# Initial-Boundary value problem for 1D scalar balance laws with strictly
convex flux

Manas R. Sahoo1, Abhrojyoti Sen2 and Manish Singh1

###### Abstract.

A Lax-Oleĭnik type explicit formula of a scalar balance law has been recently
obtained for the pure initial value problem[1]. In this article, by
introducing a suitable functional, we establish a Lax-Oleĭnik type formula for
the initial-boundary value problem. The obtained explicit formula satisfies
the initial condition in the strong sense and the boundary condition in the
sense of Bardos, le Roux and Nédélec[2]. Finally, by introducing
characteristic triangles, we constructed generalized characteristics and
showed that the obtained solution is entropy admissible.

###### Key words and phrases:

Balance laws; Hamilton-Jacobi equation; Explicit formula; $h$-curves; Boundary
value problems; Lax-Oleĭnik formula; Characteristic triangles; Generalised
characteristics.
1School of Mathematical Sciences, National Institute of Science Education and
Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, P.O. Jatni,
Khurda, Odisha 752050, India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Department of Mathematics, Indian Institute of Science Education and
Research, Dr. Homi Bhabha Road, Pune 411008, India. Email:
<EMAIL_ADDRESS>
###### 2020 Mathematics Subject Classification:

35D30, 35F31, 35L04, 35L65, 35L67.

##  1\. Introduction

We are interested to study the initial-boundary value problem for the scalar
balance laws in one space dimension,

(1.1)
$\displaystyle\begin{cases}&u_{t}+f(u)_{x}=\alpha(t)u,\,\,\,\,\,\,x>0,\,\,t>0,\\\
&u(x,0)=u_{0}(x),\\\ &u(0,t)=u_{b}(t),\end{cases}$

where $f$ is a $C^{2}(\mathbb{R};\mathbb{R})$ strictly convex function with
superlinear growth, i.e.

$\displaystyle{\lim_{|u|\to\infty}}\frac{f(u)}{|u|}=\infty,$

$\alpha\in\mathbb{L}^{\infty}([0,\infty);\mathbb{R})$, $u_{0}$ and $u_{b}$ are
bounded measurable functions. The Cauchy problem for the scalar conservation
laws (when $\alpha=0$) in one and higher dimension has been studied
extensively (see[17, 23, 3]) and references therein. One of the important
aspects here was to obtain an explicit formula. In this direction the first
work was done by E. Hopf [12], where he considered the Cauchy problem for the
Burgers equation, i.e. when $f(u)=\frac{u^{2}}{2}.$ Using the vanishing
viscosity method, Hopf obtained an explicit formula for the initial value
problem in one space dimension. Later, Lax [15] generalized this formula for
any strictly convex flux function $f$, with superlinear growth. Joseph and
Gowda [13] studied (1.1) when the flux function is strictly convex and
$\alpha(t)=0$. Recently, Adimurthi et al. [1] extended the Lax-Oleĭnik formula
to the scalar balance laws (1.1) for the pure initial value problem, where the
source field is linear. Now the next question is to obtain an explicit formula
for the equation (1.1) adjoined with initial and boundary values. This paper
aims to obtain a Lax-Oleĭnik type explicit representation of the solution for
the aforesaid initial-boundary value problem (1.1). We observe that the
behaviour of the characteristic curves for this case is completely different
compared to the case $\alpha=0.$ We will describe this phenomenon in detail in
the later sections.

For the initial-boundary value problem, boundary condition may not hold for
any arbitrary boundary data. We show that the derived explicit formula for the
solution satisfies the initial condition in the strong sense and the weak form
of the boundary condition in the sense of Bardos- le Roux- Nédélec [2] when
the initial data is bounded measurable, and boundary data is continuous.

Let $u_{b}(t)$ be any given continuous boundary data, then the weak form of
boundary condition in the sense of Bardos- le Roux- Nédélec for the solution
$u(x,t)$ is the following:

$\sup_{k\in
I(u(0+,t),u_{b}(t))}Sgn\big{[}\big{(}u(0+,t)-k\big{)}\big{(}f(u(0+,t))-f(k)\big{)}\big{]}=0,\,\,\textnormal{for
almost every $t>0$},$

where

$I(u(0+,t),u_{b}(t))=\big{[}\min\\{u(0+,t),u_{b}(t)\\},\max\\{u(0+,t),u_{b}(t)\\}\big{]}.$

For strictly convex $f,$ the above condition can be simplified [16] to an
equivalent form,

(1.2)
$\displaystyle\begin{cases}\textnormal{either}\,\,\,u(0+,t)=\bar{u_{b}}(t),\\\
\textnormal{or}\,\,\,\,f^{\prime}(u(0+,t)))\leq
0\,\,\,\textnormal{and}\,\,\,f(u(0+,t))\geq f(\bar{u_{b}}(t)).\end{cases}$

Here $\bar{u_{b}}(t)=\max\\{u_{b}(t),\lambda_{f}\\},$ where $\lambda_{f}$ is
the point where $f^{\prime}$ changes its sign. In the next section, we briefly
mention some of the earlier works studying the balance laws.

###  1.1. Background of scalar balance laws

The general form of a Cauchy problem of scalar balance laws reads

$\displaystyle u_{t}+f(u)_{x}=g(x,t,u),$ $\displaystyle u(x,0)=u_{0}(x).$

Dafermos[4] studied the structure of the solution of the above Cauchy problem
for more general flux with source field, where $f\in C^{2},f_{uu}>0,$ $g\in
C^{1}$ and also $f,g$ depends on time and space variables. However, the
authors in this paper[4] did not discuss the construction procedure of the
solution. The main tool for studying the structure of the solution is to study
the generalized characteristics associated with the solution. The author deals
with a solution $u(x,t)$ which is in the class of $BV_{loc}$ on
$\mathbb{R}\times[0,\infty)$ and $u(\cdot,t)$ has locally bounded variation in
$x$ on $\mathbb{R}$ for any fixed $t\in[0,\infty).$

Another physically important equation is the balance laws with a moving
source, for instance,

$u_{t}+f(u)_{x}=C(x)h(u).$

The authors studied the existence theory via vanishing viscosity method [6]
and large time behaviour of the solution[18], taking different appropriate
conditions on the source field $C(x)h(u).$ There is quite a number of
literature where the large time behaviour has been extensively studied,
possibly for the general source term $g(x,t,u).$ Depending on the choice of
method, suitable conditions on the flux and the source are taken in different
situations, see [5, 19, 20, 8, 9, 11, 7, 21, 24, 25, 10]. We emphasize that
the works mentioned above did not consider the construction of explicit
formulas as the presence of the source term affects the geometric structure of
the characteristic curves. Even for the case when the source depends only on
$(x,t)$ (but not on the solution), the question is not well settled because
the shape of the characteristics depends on $x$ and $t$. Next, we describe the
main contributions of this paper in detail.

###  1.2. Main results

As mentioned earlier, several authors studied the scalar balance laws from the
perspective of structure of solution, asymptotic behaviour and so on. There
was no explicit construction of solution even for the initial value problem.
Adimurthi et al.[1] first studied the Lax-Oleĭnik type formula for the pure
initial value problem. The main difficulty here is that the characteristics
are not straight lines. Here we extend the previous results of [1] to the
initial-boundary value problem. The counterpart of the conservation law (1.1)
is the Hamilton-Jacobi equation

$W_{t}+e^{-\beta(t)}f(e^{\beta(t)}W_{x})=0,$

where $\beta(t)=\int_{0}^{t}\alpha(\theta)d\theta.$ The above equation should
be adjoined with the following initial and boundary conditions.

$\displaystyle W(x,0)=\int_{0}^{x}u_{0}(y)dy,$ $\displaystyle
W_{x}(0,t)=e^{\beta(t)}u_{b}(t).$

Motivated by the works[13, 14], we consider the following functional which is
defined in Definition (2.2)

$W(x,t)=\inf_{\gamma\in\Gamma(x,t)}\Bigg{\\{}\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\gamma(0))\Bigg{\\}}.\\\
$

The minimization of the above functional exists which will be shown later and
have certain particular geometric structure. Those curves will be known as the
characteristics. Coming back to the pure initial value case[1], the
characteristic curves here are known as $h$-curves (for the definition of
$h$-curve see section $2$) are not straight lines in general.

In general for the boundary value problem, it is not possible to directly work
with all the possible $h$-curves as we observed that for certain flux $f$ and
$\alpha(t)$, the characteristic curve can cross the boundary and go out of the
prescribed domain (see example (1.1)). Not only that, as an effect of the
source term the $h$-curve may oscillate along the boundary (which is $t$-axis)
and meet the boundary countably many times (see example (1.2)). This
phenomenon is completely different from the homogeneous case $\alpha(t)=0.$
For the homogeneous case, Joseph and Gowda [13] showed that the characteristic
curve is at most of three pieces of line segments. Unlike them, here
characteristic curve may consist of multi-piece $h$-curves, in fact, we stress
that the characteristic curve consists of countably many pieces of $h$-curves
along with the boundary data. This phenomenon makes our problem significantly
different and more delicate as compared to the previous works on the initial-
boundary value problems. In this case, the proofs are also much involved and
new ideas are needed to overcome the difficulties. It will be shown that the
minimization (see Theorem(3.4)) can be split into the minimization problem of
two functionals known as the initial functional and boundary functional that
are given by

$\displaystyle
A(y,x,t)=\int_{0}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x-y,t)e^{\beta(\theta)}))d\theta+\int_{0}^{y}u_{0}(\theta)d\theta,$
$\displaystyle
B(\tau,x,t)=\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,\tau)e^{\beta(\theta)}))d\theta+W(0,\tau),$

respectively, where $f^{*}(u)$ is the convex dual of $f(u)$ and defined as
$f^{*}(u)=\displaystyle{\max_{v\in\mathbb{R}}}[uv-f(v)].$ Further, let us
define,

$\displaystyle A(x,t)=\inf_{y\in H_{I}}A(y,x,t),$ $\displaystyle
B(x,t)=\inf_{\tau\in H_{b}}B(\tau,x,t).$

See Theorem (3.4) for the definition of $H_{I},H_{b}.$ Now we state the main
result regarding the explicit formula for the solution of (1.1).

######  Theorem 1.1.

Let $W(x,t)=\min\\{A(x,t),B(x,t)\\}$ and define $W_{x}=w.$ Then

$\displaystyle
w(x,t)=\begin{cases}&h(x-y(x,t),t)\,\,\,\,\,\,\,\,\textnormal{if}\,\,\,\,A(x,t)\leq
B(x,t),\\\
&h(x,t,\tau(x,t))\,\,\,\,\,\,\,\,\,\,\,\textnormal{if}\,\,\,\,A(x,t)>B(x,t).\end{cases}$

and $w$ satisfies

$w_{t}+e^{-\beta(t)}f(e^{\beta(t)}w)_{x}=0,\,\,\textnormal{a.e.},$

where $\beta(t)=\int_{0}^{t}\alpha(\theta)d\theta.$

The next task is to show that the constructed solution satisfies boundary
condition. Since the characteristic curves can move out of the domain and may
come back inside the domain for certain time $t,$ the minimiser
$\tau(x,t)\not\to t$ always as $x\to 0+$, therefore the proof of boundary
condition is also not direct. We use dynamic programming principle when the
characteristic forms a loop near boundary.

Next, we described our solution by constructing the generalised characteristic
which is the usual way for the classical solution. For this we need to
introduce different types of characteristics triangles with an apex
$(x,t),x>0,t>0,$ depending on whether $A(x,t)<B(x,t),A(x,t)>B(x,t)$ or
$A(x,t)=B(x,t).$ More precisely, we construct a curve $\mathcal{X}(t)$ such
that $\mathcal{\dot{X}}(t)=f^{\prime}(u(\mathcal{X}(t),t)).$ This may be
useful while studying the structure of solution for the initial boundary value
problem and balance laws with discontinuous flux which will be our next goal
of research.

We finish this section by giving two examples: the first example shows that
minimization can not be taken over all $h$-curves joining any point $(x,t)$
and $(y,0)$ for a balance laws, as the $h$-curve joining these points may
leave the domain $Q$-the quarter plane.
The second example demonstrates that the minimization may be achieved by a
curve which is of countably many pieces of $h$-curves.

######  Example 1.1.

Consider the balance laws

$\displaystyle\begin{cases}&u_{t}+\Big{(}\frac{(u-60)^{2}}{2}\Big{)}_{x}=\Big{(}\frac{12t^{2}-60t+70}{4t^{3}-30t^{2}+70t+10}\Big{)}u,\\\
&u(x,0)=10.\end{cases}$

The $h$-curve $X(t)$, joining $(24,5)$ and $(0,24)$ satisfies

$\displaystyle X^{\prime}(t)=10e^{\beta(t)}-60,$ $\displaystyle
X(0)=24,X(5)=24$

where

$\beta(t)=\int_{0}^{t}\frac{12s^{2}-60s+70}{4s^{3}-30s^{2}+70s+10}ds.$

Then one can solve the above equation (1.1) to get

$X(t)=(t-1)(t-2)(t-3)(t-4).$

This crosses the $t$-axis four times, and parts of the curve lie outside the
quarter plane. It is easy to see that $\alpha(t)$ is bounded as $t\to 0$ and
$t\to\infty$ and also the roots of the denominator are either negative or
complex. Hence $\alpha(t)\in\mathbb{L}^{\infty}([0,\infty),\mathbb{R}).$

######  Example 1.2.

Consider the balance laws

$\displaystyle\begin{cases}&u_{t}+\Big{(}\frac{(u-60)^{2}}{2}\Big{)}_{x}=\alpha(t)u,\\\
&u(x,0)=0,\\\ &u(0,t)=u_{b}(t),\end{cases}$

where $\alpha(t)$ and $u_{b}(t)$ are given by

$\displaystyle\alpha(t)=\begin{cases}\frac{12[t\sin(1/t)(\sin(1/t)-\frac{1}{t}\cos(1/t))]^{2}+4(t\sin(1/t))^{3}[-\frac{1}{t^{2}}\cos(1/t)-\frac{1}{t^{3}}\sin(1/t)+\frac{1}{t^{2}}\cos(1/t)]}{60+4(t\sin(1/t))^{3}(\sin(1/t)-\frac{1}{t}\cos(1/t)))}\,&\textnormal{if}\,\,t\in\Big{(}0,\frac{1}{\pi}\Big{)},\\\
0&\textnormal{if}\,\,t\in\Big{(}\frac{1}{\pi},\infty\Big{)}.\end{cases}$
$\displaystyle
u_{b}(t)=\begin{cases}60\,\,\,&\textnormal{if}\,\,t\in\Big{(}\frac{1}{(2n-1)\pi},\frac{1}{2n\pi}\Big{)},\\\
7204n(n+1)\pi^{2}&\textnormal{if}\,\,t\in\Big{(}\frac{1}{2n\pi},\frac{1}{(2n+1)\pi}\Big{)},\\\
60&\textnormal{if}\,\,t\in\Big{(}\frac{1}{\pi},\infty\Big{)}.\end{cases}$

where $n\in\mathbb{N}.$ Invoking Lemma(3.2), one can easily check that

$\displaystyle W\Big{(}0,\frac{1}{\pi}\Big{)}$
$\displaystyle=\inf_{\gamma\in\tilde{\Gamma}(0,\frac{1}{\pi})}\Bigg{\\{}\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+v_{0}(\gamma(0))\Bigg{\\}}$
$\displaystyle=\sum_{n\in\mathbb{N}}\int^{\frac{1}{(2n-1)\pi}}_{\frac{1}{2n\pi}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\sum_{n\in\mathbb{N}}\int^{\frac{1}{2n\pi}}_{\frac{1}{(2n+1)\pi}}f(u_{b}(\theta))d\theta.$

Here $f(u)=\frac{(u-60)^{2}}{2},$ $f^{*}(u)=\frac{u^{2}}{2}+60u$ and
$\gamma(\theta)=[\theta\sin\frac{1}{\theta}]^{4}.$ Therefore the curve along
which the minimization achieved is a multi-piece $h$-curves.

