta ,\,\gamma \in {\mathbb {R}}, \end{aligned}$$

(1.1)

where $g(u)$ is any solution of the linear ordinary differential equation

$$\begin{aligned} g''(u)+\mu g(u)=\theta , \quad \mu ,\,\theta \in {\mathbb {R}}. \end{aligned}$$

(1.2)

(1.1) include the sine-Gordon, sinh-Gordon and Liouville equations, in correspondence of $\alpha =0$.

In [24], Rabelo proved that the system of the Eqs. (1.1) and (1.2) describes pseudo-spherical surfaces and possesses a zero-curvature representation with a parameter.

We consider (1.1) and assume that $\alpha \ne 0$.

When

$$\begin{aligned} \alpha =-1,\qquad \mu =0,\qquad \theta =1, \end{aligned}$$

(1.3)

(1.2) reads

$$\begin{aligned} g''(u)=1. \end{aligned}$$

(1.4)

A solution of (1.4) is

$$\begin{aligned} g(u)=\frac{u^2}{2}. \end{aligned}$$

(1.5)

Taking $\beta =0,\,\gamma >0$, substituting (1.3), and (1.5) in (1.1), we get

$$\begin{aligned} \partial _x \left( \partial _tu -\frac{1}{6}\partial _x u^3\right) =\gamma u, \end{aligned}$$

(1.6)

that was also introduced recently by Schäfer and Wayne [29] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers.

Integrating (1.6) in $x$, we gain the integro-differential formulation of (1.6) (see [27])

$$\begin{aligned} \partial _tu -\frac{1}{6} \partial _x u^3=\gamma \int ^x u(t,y) \mathrm{d}y, \end{aligned}$$

(1.7)

that is equivalent to

$$\begin{aligned} \partial _tu-\frac{1}{6} \partial _x u^3=\gamma P,\qquad \partial _x P=u. \end{aligned}$$

(1.8)

In [4, 6, 8], the authors investigated the well-posedness in classes of discontinuous functions for (1.7) or (1.8). In particular, they proved that (1.7) or (1.8) admits a unique entropy solution in the sense of the following definition:

Definition 1.1

We say that $u\in L^{\infty }((0,T)\times {\mathbb {R}}),\ T>0$, is an entropy solution of (1.7) or (1.8) if

• (i) $u$ is a distributional solution of (1.7) or equivalently of (1.8);

• (ii) for every convex function $\eta \in C^2({\mathbb {R}})$, the entropy inequality

  $$\begin{aligned} \partial _t\eta (u)+ \partial _x q(u)-\gamma \eta '(u) P\le 0, \qquad q(u)=-\int ^u \frac{\xi ^2}{2} \eta '(\xi )\, \mathrm{d}\xi , \end{aligned}$$

  (1.9)

  holds in the sense of distributions in $(0,\infty )\times {\mathbb {R}}$.

Definition 1.1 makes sense because the weak solution of (1.7) lies in $L^{\infty }$, see [4, 6, 8].

Here, we consider the case

$$\begin{aligned} \alpha =1. \end{aligned}$$

(1.10)

Taking $\mu =-1,\,\theta =0$, (1.2) reads

$$\begin{aligned} g''(u)-g(u)=0. \end{aligned}$$

(1.11)

A solution of (1.11) is

$$\begin{aligned} g(u)=e^u. \end{aligned}$$

(1.12)

Taking $\beta =0,\,\gamma =-1$, and substituting (1.10), and (1.12) in (1.1), we get

$$\begin{aligned} \partial _x \left( \partial _tu +\partial _x e^u\right) =- e^u, \end{aligned}$$

(1.13)

which is known as the exp-Rabelo equation (see [12, 28]), and describes pseudo-spherical surfaces with constant negative curvature.

Our aim is to investigate the well-posedness for the initial value problem in classes of discontinuous functions for (1.13). Therefore, we augment (1.13) with the initial datum

$$\begin{aligned} u(0,x)=u_0(x), \end{aligned}$$

(1.14)

on which we assume that

$$\begin{aligned} \sup u_0 <\infty ,\qquad \int _{{\mathbb {R}}}e^{u_0(x)} \mathrm{d}x <\infty . \end{aligned}$$

(1.15)

Integrating (1.13) in $(0,x)$, we gain the integro-differential formulation of (1.13) (see [1, 28, 31])

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+ \partial _x e^u = -\int _{0}^x e^{u(t,y)} \mathrm{d}y,&{}\qquad t>0, \ x\in {\mathbb {R}},\\ u(0,x)=u_0(x), &{}\qquad x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

(1.16)

that is equivalent to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+ \partial _x e^u= -P, &{}\qquad t>0, \ x\in {\mathbb {R}},\\ \partial _x P=e^u, &{}\qquad t>0, \ x>0,\\ P(t,0)=0,&{} \qquad t>0,\\ u(0,x)=u_0(x), &{}\qquad x\in {\mathbb {R}}. \end{array}\right. } \end{aligned}$$

(1.17)

We assume (1.15) because the unique useful conserved quantity is

$$\begin{aligned} t\longmapsto \int _{{\mathbb {R}}} e^{u(t,x)}\mathrm{d}x. \end{aligned}$$

Moreover, we prove that the weak solutions of (1.13) may not belong to $L^{\infty }$, but they are only bounded from above.