######  Remark 1.2.

If $\bar{u}_{b}$ satisfies $f(\lambda_{f}e^{\beta(t)})\leq f(\bar{u}_{b}(t))$
for almost every $t>0,$ then the characteristic curve is simpler and is at
most a three piece curve, i.e,

(1.3) $\displaystyle W(x,t)$ $\displaystyle=\min_{t\geq t_{2}\geq t_{1}\geq
0,y}\int_{t_{2}}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,t_{2})e^{\beta(\theta)}))d\theta-\int_{t_{1}}^{t_{2}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle+\int_{0}^{t_{1}}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(-y,t_{1})e^{\beta(\theta)}))d\theta+\int_{0}^{y}u_{0}(\theta)d\theta.$

In particular if $f^{\prime}(0)=0,$ the characteristic is at most of three
pieces.

###### Proof of Remark $1.2$.

Suppose for any interval $[t_{1},t_{2}]$ the minimum is attained by curve
$\gamma(\theta)$. For any given $\bar{u_{b}},$ if we show that the following

$\int_{t_{1}}^{t_{2}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta\geq-\int_{t_{1}}^{t_{2}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta$

holds, then it forces that the minimum to be hold along the boundary.
Exploiting the above inequality and using $\gamma(t_{1})=\gamma(t_{2})=0$, we
get

$\displaystyle\int_{t_{1}}^{t_{2}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta=\int_{t_{1}}^{t_{2}}\sup_{v\in\mathbb{R}}[v\dot{\gamma}(\theta)-f(v)]e^{-\beta(\theta)}d\theta\geq-\int_{t_{1}}^{t_{2}}f(\lambda_{f}e^{\beta(\theta)})e^{-\beta(\theta)}d\theta$

which gives the above Remark (1.2). ∎

In particular if $\lambda_{f}=0,$ then the remark (1.2) gives
$f(\bar{u_{b}}(\theta))\geq f(0).$ This is always true if $f^{\prime}(0)=0$
and therefore if $f^{\prime}$ changes sign at $0$, the characteristic curve
must be of three piece.

###  1.3. Organization of the paper

The rest of the article is organized as follows. In section $2$, we recollect
some well known results which will be useful to prove Theorem (1.1). In
section $3$, we prove Theorem (1.1) via variational approach. Section $4$
discusses the boundary conditions. Lastly, in section $5$ we construct
generalised characteristic and show the solution for the initial-boundary
value problem is entropy admissible in the sense of Lax.

##  2\. Preliminaries

Let us introduce a few definitions that will be used throughout the paper.
Denote the quarter plane $Q=[0,\infty)\times[0,\infty).$ We start by defining
the controlled curves.

######  Definition 2.1 (Controlled Curves).

Let $x\in\mathbb{R}$ and $0\leq s\leq t$, we then define the set of controlled
curves to be

$\displaystyle\Gamma(x,t,s)$ $\displaystyle:=\Big{\\{}\gamma:[s,t]\to
Q,\gamma(t)=x,\,\,\,\gamma\,\,\,\mbox{is a Lipschitz function}\Big{\\}},$
$\displaystyle\Gamma(x,t)$ $\displaystyle:=\Gamma(x,t,0).$

Next we define value functions.

######  Definition 2.2 (Value function).

Let $x\in\mathbb{R}$ and $0\leq s<t$, define the value functions by

(2.1) $\displaystyle W(x,t)$
$\displaystyle=\inf_{\gamma\in\Gamma(x,t)}\Bigg{\\{}\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\gamma(0))\Bigg{\\}},$
(2.2) $\displaystyle W(x,t,s)$
$\displaystyle=\inf_{\gamma\in\Gamma(x,t,s)}\left\\{\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(\gamma(s),s)\right\\},$

where $W_{0}(y)=\int_{0}^{y}u_{0}(x)\,dx.$

######  Definition 2.3 (Characteristics sets).

Define the characteristic sets $ch(x,t,s)$ and $ch(x,t)$ by

$\displaystyle ch(x,t,s)$
$\displaystyle:=\\{\gamma\in\Gamma(x,t,s):\gamma\text{ is a minimiser in
}\eqref{HL1}\\},$ $\displaystyle ch(x,t)$
$\displaystyle:=\\{\gamma\in\Gamma(x,t):\gamma\text{ is a minimiser in
}\eqref{HL}\\}.$

Next we define the extreme characteristic points, $y_{*},\ y^{*}$ for the
initial functional and $\tau_{*},\ \tau^{*}$ for the boundary functional.

######  Definition 2.4 (Extreme Characteristics points).

Let

$\displaystyle y^{*}(x,t,s)$ $\displaystyle:=\max\left\\{\gamma(s):\gamma\in
ch(x,t,s)\right\\},$ $\displaystyle y_{*}(x,t,s)$
$\displaystyle:=\min\left\\{\gamma(s):\gamma\in ch(x,t,s)\right\\},$
$\displaystyle y^{*}(x,t)$ $\displaystyle:=\max\left\\{\gamma(0):\gamma\in
ch(x,t)\right\\},$ $\displaystyle y_{*}(x,t)$
$\displaystyle:=\min\left\\{\gamma(0):\gamma\in ch(x,t)\right\\}.$

Similarly, for the boundary functional we define,

######  Definition 2.5 (Extreme Characteristics points).

Let

$\displaystyle\tau^{*}(x,t,s)$ $\displaystyle:=\max\left\\{\gamma(s):\gamma\in
ch(x,t,s)\right\\},$ $\displaystyle\tau_{*}(x,t,s)$
$\displaystyle:=\min\left\\{\gamma(s):\gamma\in ch(x,t,s)\right\\},$
$\displaystyle\tau^{*}(x,t)$ $\displaystyle:=\max\left\\{\gamma(0):\gamma\in
ch(x,t)\right\\},$ $\displaystyle\tau_{*}(x,t)$
$\displaystyle:=\min\left\\{\gamma(0):\gamma\in ch(x,t)\right\\}.$

We finish this section by recalling [1] the definition of $h$-curves.

######  Definition 2.6 ($h$-curve).

The curve joining from $(x_{1},t_{1})$ to $(x_{2},t_{2}),$ where $0\leq
t_{2}<t_{1}$ and satisfying the differential equation
$X^{\prime}(\theta)=f^{\prime}(y_{0}e^{\beta(\theta)})$ for the unique $y_{0}$
is called a $h$-curve. For $x_{2}=0,$ we denote the unique point $y_{0}$ as
$h(x_{1},t_{1},t_{2})$ and for $t_{2}=0,$ we denote the unique point $y_{0}$
as $h(x_{1}-x_{2},t_{1}).$

##  3\. Proof of Theorem (1.1)

In this section, we give the proof of Theorem (1.1). We start with the dynamic
programming principle for the initial-boundary value problem. For
$\gamma\in\Gamma(x,t,s)$, let us denote

$\displaystyle J(\gamma,x,t)=\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(\gamma(s),s).$

######  Theorem 3.1.

(Dynamics programming principle) The value function $W(x,t)$ satisfies the
dynamics programming principle, i.e.,

$W(x,t)=W(x,t,s).$

###### Proof.

Let $\gamma:[0,t]\to Q$ be any Lipschitz continuous curve satisfying
$\gamma(t)=x$. Then

$\displaystyle J(\gamma,x,t)=$ $\displaystyle\int_{\\{\gamma(\theta)\neq
0\\}\cap[s,t]\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\cap[s,t]\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle+\int_{\\{\gamma(\theta)\neq
0\\}\cap[0,s]\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}\cap[0,s]\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\gamma(0))$
$\displaystyle\geq\int_{\\{\gamma(\theta)\neq
0\\}\cap[s,t]\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\cap[s,t]\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle+W(\gamma(s),s).$

Therefore

$\displaystyle
J(\gamma,x,t)\geq\inf_{\gamma\in\Gamma(x,t,s)}\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(\gamma(s),s).$

Taking infimum over all $\gamma\in\Gamma(x,t,s),$ we have

(3.1) $W(x,t)\geq W(x,t,s).$

Now from (2.2)

$\displaystyle
W(x,t,s)=\inf_{\gamma\in\Gamma(x,t,s)}\int_{\\{\gamma(\theta)\neq
0\\}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(\gamma(s),s).$

Now for any $\epsilon>0$ there exists a $\gamma_{\epsilon}\in\Gamma(x,t,s)$
such that

(3.2) $\displaystyle\int_{\\{\gamma_{\epsilon}(\theta)\neq
0\\}}f^{*}(\dot{\gamma_{\epsilon}}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma_{\epsilon}(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(\gamma_{\epsilon}(s),s)$
$\displaystyle\leq W(x,t,s)+\epsilon.$

Again for $\epsilon^{\prime}>0$ there exists a
$\gamma_{\epsilon}^{\prime}\in\Gamma(x,t,s)$ such that
$\gamma_{\epsilon^{\prime}}(s)=\gamma_{\epsilon}(s)$ and

(3.3) $\displaystyle\int_{\\{\gamma_{\epsilon^{\prime}}(\theta)\neq
0\\}}f^{*}(\dot{\gamma_{\epsilon^{\prime}}}(\theta))e^{-\beta(\theta)}d\theta-\int_{\\{\gamma_{\epsilon^{\prime}}(\theta)=0\\}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(\gamma_{\epsilon^{\prime}}(s),s)$
$\displaystyle\leq W(\gamma_{\epsilon}(s),s)+\epsilon^{\prime}.$

Let us define the curve

$\displaystyle\bar{\gamma}(\theta)=\begin{cases}\gamma_{\epsilon}(\theta)\,\,\,&if\,\,\,\theta\in[s,t],\\\
\gamma_{\epsilon^{\prime}}(\theta)\,\,\,&if\,\,\,\theta\in[0,s].\end{cases}$

Combining the above equations (3.2) and (3.3), we get

$J(\bar{\gamma},x,t)\leq W(x,t,s)+\epsilon+\epsilon^{\prime}.$

Since $\epsilon$ and $\epsilon^{\prime}$ are arbitrary, we get

(3.4) $W(x,t)\leq W(x,t,s).$

The inequalities (3.1) and (3.4) give dynamic programming principle. ∎

In the next lemma, using the convexity of $f^{*},$ we show that the minimiser
attend through $h$-curve.

######  Lemma 3.2.

Suppose $f$ is a strictly convex function with superlinear growth,
$\displaystyle{\lim_{|u|\to\infty}}\frac{f(u)}{|u|}=\infty$. Let $S$ be the
set of all Lipschitz continuous curves in $\mathbb{R}\times[0,\infty)$ joining
$(x_{1},t_{1})$ to $(x_{2},t_{2})$ with $t_{1}<t_{2},$ then

(3.5) $\inf_{\gamma\in
S}\int_{t_{1}}^{t_{2}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta=\int_{t_{1}}^{t_{2}}f^{*}(X^{\prime}(\theta))e^{-\beta(\theta)}d\theta,$

where $X(\theta)$ is a curve joining $(x_{1},t_{1})$ to $(x_{2},t_{2})$ and
satisfying

$X^{\prime}(\theta)=f^{\prime}(y_{0}e^{\beta(\theta)}),$

for a unique $y_{0}\in\mathbb{R}.$

###### Proof.

For the existence of such a curve $X$, consider a real valued function

$y_{0}\mapsto
x_{1}+\int_{t_{1}}^{t_{2}}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta.$

Since $f$ has superlinear growth,
$x_{0}+\int_{t_{1}}^{t_{2}}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta\to\pm\infty$
as $y_{0}\to\pm\infty.$ Then by intermediate value theorem, there exists a
$x_{2}$ such that

$x_{1}+\int_{t_{1}}^{t_{2}}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta=x_{2}.$

Taking

$X(s)=x_{1}+\int_{t_{1}}^{s}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta,\,\,s\in[t_{1},t_{2}]$

serves the purpose. The uniqueness of $y_{0}$ follows from the fact that
$f^{\prime}$ is strictly increasing. Next we show (3.5). As $f^{*}$ is
strictly convex we have

(3.6) $f^{*}(x)\geq f^{*}(y)+f^{*\prime}(y)(x-y).$

Suppose $\gamma\in S$. Inserting $x=\dot{\gamma}(\theta)$ and
$y=\dot{X}(\theta)$ in the above equation (3.6), we get

$e^{-\beta(\theta)}f^{*}(\dot{\gamma}(\theta))\geq
e^{-\beta(\theta)}f^{*}(f^{\prime}(y_{0}e^{\beta(\theta)}))+y_{0}(\dot{X}(\theta)-\dot{\gamma}(\theta)).$

Now integrating the above inequality on $[t_{1},t_{2}],$ we get

$\int_{t_{1}}^{t_{2}}e^{-\beta(\theta)}f^{*}(\dot{\gamma}(\theta))d\theta\geq\int_{t_{1}}^{t_{2}}e^{-\beta(\theta)}f^{*}(f^{\prime}(y_{0}e^{\beta(\theta)}))d\theta+\int_{t_{1}}^{t_{2}}y_{0}(\dot{X}(\theta)-\dot{\gamma}(\theta))d\theta.$

As the last term
$\int_{t_{1}}^{t_{2}}y_{0}(\dot{X}(\theta)-\dot{\gamma}(\theta))d\theta=0$, we
obtained (3.5). ∎

######  Remark 3.3.

The inequality

$\int_{t_{1}}^{t_{2}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta\geq\int_{t_{1}}^{t_{2}}f^{*}(\dot{X}(\theta))e^{-\beta(\theta)}d\theta$

is strict if the Lebesgue measure of the set
$\\{\theta|\dot{\gamma}(\theta)\neq\dot{X}(\theta)\\}>0.$

The next theorem states that the value function $W(x,t)$ can be obtained by
minimising over all $h$-curves lying inside the quarter plane.

######  Theorem 3.4.

Define

$H_{b}(x,t)=\left\\{\tau|\int_{\tau}^{s}f^{\prime}(h(x,t,\tau)e^{\beta(\theta)})d\theta\geq
0,\,\,\,0\leq\tau\leq s\leq t\right\\}$

and

$H_{I}(x,t)=\left\\{y|\int_{0}^{\tau}f^{\prime}(h(x-y,t)e^{\beta(\theta)})d\theta\geq
0,\,\,\,0\leq\tau\leq t\right\\}.$

Then

(3.7) $W(x,t)=\min\left\\{A(x,t),B(x,t)\right\\}.$

Here $A(x,t)$ and $B(x,t)$ are defined as

$A(x,t)=\inf_{y\in H_{I}}A(y,x,t),$ $B(x,t)=\inf_{\tau\in H_{b}}B(\tau,x,t)$

where

$A(y,x,t)=\int_{0}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x-y,t)e^{\beta(\theta)}))d\theta+U_{0}(y)$

and

$B(\tau,x,t)=\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,\tau)e^{\beta(\theta)}))d\theta+W(0,\tau).$

###### Proof.

Let $\gamma:[0,t]\to Q$ be a Lipschitz continuous curve joining $(x,t)$ to
$(y,0)$ with Lipschitz constant $M.$ First assume $0\leq\tau<t$ be the point
such that $\gamma(\tau)=0$ and $\gamma(\theta)>0$ for $\tau<\theta\leq t.$
Step 1: We show that the $h$-curve joining the nearby points $(x_{1},t_{1})$
and $(x_{2},t_{2})$ (with $x_{i},t_{i}>0$ for $i=1,2$) lying on $\gamma$
completely stays in the interior of $Q.$ To show this let
$X(s)=x_{1}+\int_{t_{1}}^{s}f^{\prime}(y_{0}e^{\beta(s)})ds,\,t_{1}\leq s\leq
t_{2}$ be the $h$-curve joining two points $(x_{1},t_{1})$ and
$(x_{2},t_{2}).$ Then we have

$x_{2}=x_{1}+\int_{t_{1}}^{t_{2}}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta.$

Clearly

$\displaystyle
x_{2}-x_{1}=\int_{t_{1}}^{t_{2}}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta\geq
f^{\prime}\big{(}y_{0}e^{\displaystyle{\min_{\theta\in[t_{1},t_{2}]}}\beta(\theta)}\big{)}(t_{2}-t_{1}),$
$\displaystyle
x_{2}-x_{1}=\int_{t_{1}}^{t_{2}}f^{\prime}(y_{0}e^{\beta(\theta)})d\theta\leq
f^{\prime}\big{(}y_{0}e^{\displaystyle{\max_{\theta\in[t_{1},t_{2}]}}\beta(\theta)}\big{)}(t_{2}-t_{1}).$

From the above inequalities, we have

(3.8)
$\frac{(f^{\prime})^{-1}\big{(}\frac{x_{2}-x_{1}}{t_{2}-t_{1}}\big{)}}{e^{\displaystyle{\max_{\theta\in[t_{1},t_{2}]}}\beta(\theta)}}\leq
y_{0}\leq\frac{(f^{\prime})^{-1}\big{(}\frac{x_{2}-x_{1}}{t_{2}-t_{1}}\big{)}}{e^{\displaystyle{\min_{\theta\in[t_{1},t_{2}]}}\beta(\theta)}}.$

Hence we get from (3.8) we get

$|y_{0}|\leq\max\Bigg{\\{}\frac{|(f^{\prime})^{-1}\big{(}\frac{x_{2}-x_{1}}{t_{2}-t_{1}}\big{)}|}{e^{\displaystyle{\max_{\theta\in[t_{1},t_{2}]}}\beta(\theta)}},\frac{|(f^{\prime})^{-1}\big{(}\frac{x_{2}-x_{1}}{t_{2}-t_{1}}\big{)}|}{e^{\displaystyle{\min_{\theta\in[t_{1},t_{2}]}}\beta(\theta)}}\Bigg{\\}}.$

This shows that if $(x_{1},t_{1})$ and $(x_{2},t_{2})$ are nearby points, then
the $h$-curve joining these two points completely lies inside the quarter
plane.
Step 2: Let $X_{\tau}$ be the $h-$curve joining the points $(x,t)$ to
$(\gamma(\tau),\tau).$ From the definition of $h-$function,

$X_{\tau}(s)=\gamma(\tau)+\int_{\tau}^{s}f^{\prime}(y_{0}(\tau)e^{\beta(\theta)})d\theta,\,\,\,\tau\leq
s\leq t.$

In this step, we show that the function $y_{0}(\tau)$ is a continuous function
of $\tau.$

To show this note that
$x-\gamma(\tau)=\int_{\tau}^{t}f^{\prime}(y_{0}(\tau)e^{\beta(\theta)})d\theta$.
Form the previous arguments in step 1, we have $y_{0}(\tau)$ is bounded. For
any sequence $\tau_{n}\to\bar{\tau},$ there exists a subsequence
$\tau_{n_{k}}$ such that $y_{0}(\tau_{n_{k}})\to\alpha.$ Now passing to the
limit in the equation,

$x-\gamma(\tau_{n_{k}})=\int_{\tau_{n_{k}}}^{t}f^{\prime}(y_{0}(\tau_{n_{k}})e^{\beta(\theta)})d\theta,$

we get

$x-\gamma(\bar{\tau})=\int_{\bar{\tau}}^{t}f^{\prime}(\alpha
e^{\beta(\theta)})d\theta.$

By the uniqueness of $y_{0}$, we have $y_{0}(\bar{\tau})=\alpha.$ This proves
the continuity of $y_{0}.$
Step 3: From step 1, we get that for $\tau$ near to $t$, there exists a
$h$-curve in $Q$ joining $(x,t)$ to $(\gamma(\tau),\tau)$ which stays in the
interior of quarter plane $Q$. Therefore the set

$\Big{\\{}\tau|\textnormal{$\exists$ a $h-$curve in $Q$ joining $(x,t)$ to
$(\gamma(\tau),\tau)$}\Big{\\}}$

is non-empty. Now define

$\tau^{\prime}=\displaystyle{\sup_{0\leq\tau\leq
t}}\Big{\\{}\tau|\textnormal{$\exists$ a $h-$curve in $Q$ joining $(x,t)$ to
$(\gamma(\tau),\tau)$}\Big{\\}}.$

If $\tau^{\prime}>\tau$, then by the continuity of $y_{0}$ (by step 2), the
$h$-curve joining $(x,t)$ to $(\gamma(\tau^{\prime}),\tau^{\prime})$ will
touch the boundary at some point $(0,\tau_{1})$. Then by the Lemma (3.2)

$\int_{\tau_{1}}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,\tau_{1})e^{\beta(\theta)}))d\theta+w(0,\tau_{1})\leq
J(\gamma,x,t).$

If $\tau^{\prime}=\tau,$ then the $h$-curve joining $(x,t)$ to $(0,\tau)$
completely lies in $Q.$ Then by the Lemma (3.2)

$\int_{\tau_{1}}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,\tau)e^{\beta(\theta)}))d\theta+w(0,\tau)\leq
J(\gamma,x,t).$

If $\gamma(\theta)>0$ for $0<\theta\leq t,$ then the steps 1-3 can be followed
to obtain either there is a $h-$ curve joining $(x,t)$ to $(y,0)$ and touching
$t$-axis at $(0,\tau_{1})$ or the $h$-curve joining $(x,t)$ to $(y,0)$
completely lies in $Q$. In that case either

$\int_{\tau_{1}}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,\tau_{1})e^{\beta(\theta)}))d\theta+w(0,\tau_{1})\leq
J(\gamma,x,t)$

or

$\int_{0}^{t}f^{\prime}(h(x-y,t)e^{\beta(\theta)})d\theta+U_{0}(y)\leq
J(\gamma,x,t).$

This completes the proof of the theorem. ∎

######  Theorem 3.5.

(Characterization of $W(0,t)$) The value function
$W(0,t)=\displaystyle{\min_{\gamma}}\ J(\gamma,0,t)$ is achieved along a
Lipschitz continuous path $\gamma(\theta),$ which has the following property.
Either $\gamma(\theta)=0$ or if $\gamma(\theta)\neq 0$ on $[t_{1},t_{2}]$,
then $\gamma:[t_{1},t_{2}]\to Q$ is a $h$-curve.

###### Proof.

Let $\gamma_{0}$ be a fixed Lipschitz curve joining $(0,t)$ to $(y,0).$ For
each Lipschitz curve $\eta:[0,t]\to Q,$ define the set

$D_{\eta}=\\{\theta\in[0,t]:\eta(\theta)\neq\gamma_{0}(\theta)\\}.$

Now consider the set of all Lipschitz $\eta,$

$S=\Big{\\{}\eta:[0,t]\to Q:|\eta^{\prime}(t_{0})|\leq M\,\,\textnormal{for
any}\,t_{0}\in D_{\eta}\,\,\textnormal{and $t_{0}$ be a Lebesgue point of
$\eta^{\prime}(\theta)$}\Big{\\}},$

where

$M=\max\Big{(}\max_{\theta_{1},\theta_{2}\in[0,t]}f^{\prime}\Big{(}e^{\beta(\theta_{1})-\beta(\theta_{2})}\lambda_{f}\Big{)},\frac{y}{t}\Big{)}.$

Let $\eta_{i}$ be a sequence such that

$\lim_{i\to\infty}J(\eta_{i},x,t)=\inf_{\eta\in S}J(\eta,x,t).$

Since $\eta_{i}$ are members of $S,$
$|\eta_{i}^{\prime}(\theta)|\leq\max\Big{\\{}M,\textnormal{Lipschitz constant
of}\,\ \gamma_{0}\Big{\\}}$ _a.e_. Then by Arzela-Ascoli we have a subsequence
(still denoted by $\eta_{i}$) such that $\eta_{i}\to\eta_{0}$ uniformly.

Step 1: We first prove that $\eta_{0}\in S.$
To see this, suppose
$(a,b)\subset\\{\theta\in[0,t]:\eta_{0}(\theta)\neq\gamma_{0}(\theta)\\}.$
Then there exists a $i_{0}$ such that
$\eta_{i}(\theta)\neq\gamma_{0}(\theta),\,\,\theta\in(a,b)$ for $i\geq i_{0}.$
Therefore $|\eta_{i}^{\prime}(\theta)|\leq M.$ Now for any test function
$\phi,$ we have

$\displaystyle\Big{|}\int_{a}^{b}\eta_{i}^{\prime}(\theta)\phi(\theta)\Big{|}\leq
M\int_{a}^{b}|\phi(\theta)|d\theta\implies$
$\displaystyle\Big{|}\int_{a}^{b}\eta_{i}(\theta)\phi^{\prime}(\theta)\Big{|}\leq
M\int_{a}^{b}|\phi(\theta)|d\theta$ $\displaystyle\implies$
$\displaystyle\Big{|}\int_{a}^{b}\eta_{0}(\theta)\phi^{\prime}(\theta)\Big{|}\leq
M\int_{a}^{b}|\phi(\theta)|d\theta\implies$
$\displaystyle\Big{|}\int_{a}^{b}\eta_{0}^{\prime}(\theta)\phi(\theta)\Big{|}\leq
M\int_{a}^{b}|\phi(\theta)|d\theta.$

Let $t_{0}\in(a,b)$ is a Lebesgue point of $\eta_{0}$. Now choose $\phi$ as a
suitable _Friedrichs mollifier_ converging to $\delta_{t_{0}}$ to arrive at
the result that $|\eta_{0}^{\prime}(t_{0})|\leq M.$ This proves the claim.
Step 2: Next we show that $\eta_{0}$ is a minimizer. Note that $f^{*}$
satisfies

$f^{*}(x)\geq f^{*}(y)+\big{(}f^{*}\big{)}^{\prime}(y)(x-y).$

Taking $x=\dot{\eta_{i}}(\theta)$ and $y=\dot{\eta_{0}}(\theta),$ multiplying
with $e^{-\beta(\theta)}$ and integrating over ${\eta_{i}\neq 0}$ we get

$\displaystyle\int_{\eta_{i}\neq 0}f^{*}(\dot{\eta_{i}}(\theta))$
$\displaystyle\geq\int_{\eta_{i}\neq
0}f^{*}(\dot{\eta_{0}}(\theta))+\int_{\eta_{i}\neq
0}\big{(}f^{*}\big{)}^{\prime}(\dot{\eta_{0}}(\theta))(\dot{\eta_{i}}(\theta)-\dot{\eta_{0}}(\theta)).$

Again adding the $-\int_{\eta_{i}=0}f(\bar{u_{b}}(\theta))d\theta$ on both
sides we get

(3.9) $\displaystyle\int_{\eta_{i}\neq 0}f^{*}(\dot{\eta_{i}}(\theta))-$
$\displaystyle\int_{\eta_{i}=0}f(\bar{u_{b}}(\theta))d\theta+W_{0}(\eta_{i}(0))\geq\int_{\eta_{i}\neq
0}f^{*}(\dot{\eta_{0}}(\theta))$ $\displaystyle+\int_{\eta_{i}\neq
0}f^{*\prime}(\dot{\eta_{0}}(\theta))(\dot{\eta_{i}}(\theta)-\dot{\eta_{0}}(\theta))-\int_{\eta_{i}=0}f(\bar{u_{b}}(\theta))d\theta+W_{0}(\eta_{i}(0)).$

As $\dot{\eta}_{i}$ convergence weakly to $\dot{\eta}_{0}$, applying of the
$\liminf$ as $i\to\infty$ in the above equation (3.9) yields

$\inf_{\eta\in S}J(\eta,x,t)\geq\int_{\eta_{0}\neq
0}f^{*}(\dot{\eta_{i}}(\theta))-\int_{\eta_{0}=0}f(\bar{u_{b}}(\theta))d\theta+W_{0}(\eta_{i}(0))=J(\eta_{0},x,t).$