Therefore, to have the well-posedness of weak solution, we have to consider the following definition of the entropy:

Definition 1.2

A pair of functions $(\eta , q)$ is called an entropy–entropy flux pair if $\eta :{\mathbb {R}}\rightarrow {\mathbb {R}}$ is a $C^2$ function and $q :{\mathbb {R}}\rightarrow {\mathbb {R}}$ is defined by

$$\begin{aligned} q(u)=\int _{0}^{u} \eta '(\xi )f'(\xi )\mathrm{d}\xi . \end{aligned}$$

An entropy–entropy flux pair $(\eta ,\, q)$ is called convex/compactly supported if, in addition, $\eta$ is convex/compactly supported.

In light of (1.15) and Definition 1.2, we give the following definition of solution:

Definition 1.3

We say that $u$, such that

$$\begin{aligned} u\in L^\infty _{loc}((0,\infty )\times {\mathbb {R}}),\qquad \sup u(t,\cdot )<\infty ,\quad \int _{{\mathbb {R}}} e^{u(t,x)} dx< \infty , \quad t>0, \end{aligned}$$

(1.18)

is an entropy solution of the initial value problem (1.13) and (1.14) if

• (i) $u$ is a distributional solution of (1.16) or equivalently of (1.17);

• (ii) for every convex function $\eta \in C^2({\mathbb {R}})$, the entropy inequality

  $$\begin{aligned} \partial _t\eta (u)+ \partial _x q(u)+\eta '(u) \int _{0}^x e^{u}\mathrm{d}y\le 0, \qquad q(u)=\int ^u e^{\xi } \eta '(\xi )\, \mathrm{d}\xi , \end{aligned}$$

  (1.19)

  holds in the sense of distributions in $(0,\infty )\times {\mathbb {R}}$.

The main result of this paper is the following theorem.

Theorem 1.1

Let $T>0$ be given and assume (1.15). The initial value problems (1.13) and (1.14) possess a unique entropy solution $u$ in the sense of Definition 1.3. Moreover, if $u$ and $w$ are two entropy solutions of (1.13) in the sense of Definition 1.3, the following inequality holds

$$\begin{aligned} \left\| u(t,\cdot )-w(t,\cdot ) \right\| _{L^1(-R,R)}\le e^{C(T) t}\left\| u(0,\cdot )-w(0,\cdot ) \right\| _{L^1(-R-C(T)t,R+C(T)t)}, \end{aligned}$$

(1.20)

for almost every $0<t<T$, $R>0$, and some suitable constant $C(T)>0$ that depends only on $R,\,T, \, \sup u(0,\cdot ),\, \sup w(0,\cdot )$.

The existence argument is based on passing to the limit using the compensated compactness argument of [30] in a vanishing viscosity approximation of (1.17) (see Sect. 2). Moreover, we argue as in [6, 8, 21] for the uniqueness and stability of the solutions of (1.17).

The paper is organized as follows. In Sect. 2, we prove several a priori estimates on a vanishing viscosity approximation of (1.17). Those play a key role in the proof of our main result, that is given in Sect. 3.

2 Vanishing viscosity approximation

Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1.17).

Fix a small number $\varepsilon >0$, and let $u_\varepsilon =u_\varepsilon (t,x)$ be the unique classical solution of the following mixed problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu_\varepsilon +\partial _x e^{u_\varepsilon }=-P_{\varepsilon }+ \varepsilon \partial _{xx}^2\left( e^{u_\varepsilon }\right) ,&{}\quad t>0,\ x\in {\mathbb {R}},\\ \partial _x P_{\varepsilon }=e^{u_\varepsilon },&{}\quad t>0,\ x\in {\mathbb {R}},\\ P_{\varepsilon }(t,0)=0,&{}\quad t>0,\\ u_\varepsilon (0,x)=u_{\varepsilon ,0}(x),&{}\quad x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

(2.1)

where $u_{\varepsilon ,0}$ is a $C^\infty (0,\infty )$ approximation of $u_{0}$ such that

$$\begin{aligned}&u_{0,\varepsilon }\rightarrow u_{0},\quad \text {a.e. and in }L^p_{loc}({\mathbb {R}}),\, 1\le p< \infty ,\nonumber \\&\sup u_{\varepsilon ,0}\le \sup u_{0}, \quad \varepsilon >0,\nonumber \\&\int _{{\mathbb {R}}}e^{u_{\varepsilon ,0}(x)}\mathrm{d}x \le \int _{{\mathbb {R}}} e^{u_{0}(x)} \mathrm{d}x,\quad \varepsilon >0. \end{aligned}$$

(2.2)

Clearly, (2.1) is equivalent to the integro-differential problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu_\varepsilon +\partial _x e^{u_\varepsilon }=-\int _{0}^x e^{u_\varepsilon (t,y)}\mathrm{d}y+ \varepsilon \partial _{xx}^2\left( e^{u_\varepsilon }\right) ,&{}\quad t>0,\ x\in {\mathbb {R}},\\ u_\varepsilon (0,x)=u_{\varepsilon ,0}(x),&{}\quad x\in {\mathbb {R}}. \end{array}\right. } \end{aligned}$$

(2.3)

The existence of solutions of (2.1) can be obtained by fixing a small number $\delta >0$ and considering the further approximation of (2.1) (see [5, 7])

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu_{\varepsilon ,\delta }+\partial _x e^{u_{\varepsilon ,\delta }}=- P_{\varepsilon ,\delta }+\varepsilon \partial _{xx}^2\left( e^{u_{\varepsilon ,\delta }}\right) ,&{}\quad t>0,\ x\in {\mathbb {R}},\\ -\delta \partial _{xx}^2P_{\varepsilon ,\delta }+\partial _x P_{\varepsilon ,\delta }=e^{u_{\varepsilon ,\delta }},&{}\quad t>0,\ x\in {\mathbb {R}},\\ P_{\varepsilon ,\delta }(t,0)=0,&{}\quad t>0,\\ u_{\varepsilon ,\delta }(0,x)=u_{\varepsilon ,0}(x),&{}\quad x\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

and then sending $\delta \rightarrow 0$.

Observe that, multiplying (2.3) by $e^{u_\varepsilon (t,x)}$, we have

$$\begin{aligned} \partial _t\left( e^{u_\varepsilon }\right) +\frac{1}{2}\partial _x \left( e^{2u_\varepsilon }\right) = -e^{u_\varepsilon }\int _{0}^x e^{u_\varepsilon (t,y)}\mathrm{d}y +\varepsilon e^{u_\varepsilon }\partial _{xx}^2\left( e^{u_\varepsilon }\right) . \end{aligned}$$

(2.4)

Introducing the notation

$$\begin{aligned} v_{\varepsilon }(t,x)= e^{u_\varepsilon (t,x)}>0, \end{aligned}$$

(2.5)

(2.4) reads

$$\begin{aligned} \partial _tv_{\varepsilon }+\frac{1}{2} \partial _x v_{\varepsilon }^2= -v_{\varepsilon }\int _{0}^x v_{\varepsilon }(t,y) \mathrm{d}y +\varepsilon v_{\varepsilon }\partial _{xx}^2v_{\varepsilon }. \end{aligned}$$

(2.6)

It follows from (2.5) and $u_\varepsilon (t,\pm \infty )=-\infty$ that

$$\begin{aligned} v_{\varepsilon }(t,\pm \infty )=0. \end{aligned}$$

(2.7)

Moreover, from (2.2) and (2.5), we get

$$\begin{aligned} \left\| v_{\varepsilon ,0} \right\| _{L^{\infty }({\mathbb {R}})}\le e^{\sup u_{0}},\quad \int _{{\mathbb {R}}}v_{\varepsilon ,0}(x)\mathrm{d}x \le \int _{{\mathbb {R}}} e^{u_{0}(x)} \mathrm{d}x, \quad \varepsilon >0. \end{aligned}$$

(2.8)

Let us prove some a priori estimates on $v_{\varepsilon }$, and, hence on $u_\varepsilon$.