Step 3: In this final step, we will show that $|\eta_{0}^{\prime}(\theta)|\leq
M$ _a.e._ on $[0,t].$
Suppose this is not the case. Then there exists an interval $[a,b]$ where
$\eta_{0}(\theta)\neq 0$ and $a$ is the point such that any interval
$(a,a+\delta)$ where $|\eta_{0}^{\prime}(t)|>M$ has positive measure. Suppose
there is a $h$-curve $h_{1}$ joining $(\eta_{0}(a),a)$ and
$(\eta_{0}(\theta_{1}),\theta_{1})$, $a<\theta_{1}\leq b$ touches $t$-axis at
$\theta_{2}$. further note that
$h_{1}^{\prime}(\theta)=f^{\prime}(y_{0}e^{\beta(\theta)}),$ for some $y_{0}.$
Therefore $f^{\prime}(y_{0}e^{\beta(\theta_{2})})=0$. Then
$y_{0}=\lambda_{f}e^{-\beta(\theta_{2})},$ which gives
$|h_{1}^{\prime}(\theta)|\leq M.$ Now the new curve $\eta_{1}$

$\eta_{1}(\theta)=\begin{cases}h_{1}(\theta),\,\,a\leq\theta\leq\theta_{1}\\\
\eta_{0}(\theta),\,\,\theta_{1}\leq\theta\leq b.\end{cases}$

is an element of $S$ and satisfies $J(\eta_{1},x,t)<J(n_{0},x,t)$ which is a
contradiction.
Suppose such a curve $h_{1}$ does not exist. Then the $h$-curve joining
$(\eta_{0}(a),a)$ and $(\eta_{0}(b),b)$ completely lies in $Q.$ Now if there
exists a point $(\eta_{0}(t_{1}),t_{1}),t_{1}>b,\eta_{0}(t_{1})\neq 0$ such
that the $h$-curve joining $(\eta_{0}(a),a)$ and $(\eta_{0}(t_{1}),t_{1})$
touches $t$-axis, then the previous argument gives rise to a contradiction. If
such a point does not exist, then choose a point $t_{2}$ such that
$\eta_{0}(\theta)\neq 0$ for $a\leq\theta<t_{2}$, $\eta_{0}(t_{2})=0$ and the
$h$-curve joining $(\eta_{0}(a),a)$ and $(t_{2},0)$ completely lies in $Q.$
If there exists a point $t_{3}$ such that the $h$-curve from $(0,t_{2})$ to
$(\eta_{0}(t_{3}),t_{3})$ touching t-axis, then the same argument as before
leads to a contradiction. If there exists no such point $t_{3}$, then there
exists a point $t_{4}$ such that $\eta_{0}(t_{4})=0$ or
$(\eta_{0}(t_{4}),t_{4})=(y,0)$. Either cases lead to the contradiction that
$\eta_{0}$ is the minimizer. Next, as $f^{*}$ has superlinear growth, one can
deduce that for a given $t>0$, $y_{0}$ is bounded. Therefore the minimization
of the functional $J(\eta,0,t)$ on the set of Lipschitz curves $\eta$ joining
$(0,t)$ to $(y,0),y\geq 0$ in the quarter plane reduces to the minimization on
the subset of curves whose slopes are bounded by a constant that depend only
on $t$; denote it again as $M.$

Therefore, denoting $D_{M}=\Big{\\{}\eta:[0,t]\to Q\,\,\,\textnormal{Lipschitz
curve}\,\,:|\eta^{\prime}(\theta)|\leq M\,\,\,\textnormal{for
\emph{a.e}}\,\,\,\theta\in[0,t]\Big{\\}},$ we have

$\inf_{\eta}J(\eta,0,t)=\inf_{D_{M}}J(\eta,0,t).$

Then following step 2 we get the existence of a minimizer $\gamma$. Now
applying the Lemma (3.2), theorem follows. ∎

######  Theorem 3.6.

$W(x,t)$ is locally Lipschitz continuous for $x\geq 0$ and $t>0$.

###### Proof.

First we show that $W(0,t)$ is a Lipschitz continuous function. For
$0<t_{1}<t_{2},$ note that $W(0,t_{2})\leq
W(0,t_{1})-\int_{t_{1}}^{t_{2}}f(\bar{u}_{b}(\theta))d\theta.$ This implies

(3.10)
$W(0,t_{2})-W(0,t_{1})\leq\int_{t_{1}}^{t_{2}}f(\bar{u}_{b}(\theta))d\theta.$

By the previous theorem(3.5), there exists a curve $\gamma:[0,t_{2}]\to
Q,\,\,\gamma(t_{2})=0$ and $\dot{\gamma}$ bounded such that

$W(0,t_{2})=\int_{\gamma\neq
0}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta-\int_{\gamma=0}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\gamma(0)).$

Define a new curve $\tilde{\gamma}:[0,t_{1}]\to Q$ by

$\tilde{\gamma}(\theta)=\gamma(\theta+t_{2}-t_{1}).$

Now

(3.11) $\displaystyle W(0,t_{1})-W(0,t_{2})$ $\displaystyle\leq
J(\tilde{\gamma},0,t_{1})-J(\gamma,0,t_{2})$
$\displaystyle\leq\int_{\tilde{\gamma}\neq
0}f^{*}(\dot{\tilde{\gamma}}(\theta))e^{-\beta(\theta)}-\int_{\tilde{\gamma}=0}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\tilde{\gamma}(0))$
$\displaystyle-\Big{[}\int_{\gamma\neq
0}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}-\int_{\gamma=0}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\gamma(0))\Big{]}$
$\displaystyle\leq\int_{\gamma\neq 0,\,\,\theta\geq
t_{2}-t_{1}}f^{*}(\dot{\gamma}(\theta))[e^{-\beta(\theta-
t_{2}+t_{1})}-e^{-\beta(\theta)}]d\theta$ $\displaystyle+\int_{\gamma\neq
0,\,\,\theta<t_{2}-t_{1}}f^{*}(\dot{\gamma}(\theta))e^{\beta(\theta)}$
$\displaystyle+\int_{\gamma=0,\,\,\,\theta\geq
t_{2}-t_{1}}[f(\bar{u_{b}}(\theta-t_{2}+t_{1}))e^{-\beta(\theta-
t_{2}+t_{1})}-f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}]d\theta$
$\displaystyle+\int_{\gamma=0,\,\,\,\theta\leq
t_{2}-t_{1}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W_{0}(\tilde{\gamma}(0))-W_{0}(\gamma(0)).$

Since $\dot{\gamma}$ is bounded, combining (3.10)-(3.11), we get

$|W(0,t_{1})-W(0,t_{2})|\leq C|t_{1}-t_{2}|,$

where $C$ depending only on $\gamma$ and $\bar{u_{b}}.$

Next we prove that $B(x,t)$ is locally Lipschitz continuous and the proof of
Lipschitz continuity of $A(x,t)$ is available[1]. This implies
$W(x,t)=\min\\{A(x,t),B(x,t)\\}$ is locally Lipschitz continuous. Let us first
prove that $B(x,t)$ is Lipschitz continuous in $t.$ Let $t_{0}>0$ be a fixed
point. Let $t_{1},t_{2}\in(\frac{t_{0}}{2},\frac{3t_{0}}{2})$ with
$t_{1}<t_{2}.$ Further for any $t\in(\frac{t_{0}}{2},\frac{3t_{0}}{2})$, there
corresponds a $h$-curve $\eta_{t}$ along which $B(x,t)$ achieves its minimum.
As $\dot{\eta}_{t}$s’ are uniformly bounded, there exists $M>0$
$|(f^{*})(\dot{\eta}_{t})|<M$. First we have

(3.12) $B(x,t_{2})-B(x,t_{1})\leq J(\tilde{\gamma},x,t_{2})-J(\gamma,x,t_{1})$

where $\gamma:[\tau,t_{1}]\to\mathbb{R}^{+}\times\mathbb{R}^{+}$ is a
Lipschitz continuous curve such that $\gamma(t_{1})=x,\gamma(\tau)=0$ and
$J(\gamma,x,t_{1})=B(x,t_{1}).$ Also,
$\tilde{\gamma}:[\tau+(t_{2}-t_{1}),t_{2}]\to\mathbb{R}^{+}\times\mathbb{R}^{+}$
defined as

$\tilde{\gamma}(\theta)=\gamma(\theta+t_{1}-t_{2})$

such that $\tilde{\gamma}(t_{2})=x.$ Now from $\eqref{lip1}$ we have

$\displaystyle B(x,t_{2})-B(x,t_{1})\leq$
$\displaystyle\int_{\tau+(t_{2}-t_{1})}^{t_{2}}f^{*}(\dot{\gamma}(\theta+t_{1}-t_{2}))e^{-\beta(\theta)}d\theta-\int_{\tau}^{t_{1}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle+W(0,\tau+t_{2}-t_{1})-W(0,\tau).$

To estimate $B(x,t_{1})-B(x,t_{2}),$ we assume $B(x,t_{2})=J(\gamma,x,t_{2}).$
Then following the previous steps and using a change of variable we get

$\displaystyle|B(x,t_{2})-B(x,t_{1})|$
$\displaystyle\leq\int_{\tau}^{t_{1}}|f^{*}(\dot{\gamma}(\xi))[e^{-\beta(\xi+t_{2}-t_{1})}-e^{\beta(\xi)}]|d\xi+\int_{\tau}^{\tau+(t_{2}-t_{1})}|f(u_{b}(\theta))e^{-\beta(\theta)}|d\theta$
$\displaystyle\leq
M|t_{1}-t_{2}|+K|t_{2}-t_{1}|\leq\max\\{M,K\\}|t_{1}-t_{2}|.$

Next we shall show that $B(x,t)$ is locally Lipschitz in $x>0.$ Let $x_{0}>0$
be a fixed point. Let $x_{1},x_{2}\in(\frac{x_{0}}{2},\frac{3x_{0}}{2})$ with
$x_{1}<x_{2}.$ Further for any $x\in(\frac{x_{0}}{2},\frac{3x_{0}}{2})$, there
corresponds a $h$-curve $\eta_{x}$ along which $W(x,t)$ achieves its minimum.
As $\dot{\eta}_{x}$s’ are uniformly bounded, there exists $M>0$ such that for
$r\in(0,1),$ $|(f^{*})^{\prime}(r\dot{\eta})|<M.$
Suppose $\gamma:[\tau,t]\to\mathbb{R^{+}}\times\mathbb{R^{+}}$ such that
$\gamma(t)=x_{2}$ and

$B(x_{2},t)=\int_{\tau}^{t}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau).$

Define the new curve
$\tilde{\gamma}:[\tau,t]\to\mathbb{R^{+}}\times\mathbb{R^{+}}$ as

$\tilde{\gamma}(\theta)=\frac{x_{1}}{x_{2}}\gamma(\theta).$

Then $\tilde{\gamma}(\tau)=0$ and $\tilde{\gamma}(t)=x_{1}.$ We calculate

$\displaystyle B(x_{1},t)-B(x_{2},t)$
$\displaystyle\leq\int_{\tau}^{t}f^{*}(\dot{\tilde{\gamma}}(\theta))-f^{*}(\dot{{\gamma}}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle=\int_{\tau}^{t}f^{*\prime}(\xi_{\theta})e^{-\beta(\theta)}(x_{1}-x_{2})\frac{\dot{\gamma}(\theta)}{x_{2}}d\theta.$

Since $\dot{\gamma}(\theta)$ is bounded (because $\gamma$ is a $h$-curve)

(3.13) $B(x_{1},t)-B(x_{2},t)\leq
Me^{\displaystyle{\max_{\theta\in[\tau,t]}(-\beta(\theta))}}(x_{1}-x_{2}).$

To estimate $B(x_{2},t)-B(x_{1},t),$ first suppose

$B(x_{1},t)=\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}(\dot{\gamma}(\theta))d\theta+W(0,\tau),$

where $\gamma:[\tau,t]\to Q$ is a $h$-curve, $\gamma(t)=x_{1}.$ Now define a
curve

$\tilde{\gamma}(\theta)=\begin{cases}\gamma(\theta)+x_{2}-x_{1}\,\,\,\,if\,\,\,\theta\in[\tau,t],\\\
\theta-\tau+x_{2}-x_{1}\,\,\,\,if\,\,\,\theta\in[\tau-(x_{2}-x_{1}),\tau].\end{cases}$

Now

(3.14) $\displaystyle B(x_{2},t)-B(x_{1},t)\leq$
$\displaystyle\int_{\tau-(x_{2}-x_{1})}^{t}e^{-\beta(\theta)}f^{*}(\dot{\tilde{\gamma}}(\theta))d\theta+W(0,\tau-(x_{2}-x_{1}))$
$\displaystyle-\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}(\dot{\gamma}(\theta))d\theta-W(0,\tau)$
$\displaystyle\leq\Big{|}\int_{\tau-(x_{2}-x_{1})}^{t}e^{-\beta(\theta)}f^{*}(1)\Big{|}+|W(0,\tau-(x_{2}-x_{1}))-W(0,\tau)|$
$\displaystyle\leq C|x_{2}-x_{1}|,\,\,\,\,\,\,\,\textnormal{by the Lipschitz
continuity of $w$ in $t>0.$ }$

Combining the inequalities (3.13)-(3.14) we get the Lipschitz continuity of
$B(x,t)$ in the $x$ variable. ∎

######  Lemma 3.7.

(Existence of minimizer) There exists $\bar{y}\in H_{I}$ such that

$A(x,t)=\int_{0}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x-\bar{y},t)e^{\beta(\theta)}))d\theta+U_{0}(\bar{y}),$

and also $\bar{\tau}\in H_{b}$ such that

$B(x,t)=\int_{\bar{\tau}}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,\bar{\tau})e^{\beta(\theta)}))d\theta+W(0,\bar{\tau}).$

###### Proof.

Choose the sequence $y_{n}\in H_{I}$ such that

$\int_{0}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x-{y_{n}},t)e^{\beta(\theta)}))d\theta\to
A(x,t)$

as $n\to\infty$. From (3.8), we note that $\\{y_{n}\\}$ is a bounded
sequences.

Hence there exists convergent subsequence $y_{n_{k}}$ of $y_{n}$ with
$y_{n_{k}}\to\bar{y}.$ By continuity of $h$, we get $h(x-y_{n_{k}})\to
h(x-\bar{y},t).$ Thus by dominated convergence, we have

$\displaystyle\int_{0}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x-y_{n_{k}},t)e^{\beta(\theta)}))d\theta+U_{0}(y_{n_{k}})\to\int_{0}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x-\bar{y},t)e^{\beta(\theta)}))d\theta+U_{0}(\bar{y})$

Since $W(0,\bar{\tau})$ is Lipschitz continuous, similar arguments can be used
for the functional $B(x,t).$ ∎

Next lemma shows the non-intersecting property (NIP) for
$W(x,t)=\min\\{A(x,t),B(x,t)\\}.$ For $W(x,t)=A(x,t),$ NIP is already proved
in [1].