Lemma 2.1

Let $T>0$ be given and assume (2.2). We have that

$$\begin{aligned} \left\| v_{\varepsilon } \right\| _{L^{\infty }((0,\infty )\times {\mathbb {R}})} \le e^{\sup u_{0}},\quad \varepsilon >0. \end{aligned}$$

(2.9)

In particular, we get

$$\begin{aligned} \sup u_\varepsilon (t,\cdot ) \le \sup u_{0}, \quad t>0. \end{aligned}$$

(2.10)

Proof

We begin by observing that, from (2.5) and (2.6), we have

$$\begin{aligned} \partial _tv_{\varepsilon }+ \partial _x \left( \frac{v_{\varepsilon }^2}{2}\right) -\varepsilon \partial _{xx}^2\left( \frac{v_{\varepsilon }^2}{2}\right) \le 0. \end{aligned}$$

(2.11)

Therefore, a supersolution of (2.6) satisfies the following ordinary differential equation

$$\begin{aligned} \frac{\mathrm{d}z}{\mathrm{d}t}=0, \quad z(0)= e^{\sup u_{0}}, \end{aligned}$$

that is

$$\begin{aligned} z(t)= e^{\sup u_{0}}. \end{aligned}$$

(2.12)

It follows from the comparison principle for parabolic equation and (2.5) that

$$\begin{aligned} 0< v_{\varepsilon }(t,x) \le e^{\sup u_{0}}, \end{aligned}$$

(2.13)

which gives (2.9).

Finally, (2.10) follows from (2.5) and (2.13). $\square$

Lemma 2.2

Let $\alpha \ge 0$ and $T>0$ be given and assume (2.2). For each $t>0$, we have

$$\begin{aligned}&\int _{{\mathbb {R}}} v_{\varepsilon }^{\alpha +1} +\varepsilon (\alpha +1)^2\int _{0}^{t}\!\!\!\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha }(\partial _x v_{\varepsilon })^2\mathrm{d}s\mathrm{d}x\nonumber \\&\qquad +(\alpha +1)\int _{0}^t\!\!\!\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\left( \int _{0}^{x}v_{\varepsilon }\mathrm{d}y\right) \mathrm{d}s\mathrm{d}x \le \left( e^{\sup u_{0}}\right) ^{\alpha }\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x. \end{aligned}$$

(2.14)

In particular, we get

$$\begin{aligned} \int _{{\mathbb {R}}} e^{(\alpha +1)u_\varepsilon (t,x)} \mathrm{d}x&\le \left( e^{\sup u_{0}}\right) ^{\alpha }\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x,\nonumber \\ \varepsilon (\alpha +1)^2\int _{0}^{t}\!\!\!\int _{{\mathbb {R}}}e^{\alpha u_\varepsilon (t,x)}\left( \partial _x e^{u_\varepsilon (t,x)}\right) ^2\mathrm{d}s\mathrm{d}x&\le \left( e^{\sup u_{0}}\right) ^{\alpha }\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x,\nonumber \\ (\alpha +1)\int _{0}^t\!\!\!\int _{{\mathbb {R}}}e^{(\alpha +1)u_\varepsilon (t,x)}\left( \int _{0}^{x}e^{u_\varepsilon (t,x)} \mathrm{d}y\right) \mathrm{d}s\mathrm{d}x&\le \left( e^{\sup u_{0}}\right) ^{\alpha }\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x. \end{aligned}$$

(2.15)

Proof

Multiplying (2.6) by $v_{\varepsilon }^{\alpha }$, we have

$$\begin{aligned} v_{\varepsilon }^{\alpha } \partial _tv_{\varepsilon }+ v_{\varepsilon }^{\alpha +1}\partial _x v_{\varepsilon }= -v_{\varepsilon }^{\alpha +1}\int _{0}^{x} v_{\varepsilon }\mathrm{d}y+\varepsilon v_{\varepsilon }^{\alpha +1}\partial _{xx}^2v_{\varepsilon }. \end{aligned}$$

It follows from (2.5), (2.6) and an integration on ${\mathbb {R}}$ that

$$\begin{aligned} \frac{1}{\alpha +1}\frac{d}{\mathrm{d}t}\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}=&\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha }\partial _tv_{\varepsilon }\\ =&\varepsilon \int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\partial _{xx}^2v_{\varepsilon }\mathrm{d}x - \int _{{\mathbb {R}}} v_{\varepsilon }^{\alpha +1}\partial _x v_{\varepsilon }-\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\left( \int _{0}^{x} v_{\varepsilon }\mathrm{d}y\right) \mathrm{d}x\\ =&-\varepsilon (\alpha +1)\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha }(\partial _x v_{\varepsilon })^2 \mathrm{d}x -\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\left( \int _{0}^{x} v_{\varepsilon }dy\right) \mathrm{d}x, \end{aligned}$$

that is,

$$\begin{aligned} \frac{d}{\mathrm{d}t}\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}+\varepsilon (\alpha +1)^2\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha }(\partial _x v_{\varepsilon })^2 \mathrm{d}x+(\alpha +1)\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\left( \int _{0}^{x} v_{\varepsilon }\mathrm{d}y\right) \mathrm{d}x= 0. \end{aligned}$$

(2.16)

An integration on $(0,t)$ gives

$$\begin{aligned}&\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\mathrm{d}x + \varepsilon (\alpha +1)^2\int _{0}^{t}\!\!\!\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha }(\partial _x v_{\varepsilon })^2 \mathrm{d}s\mathrm{d}x\nonumber \\&\qquad +(\alpha +1)\int _{0}^{t}\!\!\!\int _{{\mathbb {R}}}v_{\varepsilon }^{\alpha +1}\left( \int _{0}^{x} v_{\varepsilon }\mathrm{d}y\right) \mathrm{d}x=\int _{{\mathbb {R}}}v_{\varepsilon ,0}^{\alpha +1} \mathrm{d}x. \end{aligned}$$

(2.17)

From (2.5) and (2.8),

$$\begin{aligned} \int _{{\mathbb {R}}}v_{\varepsilon ,0}^{\alpha +1}\mathrm{d}x \le \left\| v_{\varepsilon ,0} \right\| ^{\alpha }_{L^{\infty }((0,\infty )\times {\mathbb {R}})}\int _{{\mathbb {R}}} v_{\varepsilon ,0}\mathrm{d}x \le \left( e^{\sup u_{0}}\right) ^{\alpha }\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x. \end{aligned}$$

(2.18)

Therefore, (2.17) and (2.18) give (2.14).

Finally, (2.15) follows from (2.6) and (2.14) $\square$

3 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. We begin with the following result

Lemma 3.1

Fix $T>0$. There exists a subsequence $\{v_{\varepsilon _{k}}\}_{k\in {\mathbb {N}}}$ of $\{v_{\varepsilon }\}_{\varepsilon >0}$ and a limit function $v\in L^{\infty }((0,\infty )\times {\mathbb {R}})$ such that

$$\begin{aligned} v_{\varepsilon _{k}}\rightarrow v \text { a.e. and in }L^{p}_{loc}((0,\infty )\times {\mathbb {R}}), 1\le p<\infty . \end{aligned}$$

(3.1)

In particular, we have

$$\begin{aligned} u_{\varepsilon _{k}}\rightarrow \log v= u \text { a.e. and in } L^{p}_{loc}((0,\infty )\times {\mathbb {R}}), 1\le p<\infty \end{aligned}$$

(3.2)

Proof

Let $\eta :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be any convex $C^2$ entropy function, and $q:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be the corresponding entropy flux defined by $q'(v)=v\eta '(v)$. By multiplying (2.6) with $\eta '(v_{\varepsilon })$ and using the chain rule, we get

$$\begin{aligned} \partial _t\eta (v_{\varepsilon })+\partial _x q(v_{\varepsilon }) =&\underbrace{\varepsilon \partial _x \left( \eta '(v_{\varepsilon })v_{\varepsilon }\partial _x v_{\varepsilon }\right) }_{=:{\mathcal {L}}_{1,\varepsilon }} \, \underbrace{-\varepsilon \eta ''(v_{\varepsilon })v_{\varepsilon }\left( \partial _x u_\varepsilon \right) ^2}_{=: {\mathcal {L}}_{2,\varepsilon }}\\ \,&\underbrace{-\varepsilon \eta '(v_{\varepsilon })\left( \partial _x v_{\varepsilon }\right) ^2}_{=: {\mathcal {L}}_{3,\varepsilon }} \, \underbrace{-\eta '(v_{\varepsilon })v_{\varepsilon }\int _{0}^{x} v_{\varepsilon }\mathrm{d}y}_{=: {\mathcal {L}}_{4,\varepsilon }}, \end{aligned}$$

where ${\mathcal {L}}_{1,\varepsilon }$, ${\mathcal {L}}_{2,\varepsilon }$, ${\mathcal {L}}_{3,\varepsilon }$, ${\mathcal {L}}_{4,\varepsilon }$ are distributions. Let us show that

$$\begin{aligned} {\mathcal {L}}_{1,\varepsilon }\rightarrow 0 \text { in }H^{-1}((0,T)\times {\mathbb {R}}), T>0. \end{aligned}$$

By Lemmas 2.1 and 2.2 in correspondence of $\alpha =0$,

$$\begin{aligned} \left\| \varepsilon \eta '(v_{\varepsilon })v_{\varepsilon }\partial _x v_{\varepsilon } \right\| ^2_{L^2((0,T)\times {\mathbb {R}})}&\le \varepsilon ^2\left\| \eta ' \right\| ^2_{L^{\infty }(I)}\left\| v_{\varepsilon } \right\| _{L^{\infty }((0,\infty )\times {\mathbb {R}})} \int _{0}^{T}\left\| \partial _x v_{\varepsilon }(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\le \varepsilon \left\| \eta ' \right\| ^2_{L^{\infty }(T)}e^{\sup u_{0}}\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x\rightarrow 0, \end{aligned}$$

where

$$\begin{aligned} I=\left( 0, e^{\sup u_{0}}\right) . \end{aligned}$$

We claim that

$$\begin{aligned} \{{\mathcal {L}}_{2,\varepsilon }\}_{\varepsilon >0}\text { is uniformly bounded in } L^1((0,T)\times {\mathbb {R}}), T>0. \end{aligned}$$