######  Lemma 3.8.

(Non-intersecting property) If at the point $(x,t)$ for $x\geq 0,t>0$,
$W(x,t)=B(x,t)$ and

$B(x,t)=\int_{\tau}^{t}f^{*}(\dot{\gamma}(\theta))e^{\beta(\theta)}))d\theta+W(0,{\tau}),$

where $\gamma$ is the h-curve joining $(x,t)$ to $(0,\tau)$ and
$\gamma(\theta)>0$ for $\tau<\theta\leq t.$ Suppose $(\bar{x},\bar{t})$ is a
point on $\gamma$, then

$\int_{\tau}^{\bar{t}}f^{*}(\dot{\gamma}(\theta))e^{\beta(\theta)}))d\theta+w(0,{\tau})<J({\tilde{\gamma},\bar{x},\bar{t}})$

for any $\tilde{\gamma}$ in $Q$ which is different from $\gamma$ in the
interval $[\tau,\bar{t}]$. Therefore the minimisers corresponding to the
points $(x_{1},t)$ and $(x_{2},t)$ where $x_{1}\neq x_{2},$ will not
intersect.

###### Proof.

From the previous lemma, we have

$W(x,t)=\int_{\tau}^{t}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau)$

where $\gamma$ is a $h$-curve and $\tau=\max\\{\theta|{\gamma}(\theta)=0\\}.$
Suppose at the point $(\bar{x},\bar{t})$ on $\gamma,$ there exists a $h$-curve
$\tilde{\gamma}$ different from $\gamma$ and
$\tau^{\prime}=\max\\{\theta|\tilde{\gamma}(\theta)=0\\}$ such that

$W(\bar{x},\bar{t})=\int_{\tau^{\prime}}^{\bar{t}}f^{*}(\dot{\tilde{\gamma}}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau^{\prime}).$

Since $\gamma$ is the minimiser at $(x,t),$ we have

$\displaystyle W(x,t)$
$\displaystyle=\int_{\tau}^{t}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau)$
$\displaystyle=\int_{\tau}^{\bar{t}}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau)+\int_{\bar{t}}^{t}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle\geq
W(\bar{x},\bar{t})+\int_{\bar{t}}^{t}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle=\int_{\tau^{\prime}}^{\bar{t}}f^{*}(\dot{\tilde{\gamma}}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau^{\prime})+\int_{\bar{t}}^{t}f^{*}(\dot{\gamma}(\theta))e^{-\beta(\theta)}d\theta.$

Now define

$\displaystyle\Gamma(\theta)=\begin{cases}\gamma(\theta)\,\,\,\,&\textnormal{if}\,\,\,\theta\in[\bar{t},\,\,t],\\\
\tilde{\gamma}(\theta)\,\,\,\,&\textnormal{if}\,\,\,\theta\in[\tau^{\prime},\bar{t}].\end{cases}$

Then we get

$W(x,t)=\int_{\tau^{\prime}}^{t}f^{*}(\dot{\Gamma}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau^{\prime}).$

Define

$\tau^{\prime\prime}=\sup_{\tau^{\prime}\leq\theta\leq
t}\Big{\\{}\theta\,\,\,|\,\,\,\textnormal{there exists a $h$-curve
$\Gamma_{h}$ in $Q$
joining}(x,t)\,\,\textnormal{to}\,\,(\Gamma(\theta),\theta)\Big{\\}}.$

By definition $\Gamma_{h}$ intersects the $t$-axis. Then we define

$\displaystyle\bar{\Gamma}(\theta)=\begin{cases}\Gamma_{h}(\theta)\,\,\,\,&\textnormal{if}\,\,\,\theta\in[\tau^{\prime\prime},\,\,t]\\\
\Gamma(\theta)\,\,\,\,&\textnormal{if}\,\,\,\theta\in[\tau^{\prime},\tau^{\prime\prime}]\end{cases}$

Then by Remark (3.3) we have

$W(x,t)=\int_{\tau^{\prime}}^{t}f^{*}(\dot{\Gamma}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau^{\prime})>\int_{\tau^{\prime}}^{t}f^{*}(\dot{\bar{\Gamma}}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau^{\prime})$

which is a contradiction. This completes the proof of non-intersecting
property. ∎

Next we collect some properties of the minimizers in the following lemma.

######  Lemma 3.9.

With the definitions (2.4)-(2.5), we have the following:

  1. (1)

$\tau_{*}(x,t)$ and $\tau^{*}(x,t)$ are, for fixed $x$, monotonically
increasing in $t$ and for fixed $t$ monotonically decreasing in $x$. Moreover
we have for $t_{1}<t_{2}$ that $\tau^{*}(x,t_{1})\leq\tau_{*}(x,t_{2})$ and
for $x_{1}<x_{2}$ that $\tau_{*}(x_{1},t)\geq\tau^{*}(x_{2},t)$.

  2. (2)

$y_{*}(x,t)$ and $y^{*}(x,t)$ are, for fixed $t$ monotonically increasing in
$x$ and for $x_{1}<x_{2}$ we have $y^{*}(x_{1},t)\leq y_{*}(x_{2},t)$.

  3. (3)

$y_{*}(x,t)$ is lower semicontinuous and $y^{*}(x,t)$ is upper semicontinuous.

  4. (4)

$\tau_{*}(x,t)$ is lower semicontinuous and $\tau^{*}(x,t)$ is upper
semicontinuous.

###### Proof.

Proof of these properties follow from the non-intersecting property. ∎

Now we are ready to give the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Step 1: Let $(x,t)$ be a point of continuity for $y_{*},\tau_{*},y^{*}$ and
$\tau^{*}.$ Using the dynamics programming principle, there exist a level
$\tau>0$ and a neighbourhood $N_{I}$ of $x$ and $N_{B}$ of $t$ such that
$W(y,s),\,\,(y,s)\in N_{I}\times N_{B},$ can be expressed as follows:

$W(y,s)=\min_{z\in
H_{I}}\Big{\\{}W(z,\tau)+\int_{\tau}^{t}f^{*}(f^{\prime}(h(y-z,t,\tau))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta\Big{\\}}.$

Define

$W(y,z,t)=W(z,\tau)+\int_{\tau}^{t}f^{*}(f^{\prime}(h(y-z,t,\tau))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta.$

Let $x_{1}<x<x_{2}$ and $x_{1},x_{2}\in N_{I},$ now we calculate

$\displaystyle W(x_{1},t)-W(x_{2},t)$
$\displaystyle=W(y(x_{1},t,\tau),x_{1},t)-W(y(x_{2},t,\tau),x_{2},t)$
$\displaystyle=W(y(x_{1},t,\tau),x_{1},t)-W(y(x_{1},t,\tau),x_{2},t)+W(y(x_{1},t,\tau),x_{2},t)-W(x_{2},y(x_{2},t,\tau),x_{2},t).$

As the second term is non-negative, we get

(3.15) $\displaystyle W(x_{1},t)-W(x_{2},t)$ $\displaystyle\geq
W(y(x_{1},t,\tau),x_{1}t)-W(y(x_{1},t,\tau),x_{2}t)$
$\displaystyle\geq\int_{\tau}^{t}[f^{*}(f^{\prime}(h(x_{1}-y(x_{1},t,\tau),t,\tau))e^{\beta(\theta)})-f^{*}(f^{\prime}(h(x_{2}-y(x_{1},t,\tau),t,\tau))e^{\beta(\theta)})]e^{-\beta(\theta)}d\theta.$

Similarly

$\displaystyle W(x_{1},t)-W(x_{2},t)$
$\displaystyle=W(y(x_{1},t,\tau),x_{1}t)-W(y(x_{2},t,\tau),x_{2},t)$
$\displaystyle=W(y(x_{1},t,\tau),x_{1},t)-W(y(x_{2},t,\tau),x_{1},t)+W(y(x_{2},t,\tau),x_{1},t)-W(y(x_{2},t,\tau),x_{2},t).$

As the first term is not positive, we get

(3.16) $\displaystyle W(x_{1},t)-W(x_{2},t)$ $\displaystyle\leq
W(y(x_{2},t,\tau),x_{1}t)-W(y(x_{2},t,\tau),x_{2}t)$
$\displaystyle\leq\int_{\tau}^{t}[f^{*}(f^{\prime}(h(x_{1}-y(x_{2},t,\tau),t,\tau))e^{\beta(\theta)})-f^{*}(f^{\prime}(h(x_{2}-y(x_{2},t,\tau),t,\tau))e^{\beta(\theta)})]e^{-\beta(\theta)}d\theta.$

Step 2: In the second step we simplify both the expressions (LABEL:ww4.11)
and(3.16). Let us calculate

$\displaystyle\frac{\partial}{\partial
z}\Big{[}e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(x-z,t,\tau)e^{\beta(\theta)}\big{)}\Big{)}\Big{]}$
$\displaystyle=h(x-z,t,\tau)\frac{\partial}{\partial
z}f^{\prime}\big{(}h(x-z,t,\tau)e^{\beta(\theta)}.$

Integrating the above expression from $x$ to $y$ with respect to the variable
$z$,

$\displaystyle
e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(x-y,t,\tau)e^{\beta(\theta)}\Big{)}-e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(0,t,\tau)e^{\beta(\theta)}\Big{)}$
$\displaystyle=\int_{x}^{y}h(x-z,t,\tau)\frac{\partial}{\partial
z}f^{\prime}\big{(}h(x-z,t,\tau)e^{\beta(\theta)}\big{)}dz.$

Again integrating from $\tau$ to $t$ with respect to the variable $\theta$, we
find

(3.17)
$\displaystyle\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(x-y,t,\tau)e^{\beta(\theta)}\Big{)}d\theta$
$\displaystyle=\int_{\tau}^{t}\int_{x}^{y}h(x-z,t,\tau)\frac{\partial}{\partial
z}f^{\prime}\big{(}h(x-z,t,\tau)e^{\beta(\theta)}\big{)}dzd\theta+\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(0,t,\tau)e^{\beta(\theta)}\Big{)}d\theta$
$\displaystyle=\int_{x}^{y}\int_{\tau}^{t}h(x-z,t,\tau)\frac{\partial}{\partial
z}f^{\prime}\big{(}h(x-z,t,\tau)e^{\beta(\theta)}\big{)}d\theta
dz+\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(0,t,\tau)e^{\beta(\theta)}\Big{)}d\theta$
$\displaystyle=-\int_{x}^{y}h(x-z,t,\tau)dz-\int_{\tau}^{t}e^{-\beta(\theta)}f(h(0,t,\tau)e^{\beta(\theta)})d\theta,$

where we used the identity

$\int_{\tau}^{t}e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(0,t,\tau)e^{\beta(\theta)}\Big{)}d\theta=-\int_{\tau}^{t}e^{-\beta(\theta)}f(h(0,t,\tau)e^{\beta(\theta)})d\theta,$

which follows by using the identity $f^{*}(f^{\prime}(p)=pf^{\prime}(p)-f(p)$
and definition of $h$ function. Dividing by $x_{1}-x_{2}$ and simplifying, we
get

$\frac{1}{x_{1}-x_{2}}\int_{x_{2}-y(x_{1},t,\tau)}^{x_{1}-y(x_{1},t,\tau)}h(p,t,\tau)dp\leq\frac{W(x_{1},t)-W(x_{2},t)}{x_{1}-x_{2}}\leq\frac{1}{x_{1}-x_{2}}\int_{x_{2}-y(x_{2},t,\tau)}^{x_{1}-y(x_{2},t,\tau)}h(p,t,\tau)dp.$

Then passing to the limit as $x_{1},x_{2}\to x$ and using the monotonicity
properties of $y(\cdot,t,\tau)$, we get,

$W_{x}=h(x-y(x,t,\tau),t,\tau).$

Step 3: In this step we shall prove $W_{t}=-e^{\beta(t)}f(e^{\beta(t)}w)$. Let
$t_{1}<t<t_{2}$ and $t_{1},t_{2}\in N_{B}.$ Now we calculate

$\displaystyle W(x,t_{1})-W(x_{,}t_{2})$
$\displaystyle=W(y(x,t_{1},\tau),xt_{1})-W(y(x,t_{2},\tau),x,t_{2})$
$\displaystyle=W(y(x,t_{1},\tau),x,t_{1})-W(y(x,t_{2},\tau),x,t_{1})+W(y(x,t_{2},\tau),x,t_{1})-W(y(x,t_{2},\tau),x,t_{2}).$

As the first term is not positive, we get

(3.18) $\displaystyle W(x,t_{1})-W(x_{,}t_{2})$ $\displaystyle\leq
W(y(x,t_{2},\tau),x,t_{1})-W(y(x,t_{2},\tau),x,t_{2})$
$\displaystyle=\int_{\tau}^{t_{1}}f^{*}(f^{\prime}(h(x-y(x,t_{2},\tau),t_{1},\tau))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta$
$\displaystyle-\int_{\tau}^{t_{2}}f^{*}(f^{\prime}(h(x_{2}-y(x,t_{2},\tau),t_{2},\tau))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta.$

Dividing by $t_{1}-t_{2}$ and passing to the limit as $t_{1},t_{2}\to t,$ we
get that the right hand side of the above inequality is the derivative of

$\int_{\tau}^{r}f^{*}(f^{\prime}(h(x-y(x,t_{2},\tau),r,\tau))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta$

with respect to $r$ at the point $t.$ Now we calculate

(3.19)
$\displaystyle\frac{d}{dr}\Big{[}\int_{\tau}^{r}f^{*}(f^{\prime}(h(x-y(x,t_{2},\tau),r,\tau))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta\Big{]}$
$\displaystyle=$
$\displaystyle\frac{d}{dr}\Big{[}h(x-y(x,t_{2},\tau),r,\tau)(x-y(x,t_{2},\tau))-\int_{\tau}^{r}f\Big{(}h(x-y(x,t_{2},\tau),r,\tau))e^{\beta(\theta)}\Big{)}e^{-\beta(\theta)}d\theta\Big{]}$
$\displaystyle=$ $\displaystyle
h_{r}(x-y(x,t_{2},\tau),r,\tau)(x-y(x,t_{2},\tau))-f\Big{(}h(x-y(x,t_{2},\tau),r,\tau))e^{\beta(r)}\Big{)}e^{-\beta(r)}$
$\displaystyle-\int_{\tau}^{r}f\Big{(}h(x-y(x,t_{2},\tau),r,\tau))e^{\beta(\theta)}\Big{)}h_{r}(x-y(x,t_{2},\tau),r,\tau)d\theta.$

Therefore from (3.18) we get

$\limsup_{t_{1},t_{2}\to
t}\frac{W(x,t_{1})-W(x_{,}t_{2})}{t_{1}-t_{2}}\geq-f\Big{(}h(x-y(x,t,\tau),t,\tau))e^{\beta(t)}\Big{)}e^{-\beta(t)}.$

Similar calculation yields

$\liminf_{t_{1},t_{2}\to
t}\frac{W(x,t_{1})-W(x_{,}t_{2})}{t_{1}-t_{2}}\leq-f\Big{(}h(x-y(x,t,\tau),t,\tau))e^{\beta(t)}\Big{)}e^{-\beta(t)}.$

The above two inequalities show that the derivative of $W$ with respect to $t$
exists and

$W_{t}=-f\Big{(}h(x-y(x,t,\tau),t,\tau))e^{\beta(t)}\Big{)}e^{-\beta(t)}.$

By integration by parts,

$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}\Big{[}W\phi_{t}(x,t)+e^{-\beta(t)}f(e^{\beta(t)}W_{x})\phi(x,t)\Big{]}dxdt$
$\displaystyle=-\int_{0}^{\infty}\int_{0}^{\infty}\Big{[}W_{t}+e^{-\beta(t)}f(e^{\beta(t)}W_{x})\Big{]}\phi(x,t)dxdt=0.$

Thus $w$ satisfies $w_{t}+e^{-\beta(t)}f(e^{\beta(t)}w)_{x}=0$ in the sense of
distribution. This completes the proof.

∎

In the next section we prove the weak boundary condition.

##  4\. Boundary condition

In this section we verify the boundary condition in the sense of Bardos- le
Roux- Nédélec[2]. We assume that the boundary data $u_{b}$ is continuous.
Based upon the characterization of $W(0,t)$ in Theorem (3.5), we classify the
boundary points into three types:

######  Definition 4.1.

Let $t>0$ and $(0,t)$ be a point on the boundary $x=0$,

  1. 1.

A point $(0,t)$ is said to be a boundary point of Type-1 if

$W(0,t)=-\int_{\tau}^{t}f(\overline{u_{b}}(\theta))e^{-\beta(\theta)}d\theta+W(0,\tau)$

for some $\tau<t.$

  2. 2.

A point $(0,t)$ is said to be a boundary point of Type-2 if

$W(0,t)=-\int_{\tau}^{t}f^{*}(f^{\prime}(h(0,t,\tau)e^{\beta(\theta))})e^{-\beta(\theta)}d\theta+W(0,\tau)$

for some $\tau<t.$

  3. 3.

A point $(0,t)$ is said to be a boundary point of Type-3 if there exists a
sequence of points $\\{t_{n}\\}$ satisfying $t_{n}<t_{n+1}$and converging to
$t$ such that

$W(0,t)=\sum_{n\in\mathbb{N}}\int_{t_{2n-1}}^{t_{2n}}f^{*}(f^{\prime}(h(0,t_{2n},t_{2n-1})e^{\beta(\theta)}))e^{-\beta(\theta)}d\theta-\sum_{n\in\mathbb{N}}\int_{t_{2n}}^{t_{2n+1}}f(u_{b}(\theta))d\theta+W(0,t_{1}).$

Now we prove the weak form of boundary condition in the sense of Bardos-le
Roux- Nédélec.

######  Theorem 4.2.

The solution $u$ to (1.1) is given by

$\displaystyle
u(x,t)=\begin{cases}&e^{\beta(t)}h(x-y^{*}(x,t),t)\,\,\,\,\,\,\,\,\textnormal{if}\,\,\,\,A(x,t)\leq
B(x,t),\\\
&e^{\beta(t)}h(x,t,\tau^{*}(x,t))\,\,\,\,\,\,\,\,\textnormal{if}\,\,\,\,A(x,t)>B(x,t),\end{cases}$

and $u$ satisfies the initial condition and a weak form of boundary condition
in the sense of (1.2).

###### Proof.

For $x>0,$ there exists an $\epsilon>0$ such that for $t<\epsilon,$ we have
$A(x,t)<B(x,t).$ Then $u(x,t)$ satisfies the initial condition [1]. Here we
prove the boundary condition.

_Case-I: $A(0,t)\leq B(0,t)$:_ In this case
$u(0+,t)=e^{\beta(t)}h(-y^{*}(0,t),t).$ The $h$-curve $X(\theta)$ joining
$(0,t)$ to $(y^{*}(0,t),0)$ satisfies

$X^{\prime}(\theta)=f^{\prime}(h(-y^{*}(0,t),t)e^{\beta(\theta)}).$

Since $X(\theta)\geq 0$ and $X(t)=0,$ $X^{\prime}(t)\leq 0.$ This implies
$f^{\prime}(u(0+,t))\leq 0.$ Observe that, it is enough to show that the
boundary condition holds for a particular $W(0,\tau).$ Since

$\displaystyle\min_{\tau\in H_{b}}B(\tau,x,t)\leq$ $\displaystyle\min_{t\geq
t_{2}\geq t_{1}\geq
0,y}\int_{t_{2}}^{t}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(x,t,t_{2})e^{\beta(\theta)}))d\theta-\int_{t_{1}}^{t_{2}}f(\bar{u_{b}}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle+\int_{0}^{t_{1}}e^{-\beta(\theta)}f^{*}(f^{\prime}(h(-y,t_{1})e^{\beta(\theta)}))d\theta+\int_{0}^{y}u_{0}(\theta)d\theta$
$\displaystyle=\min_{t\geq t_{2}\geq t_{1}\geq 0,y}B(x,y,t,t_{1},t_{2}),$

we can use the functional consisting three piece $h$-curves.

Now as in the proof of Theorem $1.1$, we rewrite the functional $A(y,x,t)$ and
$B(x,y,t,t_{1},t_{2})$ in the following equivalent form:

$\displaystyle
A(y,x,t)=\int_{0}^{y}u_{0}(\theta)d\theta-\int_{\tau}^{t}e^{-\beta(\theta)}f\big{(}h(x-y,\theta)e^{\beta(\theta)}\big{)}d\theta-\int_{0}^{\tau}e^{-\beta(\theta)}f(h(x-y,\tau)e^{\beta(\theta)})d\theta$
$\displaystyle+(x-y)h(x-y,\tau),$ $\displaystyle
B(x,y,t,t_{1},t_{2})=\int_{0}^{y}u_{0}(\theta)d\theta-\int_{t_{1}}^{t_{2}}e^{-\beta(\theta)}f(\bar{u_{b}}(\theta))d\theta-\int_{\tau}^{t_{1}}e^{-\beta(\theta)}f\big{(}h(-y,\theta)e^{\beta(\theta)}\big{)}d\theta$
$\displaystyle-\int_{0}^{\tau}e^{\beta(\theta)}f\big{(}h(-y,\tau)e^{\beta(\theta)}\big{)}d\theta-
yh(-y,\tau)+\int_{0}^{x}h(\theta,t,t_{2})d\theta-\int_{t_{2}}^{t}e^{-\beta(\theta)}f(h(0,t,t_{2})e^{\beta(\theta)})d\theta.$

Since the characteristic curve joining the points $(0,t)$ to $(y^{*}(0,t),t)$
is a minimizer of the functional $A(y,x,t),$ we find

(4.1) $A(y^{*}(0,t),0,t)\leq B(0,y^{*}(0,t),t,t-\varepsilon).$

We observe that in the above inequality, $t_{1}$ is replaced by
$t-\varepsilon$ and $t_{2}$ is replaced by $t.$ Since,

$\frac{(f^{\prime})^{-1}(0)}{e^{\displaystyle{\max_{\theta\in[t_{2},t]}\beta(\theta)}}}\leq
h(0,t,t_{2})\leq\frac{(f^{\prime})^{-1}(0)}{e^{\displaystyle{\min_{\theta\in[t_{2},t]}\beta(\theta)}}},$

we have

$\lim_{t_{2}\to
t}\int_{t_{2}}^{t}e^{-\beta(\theta)}f(h(0,t,t_{2})e^{\beta(\theta)})d\theta=0.$

Hence from the inequality (4.1), we have

$\displaystyle\int_{0}^{y^{*}(0,t)}u_{0}(\theta)d\theta-\int_{\tau}^{t}e^{-\beta(\theta)}f\big{(}h(-y^{*}(0,t),\theta)e^{\beta(\theta)}\big{)}d\theta-\int_{0}^{\tau}e^{\beta(\theta)}f\big{(}h(-y^{*}(0,t),\tau)e^{\beta(\theta)}\big{)}d\theta$
$\displaystyle-y^{*}(0,t)h(-y^{*}(0,t),\tau)$ $\displaystyle\leq$
$\displaystyle\int_{0}^{y^{*}(0,t)}u_{0}(\theta)d\theta-\int_{t-\varepsilon}^{t}e^{-\beta(\theta)}f(\bar{u_{b}}(\theta))d\theta-\int_{\tau}^{t-\varepsilon}e^{-\beta(\theta)}f\big{(}h(-y^{*}(0,t),\theta)e^{\beta(\theta)}\big{)}d\theta$
$\displaystyle-\int_{0}^{\tau}e^{\beta(\theta)}f\big{(}h(-y^{*}(0,t),\tau)e^{\beta(\theta)}\big{)}d\theta-y^{*}(0,t)h(-y^{*}(0,t),\tau).$

This implies

$\int_{t-\varepsilon}^{t}e^{-\beta(\theta)}f(\bar{u_{b}}(\theta))d\theta\leq\int_{t-\varepsilon}^{t}e^{-\beta(\theta)}f\big{(}h(-y^{*}(0,t),\theta)e^{\beta(\theta)}\big{)}d\theta.$

Now dividing by $\varepsilon$ on the both sides and passing to the limit as
$\varepsilon\to 0,$ we get

$f(\bar{u_{b}}(t))\leq f(u(0+,t)).$

This proves the Bardos- le Roux- Nédélec boundary condition for the _Case-I_.
Case-II: $B(0,t)<A(0,t)$ and $\tau_{*}(x,t)$ does not converge to $t$: In this
case, we use dynamic programming principle and the proof will follow as in
_Case-I_.
Case-III: $B(0,t)<A(0,t)$ and $\tau_{*}(x,t)$ converges to $t$: If $(0,t)$ is
of Type 2, then $\tau_{*}(x,t)$ does not converge to $t.$ Therefore in this
case the boundary points $(0,t)$ are not of _Type-2_. So _Case-III_ has only
two sub cases as follows.
_Sub-case 1:_ If $(0,t)$ is of Type 1, then for $(x,t)$ near to $(0,t)$, we
have

$W(x,t)=\int_{\tau_{*}(x,t)}^{t}f^{*}(f^{\prime}(h(x,t,\tau_{*}(x,t))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta-\int^{\tau_{*}(x,t)}_{\tau}f(\overline{u_{b}}(s))e^{-\beta(\theta)}d\theta+W(0,\tau).$

Given any sequence $x_{n}\to 0,$
$X^{\prime}(\tau^{*}(x_{n},t))=f^{\prime}(h(x_{n},t,\tau_{*}(x_{n},t))e^{\beta(\tau_{*}(x_{n},t))})\geq
0,$ where $X(\theta)$ is the $h-$curve joining $(x_{n},t)$ to
$(0,\tau_{*}(x_{n},t)).$ If $(x_{n},t)$ satisfies
$f^{\prime}(h(x_{n},t,\tau(x_{n},t))e^{\beta(\tau(x_{n},t))})>0,$ then there
exists a $\delta>0$ such that for
$\tau_{1}\in(\tau_{*}(x_{n},t)-\delta,\tau_{*}(x_{n},t)+\delta),$ the
$h-$curves joining $(x_{n},t)$ to $(0,\tau_{1})$ lies completely in $Q.$ Then

$W(x,t,\tau_{1})=\int_{\tau_{1}}^{t}f^{*}(f^{\prime}(h(x,t,\tau_{1})e^{\beta(\theta)})e^{-\beta(\theta)}d\theta-\int^{\tau_{1}}_{\tau}f(\overline{u_{b}}(s))e^{-\beta(s)}ds+W(0,\tau).$

attains its minimum at $\tau_{1}=\tau_{*}(x_{n},t).$ Therefore
$\frac{\partial}{\partial\tau_{1}}W(x,t,\tau_{1})=0$ at
$\tau_{1}=\tau_{*}(x_{n},t).$ This implies
$f(\bar{u}_{b}(\tau_{*}(x_{n},t)))=f(h(x_{n},t,\tau_{*}(x_{n},t))e^{\beta(\tau_{*}(x_{n},t))}).$
Since $f^{\prime}(u(0+,t))\geq 0$, we get $u(0+,t)=\bar{u}_{b}(t).$
Now if
$f^{\prime}(h(x_{n},t,\tau^{*}(x_{n},t))e^{\beta(\tau^{*}(x_{n},t))})=0$, then
$f^{\prime}(u(0+,t))=0.$ Now we prove $f(u(0+,t))=f(\bar{u}_{b}(t)).$ Since
$f^{\prime}(u(0+,t))=0,$ this implies $u(0+,t)=\lambda_{f}$. Since $f$ is
strictly convex and $f^{\prime}$ changes sign at $\lambda_{f},$ therefore $f$
is strictly increasing in the interval $[\lambda_{f},\infty)$ and by
definition of $\bar{u_{b}},$ we know $\bar{u_{b}}(t)\geq\lambda_{f}.$ So
$f(\bar{u_{b}}(t))\geq f(\lambda_{f}).$ Now to show that
$f(\bar{u_{b}}(t))\leq f(\lambda_{f}),$ first note that

$W(x,t)=\int_{\tau_{*}(x,t)}^{t}f^{*}(f^{\prime}(h(x,t,\tau_{*}(x,t))e^{\beta(\theta)})e^{-\beta(\theta)}d\theta+W(0,\tau_{*}(x,t)).$

It is clear that for any $\delta>0,$ the sets
$(\tau_{*}(x,t),\tau_{*}(x,t)+\delta$) contain points such that the $h$-curve
joining those points from $(x,t)$ completely lies in $Q.$ So for any $x,$
there exists a point $h_{x}<x,$ such that