Again by Lemmas 2.1 and 2.2 in correspondence of $\alpha =0$,

$$\begin{aligned} \left\| \varepsilon \eta ''(v_{\varepsilon })v_{\varepsilon }(\partial _x v_{\varepsilon })^2 \right\| _{L^1((0,T)\times {\mathbb {R}})}&\le \left\| \eta '' \right\| _{L^{\infty }(I)}\left\| v_{\varepsilon } \right\| _{L^{\infty }((0,\infty )\times {\mathbb {R}})}\varepsilon \int _{0}^{T}\left\| \partial _x v_{\varepsilon }(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\&\le \left\| \eta '' \right\| _{L^{\infty }(I)}e^{\sup u_{0}}\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x. \end{aligned}$$

We have that

$$\begin{aligned} \{{\mathcal {L}}_{3,\varepsilon }\}_{\varepsilon >0}\text { is uniformly bounded in } L^1((0,T)\times {\mathbb {R}}), T>0. \end{aligned}$$

Again by Lemmas 2.1 and 2.2 in correspondence of $\alpha =0$,

$$\begin{aligned} \left\| \varepsilon \eta '(v_{\varepsilon })(\partial _x v_{\varepsilon })^2 \right\| _{L^1((0,T\times {\mathbb {R}})}\le&\left\| \eta ' \right\| _{L^{\infty }(I)}\varepsilon \int _{0}^{T}\left\| \partial _x v_{\varepsilon }(s,\cdot ) \right\| ^2_{L^2({\mathbb {R}})}\mathrm{d}s\\ \le&\left\| \eta ' \right\| _{L^{\infty }(I)}\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x. \end{aligned}$$

We claim that

$$\begin{aligned} \{{\mathcal {L}}_{4,\varepsilon }\}_{\varepsilon >0}\text { is uniformly bounded in } L^1((0,T)\times {\mathbb {R}}), T>0. \end{aligned}$$

Again by Lemmas 2.1 and 2.2 in correspondence of $\alpha =0$,

$$\begin{aligned} \left\| \eta '(v_{\varepsilon })v_{\varepsilon }\int _{0}^{x} v_{\varepsilon }dy \right\| _{L^1((0,T)\times {\mathbb {R}})}\le&\left\| \eta ' \right\| _{L^{\infty }(I)}\int _{0}^{T}\!\!\!\int _{{\mathbb {R}}}v_{\varepsilon }\left( \int _{0}^{x}v_{\varepsilon }\mathrm{d}y\right) \mathrm{d}s\mathrm{d}x\\ \le&\left\| \eta ' \right\| _{L^{\infty }(I)}\int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x. \end{aligned}$$

Therefore, Murat’s lemma [23] implies that

$$\begin{aligned} \left\{ \partial _t\eta (v_{\varepsilon })+\partial _x q(v_{\varepsilon })\right\} _{\varepsilon >0} \text {lies in a compact subset of }H_{{\mathrm {loc}}}^{-1}((0,T)\times {\mathbb {R}}). \end{aligned}$$

(3.3)

The $L^{\infty }$ bound stated in Lemma 2.1, (3.3), and the Tartar’s compensated compactness method [30] give the existence of a subsequence $\{v_{\varepsilon _{k}}\}_{k\in {\mathbb {N}}}$ and a limit function $v\in L^{\infty }((0,\infty )\times {\mathbb {R}}),$ such that (3.1) holds.

(3.2) follows from (2.5) and (3.1). $\square$

Proof of Theorem 1.1

We begin by proving that $u$, defined in (3.2), is an entropy solution of (1.16) or (1.17) in the sense of Definition 1.3. Let $\phi \in C^{\infty }({\mathbb {R}}^2)$ be a positive text function with a support, and let us consider a compactly supported entropy–entropy flux pair $(\eta , q)$. We have to prove

$$\begin{aligned}&\int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}(\eta (u)\partial _t\phi +q(u)\partial _x \phi )\mathrm{d}t\mathrm{d}x - \int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}\eta '(u)\left( \int _{0}^{x}e^{u} \mathrm{d}y \right) \mathrm{d}t\mathrm{d}x\nonumber \\&\qquad + \int _{{\mathbb {R}}}\eta (u_{0}(x))\phi (0,x)\mathrm{d}x \ge 0. \end{aligned}$$