(4.2) $\displaystyle W(x,t,\tau^{*}(x,t)+h_{x})\geq W(x,t,\tau^{*}(x,t))$
$\displaystyle\implies$
$\displaystyle\int_{\tau^{*}(x,t)}^{\tau^{*}(x,t)+h_{x}}\frac{\partial}{\partial\tau}\Big{[}\int_{\tau}^{t}f^{*}(f^{\prime}(h(x,t,\tau)e^{\beta(\theta)})e^{-\beta(\theta)}d\theta\Big{]}\geq
W(0,\tau^{*}(x,t))-W(0,\tau^{*}(x,t)+h_{x})$ $\displaystyle\implies$
$\displaystyle\int_{\tau^{*}(x,t)}^{\tau^{*}(x,t)+h_{x}}f(h(x,t,\tau)e^{\beta(\tau)})e^{-\beta(\tau)}d\tau\geq
W(0,\tau^{*}(x,t))-W(0,\tau^{*}(x,t)+h_{x})$ $\displaystyle\implies$
$\displaystyle\int_{\tau^{*}(x,t)}^{\tau^{*}(x,t)+h_{x}}f(h(x,t,\tau)e^{\beta(\tau)})e^{-\beta(\tau)}d\tau\geq\int_{\tau^{*}(x,t)+h_{x}}^{\tau^{*}(x,t)}f(\bar{u}_{b}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle\implies$ $\displaystyle\lim_{x\to
0}\frac{1}{h_{x}}\int_{\tau^{*}(x,t)}^{\tau^{*}(x,t)+h_{x}}f(h(x,t,\tau)e^{\beta(\tau)})e^{-\beta(\tau)}d\tau\geq\frac{1}{h_{x}}\int_{\tau^{*}(x,t)+h_{x}}^{\tau^{*}(x,t)}f(\bar{u}_{b}(\theta))e^{-\beta(\theta)}d\theta$
$\displaystyle\implies$ $\displaystyle f(u(0+,t))\geq f(\bar{u}_{b}(t).$

_Sub-case 2:_ Suppose $(0,t)$ is of Type 3 and there are infinitely many $n$
such that $x_{n}\to 0$ for which
$\tau_{*}(x_{n},t)\in(t_{2n_{k}},t_{2n_{k}+1}],$ then the same analysis as in
_sub-case 1_ of _Case III_ can be followed.
Suppose $(0,t)$ is of Type 3 and there are infinitely many $n$ such that
$x_{n}\to 0$ for which $\tau_{*}(x_{n},t)\in(t_{2n_{k}-1},t_{2n_{k}}],$ then
$\tau_{*}(x_{n},t)=t_{2n_{k}},$ because two minimizers can not intersect
because of the non-intersecting property, lemma (3.8). Further
$f^{\prime}\Big{(}h(x,t,t_{2n_{k}})e^{\beta(t_{2n_{k}})}\Big{)}=0$ as the $h-$
curve joining $(x_{n},t)$ to $(0,t_{2n_{k}})$ and the $h-$curve joining
$(0,t_{2n_{k}})$ to $(0,t_{2n_{k-1}})$ is the single $h$-curve (since it
minimizes) and this concludes
$h(x,t,t_{2n_{k}})e^{\beta(t_{2n_{k}})}=\lambda_{f}.$ In this case
$u(0+,t)=\lambda_{f},$ so $f^{\prime}(u(0+,t))=0.$ Now by the similar argument
as (4.2) we prove $f(u(0+,t))=f(\bar{u}_{b}(t)).$ ∎

######  Remark 4.3.

We showed the Bardos- le Roux- Nédélec condition holds by assuming that the
boundary data $u_{b}$ is continuous. The same can be shown for any bounded
measurable data $u_{b}$ by considering the Lebesgue points.

##  5\. Construction of generalised characteristics

In this section we assume that for every $(x,t)\in Q,$ $W(x,t)$ is given by
(1.3) (see Remark 1.2 for a sufficient condition for this situation). With
this assumption we will interpret the explicit formulae for the balance laws
by means of generalised characteristics. Inspired by [22], we begin with the
definition of characteristic triangles.

######  Definition 5.1.

Let $x>0$, $t>0$ and $A(x,t)$, $B(x,t)$ are given as above.

  1. 1.

For $A(x,t)<B(x,t)$ we define the characteristics triangle at the point
$(x,t)$ as the region bounded by the $h$-curves joining the points
$(x,t),(y_{*}(x,t),0)$ and $(y^{*}(x,t),0).$

  2. 2.

For $A(x,t)>B(x,t)$ we define the characteristics triangle at the point
$(x,t)$ as the region bounded by the $h$-curves joining the points
$(x,t),(0,\tau_{*}(x,t),$ and $(0,\tau^{*}(x,t)).$

  3. 3.

For $A(x,t)=B(x,t)$ we define the characteristics triangle at the point
$(x,t)$ as the region bounded by the $h$-curves joining the points $(x,t)$,
$(y^{*}(x,t),0)$, $(0,\tau^{*}(x,t))$ which includes the point $(0,0)$, where
$(x,t)$ and $(y_{*}(x,t),0)$ (resp. $(\tau^{*}(x,t),0)$) are joined by a
$h$-curves for initial (resp. boundary) functional.

We denote the characteristic triangle associated with the point $(x,t)$ by
$\Delta(x,t)$.

We list some properties of characteristic triangle in the following lemma.

######  Lemma 5.2.

Let $x>0,t>0.$ Then we have the following properties of characteristic
triangle.

  1. (1)

Let $t>0$ be fixed, and $x_{1}$, $x_{2}>0$, $x_{1}\neq x_{2}$ but arbitrary.
Then the characteristic triangles associated to $(x_{1},t)$ and $(x_{2},t)$ do
not intersect in the interior of $\mathbb{R}_{+}^{2}$.

  2. (2)

If two characteristic triangles intersect in $\mathbb{R}_{+}^{2}$, then one is
contained in the other.

  3. (3)

For any time $t_{0}>0$ we have

$\bigcup_{x\in[0,\infty)}\Delta(x,t_{0})=\\{(x,t)|x\in[0,\infty),0\leq t\leq
t_{0}\\}\,.$

###### Proof.

The proof uses monotonicity properties of minimisers and follows from the
similar lines given in [22]. ∎

The next theorem is devoted to construct the generalised characteristics.

######  Theorem 5.3.

Let $t_{1}>0$ be fixed. For each $(x_{1},t_{1})$ there exists a unique
Lipschitz continuous curve $x=\mathcal{X}(t)$ such that
$\mathcal{X}(t_{1})=x_{1}$ and for any point $(x,t),t\geq t_{1}$ on the curve,
we have the following:

(i) When $A(x,t)<B(x,t)$ we have

$\displaystyle\lim_{t^{\prime\prime},t^{\prime}\to
t}\frac{\mathcal{X}(t^{\prime\prime})-\mathcal{X}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}=\begin{cases}f^{\prime}\big{(}h(x-y(x,t),t)e^{\beta(t)}\big{)}\,\,\,\,\,&\textnormal{if}\,\,\,y_{*}(x,t)=y^{*}(x,t),\\\
\frac{f\big{(}h(x-y_{*}(x,t),t)e^{\beta(t)}\big{)}-f\big{(}h(x-y^{*}(x,t),t)e^{\beta(t)}\big{)}}{h(x-y_{*}(x,t),t)e^{\beta(t)}-h(x-y^{*}(x,t),t)e^{\beta(t)}}\,\,\,\,\,&\textnormal{if}\,\,\,y_{*}(x,t)<y^{*}(x,t).\end{cases}$

(ii) When $B(x,t)<A(x,t)$ we have

$\displaystyle\lim_{t^{\prime\prime},t^{\prime}\to
t}\frac{\mathcal{X}(t^{\prime\prime})-\mathcal{X}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}=\begin{cases}f^{\prime}\big{(}h(x,t,\tau(x,t))e^{\beta(t)}\big{)}\,\,\,\,\,&\textnormal{if}\,\,\,\tau_{*}(x,t)=\tau^{*}(x,t),\\\
\frac{f\big{(}h(x,t,\tau_{*}(x,t))e^{\beta(t)}\big{)}-f\big{(}h(x,t,\tau^{*}(x,t))e^{\beta(t)}\big{)}}{h(x,t,\tau_{*}(x,t))e^{\beta(t)}-h(x,t,\tau^{*}(x,t))e^{\beta(t)}}\,\,\,\,\,&\textnormal{if}\,\,\,\tau_{*}(x,t)<\tau^{*}(x,t).\end{cases}$

(iii) When $A(x,t)=B(x,t)$ and $y^{*}(x,t)\neq 0$ or $\tau^{*}(x,t)\neq 0,$ we
have

$\lim_{t^{\prime\prime},t^{\prime}\to
t}\frac{\mathcal{X}(t^{\prime\prime})-\mathcal{X}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}=\frac{f\big{(}h(x-y^{*}(x,t),t)e^{\beta(t)}\big{)}-f\big{(}h(x,t,\tau^{*}(x,t))e^{\beta(t)}\big{)}}{h(x-y^{*}(x,t),t)e^{\beta(t)}-h(x,t,\tau^{*}(x,t))e^{\beta(t)}}.$

(iv) When $A(x,t)=B(x,t)$ and both $y^{*}(x,t)=0,\tau^{*}(x,t)=0$ we have

$\lim_{t^{\prime\prime},t^{\prime}\to
t}\frac{\mathcal{X}(t^{\prime\prime})-\mathcal{X}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}=f^{\prime}\big{(}h(x,t)e^{\beta(t)}\big{)}.$

###### Proof.

To prove $(i),$ let $t<t^{\prime}<t^{\prime\prime}$ and
$\mathcal{X}(t)=x,\mathcal{X}(t^{\prime})=x^{\prime},\mathcal{X}(t^{\prime\prime})=x^{\prime\prime}.$
First we consider the case $y_{*}(x,t)=y^{*}(x,t).$ In this case the slope
$\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}$ lies between
the slope of the characteristic curves joining the points
$\\{(x^{\prime\prime},t^{\prime\prime}),(y^{*}(x^{\prime\prime},t^{\prime\prime}),0)\\}$
and
$\\{(x^{\prime\prime},t^{\prime\prime}),(y_{*}(x^{\prime\prime},t^{\prime\prime}),0)\\}.$
Characteristic curve through the points
$\\{(x^{\prime\prime},t^{\prime\prime}),(y_{*}(x^{\prime\prime},t^{\prime\prime}),0)\\}$
is

$X_{1}(\xi)=y_{*}(x^{\prime\prime},t^{\prime\prime})+\int_{0}^{\xi}f^{\prime}\big{(}h(x^{\prime\prime}-y_{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta,$

and characteristic curve through the points
$\\{(x^{\prime\prime},t^{\prime\prime}),(y^{*}(x^{\prime\prime},t^{\prime\prime}),0)\\}$
is

$X_{2}(\xi)=y^{*}(x^{\prime\prime},t^{\prime\prime})+\int_{0}^{\xi}f^{\prime}\big{(}h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta.$

Hence

(5.1)
$\frac{X_{1}(t^{\prime\prime})-X_{1}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\geq\frac{X_{2}(t^{\prime\prime})-X_{2}(t^{\prime})}{t^{\prime\prime}-t^{\prime}},$

i.e.

$\frac{\int_{t^{\prime}}^{t^{\prime\prime}}f^{\prime}\big{(}h(x-y_{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta}{t^{\prime\prime}-t^{\prime}}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\geq\frac{\int_{t^{\prime}}^{t^{\prime\prime}}f^{\prime}\big{(}h(x-y^{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta}{t^{\prime\prime}-t^{\prime}}.$

Now passing to the limit as $t^{\prime\prime},t^{\prime}\to t$ and using the
monotonicity and semi continuity of $y_{*},y^{*}$ we obtain the first identity
of $(i).$ Now consider the case $y_{*}(x,t)<y^{*}(x,t).$ From the definition
of $A(y,x,t)$ we have

$A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})\leq
A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime})$

The above inequality can be rewritten as

(5.2)
$\displaystyle\big{[}A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}$
$\displaystyle\leq\big{[}A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}A(y_{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}.$

We expand the above inequality step by step. Firstly, using (3.17) we have

(5.3) $\displaystyle
A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})$
$\displaystyle=-\int_{x^{\prime\prime}}^{y^{*}(x^{\prime\prime},t^{\prime\prime})}h(x^{\prime\prime}-\eta,t^{\prime\prime})d\eta-\int_{0}^{t^{\prime\prime}}e^{-\beta(\eta)}f(h(0,t^{\prime\prime})e^{\beta(\eta)})d\eta$
$\displaystyle+\int_{x^{\prime\prime}}^{y^{*}(x^{\prime\prime},t^{\prime\prime})}h(x^{\prime\prime}-\eta,t^{\prime})d\eta+\int_{0}^{t^{\prime}}e^{-\beta(\eta)}f(h(0,t^{\prime})e^{\beta(\eta)})d\eta.$

Using a similar calculation of (3.17), one can obtain, for $0<\tau<t,$

$\displaystyle\int_{x}^{y}h(x-\eta,t)dz+\int_{0}^{t}e^{-\beta(\eta)}f(h(0,t)e^{\beta(\eta)})d\eta$
$\displaystyle=\int_{\tau}^{t}e^{-\beta(\eta)}f\big{(}h(x-y,\eta)e^{\beta(\eta)}\big{)}d\eta+\int_{0}^{\tau}e^{-\beta(\eta)}f(h(x-y,\tau)e^{\beta(\eta)})d\eta-(x-y)h(x-y,\tau).$

Using the above equation in (5.3), we get

$A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})=-\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}f(h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),\eta)e^{\beta(\eta)})d\eta.$

Similarly,

$A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})=-\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}f(h(x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime}),\eta)e^{\beta(\eta)})d\eta.$

Thus we find

$\displaystyle\big{[}A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}$
$\displaystyle=\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime}),\eta)e^{\beta(\eta)})-f(h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),\eta)e^{\beta(\eta)})\Big{]}d\eta.$

and after a change of variable

$\displaystyle\big{[}A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}A(y_{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime})-A(y_{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}$
$\displaystyle=\int_{x^{\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}^{x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}h(z,t^{\prime})dz-\int_{x^{\prime}-y_{*}(x^{\prime},t^{\prime})}^{x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime})}h(z,t^{\prime})dz.$

Therefore from the inequality (5.2) we get

(5.4)
$\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime}),\eta)e^{\beta(\eta)})-f(h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),\eta)e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\Big{[}\int_{x^{\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}^{x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}h(z,t^{\prime})dz-\int_{x^{\prime}-y_{*}(x^{\prime},t^{\prime})}^{x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime})}h(z,t^{\prime})dz\Big{]}}\leq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}.$

On the other hand considering the inequality

$\displaystyle\big{[}A(y_{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y_{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}$
$\displaystyle\leq\big{[}A(y_{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-A(y_{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}A(y^{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime})-A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}$

and simplifying it as before, we get

(5.5)
$\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime}),\eta)e^{\beta(\eta)})-f(h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),\eta)e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\Big{[}\int_{x^{\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}^{x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}h(z,t^{\prime})dz-\int_{x^{\prime}-y_{*}(x^{\prime},t^{\prime})}^{x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime})}h(z,t^{\prime})dz\Big{]}}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}.$

Passing to the limit as $t^{\prime\prime},t^{\prime}\to t$ in (5.4)-(5.5) we
obtain the second identity of $(i).$
For $(ii)$ we observe that the slope
$\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}$ lies between
the slope of the characteristic curve joining the points
$\\{(x^{\prime\prime},t^{\prime\prime}),(0,\tau_{*}(x^{\prime\prime},t^{\prime\prime}))\\}$
and the points
$\\{(x^{\prime\prime},t^{\prime\prime}),(0,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\\}.$
Characteristic curve through
$\\{(x^{\prime\prime},t^{\prime\prime}),(0,\tau_{*}(x^{\prime\prime},t^{\prime\prime}))\\}$
is

$Y_{1}(\xi)=\int_{\tau_{*}(x^{\prime\prime},t^{\prime\prime})}^{\xi}f^{\prime}\big{(}h(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\theta)}\big{)}d\theta,\,\,\,Y_{1}(t^{\prime\prime})=x^{\prime\prime}$

and characteristic curve through
$\\{(x^{\prime\prime},t^{\prime\prime}),(0,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\\}$
is

$Y_{2}(\xi)=\int_{\tau^{*}(x^{\prime\prime},t^{\prime\prime})}^{\xi}f^{\prime}\big{(}h(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\theta)}\big{)}d\theta,\,\,\,Y_{2}(t^{\prime\prime})=x^{\prime\prime}.$

Hence

$\frac{Y_{2}(t^{\prime\prime})-Y_{2}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\geq\frac{Y_{1}(t^{\prime\prime})-Y_{1}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}.$

This implies

$\frac{\int_{t^{\prime}}^{t^{\prime\prime}}f^{\prime}\big{(}h(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\theta)}\big{)}d\theta}{t^{\prime\prime}-t^{\prime}}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\geq\frac{\int_{t^{\prime}}^{t^{\prime\prime}}f^{\prime}\big{(}h(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\theta)}\big{)}d\theta}{t^{\prime\prime}-t^{\prime}}.$

Passing to the limit as $t^{\prime\prime},t^{\prime}\to t,$ we obtain the
first identity of $(ii).$ To prove the second identity for the case
$\tau_{*}(x,t)<\tau^{*}(x,t),$ observe that

$\displaystyle
B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime},t^{\prime}))$
$\displaystyle\leq
B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime},t^{\prime},\tau{*}(x^{\prime},t^{\prime}))$

The above inequality can be rewritten as

(5.6)
$\displaystyle\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))\big{]}$
$\displaystyle\leq\big{[}B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}B(x^{\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))\big{]}.$

Firstly, using (3.19) we have

$\displaystyle
B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))=-\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}f(h(x^{\prime\prime},\eta,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\eta)})d\eta,$

and similarly

$\displaystyle
B(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))$
$\displaystyle=-\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}f(h(x^{\prime\prime},\eta,\tau_{*}(x^{\prime},t^{\prime}))e^{\beta(\eta)})d\eta.$

Hence L.H.S of the inequality (5.6) imply

$\displaystyle\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))\big{]}$
$\displaystyle=\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime},\eta,\tau_{*}(x^{\prime},t^{\prime}))e^{\beta(\eta)})-f(h(x^{\prime\prime},\eta,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\eta)})\Big{]}d\eta.$

Now using a similar calculation as (3.19), for $t_{2}<\theta<t$ one obtains

$\displaystyle\int_{t_{2}}^{t}e^{-\beta(\theta)}f^{*}\Big{(}f^{\prime}\big{(}h(x,t,t_{2})e^{\beta(\theta)}\big{)}\Big{)}d\theta$
$\displaystyle=\int_{0}^{x}h(\eta,t,t_{2})d\eta-\int_{t_{2}}^{t}e^{-\beta(\theta)}f\big{(}h(0,t,t_{2})e^{\beta(\theta)}\big{)}d\theta.$

Applying the above equation in the R.H.S of (5.6) gives

$\displaystyle\big{[}B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}B(x^{\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))\big{]}$
$\displaystyle=\int_{x^{\prime}}^{x^{\prime\prime}}h(\eta,t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))-h(\eta,t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))d\eta.$

This implies

(5.7)
$\displaystyle\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime},\eta,\tau_{*}(x^{\prime},t^{\prime}))e^{\beta(\eta)})-f(h(x^{\prime\prime},\eta,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\int_{x^{\prime}}^{x^{\prime\prime}}h(\eta,t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))-h(\eta,t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))d\eta}\leq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}.$

Similarly from the inequality

$\displaystyle\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau_{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))\big{]}$
$\displaystyle\leq\big{[}B(x^{\prime},t^{\prime},\tau_{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau_{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))\big{]}$

we get

(5.8)
$\displaystyle\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime},\eta,\tau_{*}(x^{\prime},t^{\prime}))e^{\beta(\eta)})-f(h(x^{\prime\prime},\eta,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\int_{x^{\prime}}^{x^{\prime\prime}}h(\eta,t^{\prime},\tau_{*}(x^{\prime},t^{\prime}))-h(\eta,t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))d\eta}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}.$

Passing to the limit $t^{\prime\prime},t^{\prime}\to t$ in (5.7)-(5.8), we
obtain the second identity of $(ii).$
For case $(iii),$ we observe the inequality

$\displaystyle
B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})\leq
B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-A(y^{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime}).$

Following the previous strategy the above inequality can be written as

(5.9)
$\displaystyle\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}$
$\displaystyle\leq\big{[}B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime})-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime\prime},t^{\prime\prime}))\big{]}-\big{[}A(y^{*}(x^{\prime},t^{\prime}),x^{\prime},t^{\prime})-A(y^{*}(x^{\prime},t^{\prime}),x^{\prime\prime},t^{\prime})\big{]}.$

From the previous analysis it follows directly that

(5.10)
$\displaystyle\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime}-y^{*}(x^{\prime},t^{\prime}),\eta)e^{\beta(\eta)})-f(h(x^{\prime\prime},\eta,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\int_{x^{\prime}-y^{*}(x^{\prime},t^{\prime})}^{x^{\prime\prime}-y^{*}(x^{\prime},t^{\prime})}h(z,t^{\prime})d\eta-\frac{1}{x^{\prime\prime}-x^{\prime}}\int_{x^{\prime}}^{x^{\prime\prime}}h(\eta,t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))d\eta}\leq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}.$

Similarly from the inequality

$\displaystyle
A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime},t^{\prime}))$
$\displaystyle\leq
A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))$

and hence from

$\displaystyle\big{[}A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}B(x^{\prime\prime},t^{\prime\prime},\tau^{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))\big{]}$
$\displaystyle\leq\big{[}A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime},t^{\prime})-A(y^{*}(x^{\prime\prime},t^{\prime\prime}),x^{\prime\prime},t^{\prime})\big{]}-\big{[}B(x^{\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))-B(x^{\prime\prime},t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))\big{]}$

we get

(5.11)
$\displaystyle\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),\eta)e^{\beta(\eta)})-f(h(x^{\prime\prime},\eta,\tau^{*}(x^{\prime\prime},t^{\prime\prime}))e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\int_{x^{\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}^{x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}h(z,t^{\prime})d\eta-\frac{1}{x^{\prime\prime}-x^{\prime}}\int_{x^{\prime}}^{x^{\prime\prime}}h(\eta,t^{\prime},\tau^{*}(x^{\prime},t^{\prime}))d\eta}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}.$

Now passing to the limit as $t^{\prime\prime},t^{\prime}\to t$ in
(5.10)-(5.11) we get the identity of $(iii).$
Proof of $(iv)$ can be obtained from the following observation. When
$A(x,t)=B(x,t),$

$\frac{Y_{2}(t^{\prime\prime})-Y_{2}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}\geq\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}\geq\frac{X_{2}(t^{\prime\prime})-X_{2}(t^{\prime})}{t^{\prime\prime}-t^{\prime}}.$

Now from the definition of $X_{2}(\xi),Y_{2}(\xi)$ and using the fact
$y^{*}(x,t)=\tau^{*}(x,t)=0$ we get $(iv).$ ∎

In the next lemma we show that the constructed curve actually starts from
$x$-axis or $t$-axis. The proof is exactly same as [22], so we omit it.

######  Lemma 5.4.

There is a countable set $\mathcal{S}$ of points on the $x$\- and $t$-axis
with the following properties.

  1. (1)

For all $(\eta,0)\not\in\mathcal{S}$ there is a unique Lipschitz continuous
curve $x=\mathcal{X}(\eta,t)$, $t\geq 0$, such that $\mathcal{X}(\eta,0)=\eta$
and the characteristic triangles associated to points on the curve form an
increasing family of sets.

  2. (2)

For all $(0,\eta)\not\in\mathcal{S}$ there is a unique Lipschitz continuous
curve $x=\mathcal{Y}(\eta,t)$, $t\geq\eta$, such that
$\mathcal{Y}(\eta,\eta)=0$ and the characteristic triangles associated to
points on the curve form an increasing family of sets.

Further, for all $\eta>0$ such that $(\eta,0)$ and $(0,\eta)$ do not belong to
$\mathcal{S}$,

$\displaystyle\tfrac{\partial}{\partial t}\mathcal{X}(\eta,t)$
$\displaystyle=f^{\prime}\big{(}u(\mathcal{X}(\eta,t),t)\big{)}$ for almost
all $t>0$ $\displaystyle\tfrac{\partial}{\partial t}\mathcal{Y}(\eta,t)$
$\displaystyle=f^{\prime}\big{(}u(\mathcal{Y}(\eta,t),t)\big{)}$
$\displaystyle\text{for almost all $t>\eta$}\,,$

where the right hand side is a measurable function.

###### Proof.

Our proof follows the arguments found in [22] by appropriately changing the
characteristic lines by $h$-curves. .

∎

######  Remark 5.5.

We constructed the curve $\mathcal{X}(t)$ in the quarter plane $x>0,t>0.$ If
the curve reaches $x=0,$ then for that point $(0,t),$ it falls precisely under
the case $A(0,t)\leq B(0,t)$ and we show the boundary condition by Bardos- le
Roux- Nédélec is satisfied, since $f^{\prime}(u(0+,t))\leq 0.$

######  Remark 5.6.

Note that, for a fixed $t>0,$ if $A(x,t)=B(x,t)$ in an interval $[a(t),b(t)]$
then $\tau^{*}(x,t)=\tau_{*}(x,t)=0,\,\,y_{*}(x,t)=y^{*}(x,t)=0.$ Suppose for
any point $(x_{0},t_{0})\in[a(t),b(t)],$ $\tau^{*}(x_{0},t_{0})>0,$ then by
the non-intersecting property of $h$-curves we arrive at a contradiction. This
property corresponds to the rarefaction solution.

Next we show that the constructed solution is entropy admissible in the sense
of Lax.

######  Theorem 5.7.

Let $x>0$ and $t>0.$ If $x$ is point of discontinuity of $u(\cdot,t),$ then
the following inequality

(5.12) $f^{\prime}(u(x-,t))>\dot{\mathcal{X}}(t)>f^{\prime}(u(x+,t))$

holds for almost every $t>0.$

###### Proof.

We need to prove (5.12) for $A(x,t)<B(x,t),$ $A(x,t)>B(x,t)$ and
$A(x,t)=B(x,t).$ Note that it is enough to prove when the minimizers are not
equal, i.e $y_{*}(x,t)<y^{*}(x,t)$ or $\tau_{*}(x,t)<\tau^{*}(x,t).$ When
$A(x,t)<B(x,t)$ and $y_{*}(x,t)<y^{*}(x,t),$ observe that
$u(x-,t)=h(x-y_{*}(x,t),t)e^{\beta(t)}$ and
$u(x+,t)=h(x-y^{*}(x,t),t)e^{\beta(t)}.$ The characteristic curve through the
points
$\\{(x^{\prime\prime},t^{\prime\prime}),(y_{*}(x^{\prime\prime},t^{\prime\prime}),0)\\}$
is given by

$X_{1}(\xi)=y_{*}(x^{\prime\prime},t^{\prime\prime})+\int_{0}^{\xi}f^{\prime}\big{(}h(x^{\prime\prime}-y_{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta.$

Also the characteristic curve through the points
$\\{(x^{\prime\prime},t^{\prime\prime}),(y^{*}(x^{\prime\prime},t^{\prime\prime}),0)\\}$
is given by

$X_{2}(\xi)=y^{*}(x^{\prime\prime},t^{\prime\prime})+\int_{0}^{\xi}f^{\prime}\big{(}h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta.$

Hence from (5.1) we have

(5.13)
$\frac{\int_{t^{\prime}}^{t^{\prime\prime}}f^{\prime}\big{(}h(x-y_{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta}{t^{\prime\prime}-t^{\prime}}>\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}>\frac{\int_{t^{\prime}}^{t^{\prime\prime}}f^{\prime}\big{(}h(x-y^{*}(x^{\prime\prime},t^{\prime\prime}),t^{\prime\prime})e^{\beta(\theta)}\big{)}d\theta}{t^{\prime\prime}-t^{\prime}}.$

Now since $y_{*}(x,t)<y^{*}(x,t),$ from inequalities (5.4)-(5.5) we get

(5.14)
$\frac{x^{\prime\prime}-x^{\prime}}{t^{\prime\prime}-t^{\prime}}=\frac{\frac{1}{t^{\prime\prime}-t^{\prime}}\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\beta(\eta)}\Big{[}f(h(x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime}),\eta)e^{\beta(\eta)})-f(h(x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime}),\eta)e^{\beta(\eta)})\Big{]}d\eta}{\frac{1}{x^{\prime\prime}-x^{\prime}}\Big{[}\int_{x^{\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}^{x^{\prime\prime}-y^{*}(x^{\prime\prime},t^{\prime\prime})}h(z,t^{\prime})dz-\int_{x^{\prime}-y_{*}(x^{\prime},t^{\prime})}^{x^{\prime\prime}-y_{*}(x^{\prime},t^{\prime})}h(z,t^{\prime})dz\Big{]}}.$

Now passing to the limit as $t^{\prime\prime},t^{\prime}\to t$ in
(5.13)-(5.14) we get the desired inequality (5.12). When $B(x,t)<A(x,t),$
$u(x-,t)=h(x,t,\tau^{*}(x,t))e^{\beta(t)}$ and
$u(x+,t)=h(x,t,\tau_{*}(x,t))e^{\beta(t)}$ and the rest of the proof will be
similar for other cases also. ∎

## Acknowledgement

Manas R. Sahoo would like to acknowledge the support received from SERB-
MATRICS via Grant No. MTR/2018/000864 and Inspire Grant No.DST/
INSPIRE/04/2015/001574.

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