(3.4)

Multiplying (2.1) by $\eta '(u_\varepsilon )$, we have

$$\begin{aligned} \partial _t\eta (u_{\varepsilon _{k}})+\partial _x q(u_{\varepsilon _{k}}) +\eta '(u_{\varepsilon _{k}})\int _{0}^{x} e^{u_{\varepsilon _{k}}} \mathrm{d}dy = \varepsilon _{k}\eta '(u_{\varepsilon _{k}})\partial _{xx}^2\left( e^{u_{\varepsilon _{k}}}\right) . \end{aligned}$$

Since

$$\begin{aligned} \varepsilon _{k}\eta '(u_{\varepsilon _{k}})\partial _{xx}^2\left( e^{u_{\varepsilon _{k}}}\right) =&\partial _x \left( \varepsilon _{k}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right) -\varepsilon _{k}\eta ''(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \partial _x u_{\varepsilon _{k}}\\ =&\partial _x \left( \varepsilon _{k}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right) -\varepsilon _{k} \eta ''(u_{\varepsilon _{k}}) e^{u_{\varepsilon _{k}}}(\partial _x u_{\varepsilon _{k}})^2, \end{aligned}$$

we have

$$\begin{aligned}&\partial _t\eta (u_{\varepsilon _{k}})+\partial _x q(u_{\varepsilon _{k}}) +\eta '(u_{\varepsilon _{k}})\int _{0}^{x} e^{u_{\varepsilon _{k}}} \mathrm{d}y\nonumber \\&\quad = \partial _x \left( \varepsilon _{k}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right) -\varepsilon _{k} \eta ''(u_{\varepsilon _{k}}) e^{u_{\varepsilon _{k}}}(\partial _x u_{\varepsilon _{k}})^2\nonumber \\&\quad \le \partial _x \left( \varepsilon _{k}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right) . \end{aligned}$$

(3.5)

Multiplying (3.5) by $\phi$, an integration on $(0,\infty )\times {\mathbb {R}}$ gives

$$\begin{aligned}&\int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}(\eta (u_{\varepsilon _{k}})\partial _t\phi +q(u_{\varepsilon _{k}})\partial _x \phi )\mathrm{d}t\mathrm{d}x -\int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}\eta '(u_{\varepsilon _{k}})\left( \int _{0}^{x}e^{u_{\varepsilon _{k}}} \mathrm{d}y \right) \mathrm{d}t\mathrm{d}x\nonumber \\&\qquad + \int _{{\mathbb {R}}}\eta (u_{\varepsilon _{k}, 0}(x))\phi (0,x)\mathrm{d}x \nonumber \\&\qquad +\varepsilon _{k}\int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \partial _x \phi \mathrm{d}t\mathrm{d}x\ge 0. \end{aligned}$$

(3.6)

Let us show that

$$\begin{aligned} \varepsilon _{k}\int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \partial _x \phi \mathrm{d}t\mathrm{d}x\rightarrow 0. \end{aligned}$$

(3.7)

Fix $T>0$. From (2.14) in correspondence of $\alpha =0$, and the Hölder inequality,

$$\begin{aligned}&\varepsilon _{k}\int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}}\eta '(u_{\varepsilon _{k}})\partial _x \left( e^{u_{\varepsilon _{k}}}\right) \partial _x \phi \mathrm{d}t\mathrm{d}x\\&\quad \le \varepsilon _{k} \int _{0}^{\infty }\!\!\!\int _{{\mathbb {R}}} \vert \eta '(u_{\varepsilon _{k}}) \vert \left| \partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right| \vert \partial _x \phi \vert \mathrm{d}t\mathrm{d}x\\&\quad \le \varepsilon _{k} \left\| \eta ' \right\| _{L^{\infty }((0,\infty )\times {\mathbb {R}})}\left\| \partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right\| _{L^2({\mathrm {supp}}\,(\partial _x \phi ))}\left\| \partial _x \phi \right\| _{L^2({\mathrm {supp}}\,(\partial _x \phi ))}\\&\quad \le \varepsilon _{k} \left\| \eta ' \right\| _{L^{\infty }((0,\infty )\times {\mathbb {R}})}\left\| \partial _x \left( e^{u_{\varepsilon _{k}}}\right) \right\| _{L^2((0,T)\times {\mathbb {R}})}\left\| \partial _x \phi \right\| _{L^2((0,T)\times {\mathbb {R}})}\\&\quad \le \sqrt{\varepsilon _{k}}\left\| \eta ' \right\| _{L^{\infty }((0,\infty )\times {\mathbb {R}})}\left\| \partial _x \phi \right\| _{L^2((0,T)\times {\mathbb {R}})}\left( \int _{{\mathbb {R}}}e^{u_{0}(x)} \mathrm{d}x\right) ^{\frac{1}{2}}\rightarrow 0, \end{aligned}$$

that is (3.7).

Therefore, (3.6) follows from (2.2), (3.2), (3.6) and (3.7).

We claim that prove that $u(t,x)$ is unique and (1.20) holds. We consider two entropy solutions $u(t,x),\, w(t,x)$ be of (1.16) or (1.17) such that

$$\begin{aligned} \sup u(t,\cdot )\le & {} \sup u_{0}, \quad \sup w(t,\cdot ) \le \sup w_{0},\quad t>0,\nonumber \\ \int _{{\mathbb {R}}} e^{u_{0}(x)}\mathrm{d}x< & {} \infty ,\quad \int _{{\mathbb {R}}} e^{w_{0}(x)} \mathrm{d}x <\infty , \end{aligned}$$

(3.8)

Due to (3.8), we have

$$\begin{aligned} \left| e^{u}-e^{w}\right| \le C_{0}\vert u -w\vert , \end{aligned}$$

(3.9)

where

$$\begin{aligned} C_{0} = e^{\sup u_{0}} + e^{\sup w_{0}}. \end{aligned}$$

Arguing as in [6], Theorem$3.1$], we can prove that

$$\begin{aligned} \partial _t(\vert u - w \vert ) + \partial _x \left( (e^{u} - e^{w}\right) {\mathrm {sign}}\left( u-w\right) ) +{\mathrm {sign}}\left( u-w\right) \int _{0}^{x}\left( e^{u}-e^{w}\right) \mathrm{d}y \le 0 \end{aligned}$$

holds in sense of distributions in $(0,\infty )\times {\mathbb {R}}$, and

$$\begin{aligned}&\left\| u(t,\cdot )-w(t,\cdot ) \right\| _{I(t)}\nonumber \\&\quad \le \left\| u_{0}-w_{0} \right\| _{I(0)}-\int _{0}^{t}\!\!\int _{I(s)}{\mathrm {sign}}\left( u-w\right) \left( \int _{0}^{x}\left( e^{u}-e^{w}\right) \mathrm{d}y\right) \mathrm{d}s\mathrm{d}x, \quad 0<t<T,\qquad \qquad \end{aligned}$$

(3.10)

where

$$\begin{aligned} I(s)=[-R-C_{0}(t-s),R+C_{0}(t-s)]. \end{aligned}$$

Due to (3.9),

$$\begin{aligned}&-\int _{0}^{t}\!\!\int _{I(s)}{\mathrm {sign}}\left( u-v\right) \left( \int _{0}^{x}\left( e^{u}-e^{w}\right) dy\right) \mathrm{d}s\mathrm{d}x\nonumber \\&\quad \le \int _{0}^{t}\!\!\int _{I(s)}\left| \left( \int _{0}^{x}\left| e^{u}-e^{w}\right| \mathrm{d}y\right) \right| \mathrm{d}s\mathrm{d}x\nonumber \\&\quad \le C_{0}\int _{0}^{t}\!\!\int _{I(s)}\Big (\Big \vert \int _{I(s)}\vert u-v\vert \mathrm{d}y\Big \vert \Big )\mathrm{d}s\mathrm{d}x\nonumber \\&\quad =C_{0} \int _{0}^{t} \vert I(s)\vert \left\| u(s,\cdot )-v(s,\cdot ) \right\| _{L^{1}(I(s))}\mathrm{d}s. \end{aligned}$$

(3.11)

Moreover,

$$\begin{aligned} \vert I(s) \vert =2R+2C_{0}(t-s)\le 2R+2C_{0}t\le 2R+2C_{0}T. \end{aligned}$$

(3.12)

We consider the following continuous function:

$$\begin{aligned} G(t)=\left\| u(t,\cdot )-v(t,\cdot ) \right\| _{L^{1}(I(t))}, \quad t\ge 0. \end{aligned}$$

(3.13)

It follows from (3.10), (3.11), (3.12) and (3.13) that

$$\begin{aligned} G(t)\le G(0)+C(T)\int _{0}^{t}G(s)\mathrm{d}s, \end{aligned}$$

where $C(T)=2R+2C_{0}T$. The Gronwall inequality and (3.13) give

$$\begin{aligned} \left\| u(t,\cdot )-v(t,\cdot ) \right\| _{L^{1}(-R,R)}\le e^{C(T)t}\left\| u_{0}-v_{0} \right\| _{L^{1}(-R-C_{0}t,R+C_{0}t)}, \end{aligned}$$

that is (1.20). $\square$