\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 157, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/157\hfil Explicit solutions for a system of PDEs]
{Explicit solutions for a system of first-order partial differential equations}
\author[K. T. Joseph\hfil EJDE-2008/157\hfilneg]
{Kayyunnapara Thomas Joseph}
\address{Kayyunnapara Thomas Joseph \newline
School of Mathematics\\
Tata Institute of Fundamental Research\\
Homi Bhabha Road\\
Mumbai 400005, India}
\email{ktj@math.tifr.res.in}
\thanks{Submitted November 1, 2008. Published November 20, 2008.}
\subjclass[2000]{35A20, 35L50, 35R05}
\keywords{First order equations; boundary conditions; exact solutions}
\begin{abstract}
In this paper we construct explicit weak solutions of a system of
two partial differential equations in the quarter plane
$\{(x,t):x>0,t>0\}$ with initial conditions at $t=0$ and a
weak form of Dirichlet boundary conditions at $x=0$. This system
was first studied by LeFloch \cite{le1}, where he constructed explicit
formula for the weak solution of pure initial value problem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
LeFloch \cite{le1} constructed an explicit formula for the solution
to initial-value problem
\begin{equation}
\begin{gathered}
u_t + f(u)_x =0,\\
v_t + f'(u) v_x =0,
\end{gathered}
\label{e1.1}
\end{equation}
with initial conditions
\begin{equation}
\begin{pmatrix}
u(x,0)\\
v(x,0)
\end{pmatrix}
= \begin{pmatrix}
u_0(x)\\
v_0(x)
\end{pmatrix},
\label{e1.2}
\end{equation}
in the domain $\{(x,t) : -\infty < x < \infty, t>0 \}$, where $f(u)$
is strictly convex.
The first equation is a convex
conservation law and the Lax formula \cite{la1} gives the entropy weak
solution $u(x,t)$ when the initial data $u(x,0)=u_0(x)$ is in the
space of bounded measurable functions. The solution
$u(x,t)$ remains in the space bounded functions and is locally a $BV$
function for $t>0$. Then the second
equation for $v$ is a nonconservative scalar equation with bounded and
$BV_{loc}$ function
$f'(u)$ as coefficient and LeFloch \cite{le1} gave an explicit formula for
the
solution $v(x,t)$ satisfying initial data $v(x,0)=v_0(x)$, when $v_0$ is
Lipschitz continuous. To justify the nonconservative product
which appear in the second equation Volpert product \cite{v1} was used
and the second equation was interpreted in the sense of measures.
In this paper we study \eqref{e1.1} in the quarter plane $\{(x,t)
: x >0, t>0 \}$, supplemented with an initial condition at $t=0$
\begin{equation}
\begin{pmatrix}
u(x,0)\\
v(x,0)
\end{pmatrix}
= \begin{pmatrix}
u_0(x)\\
v_0(x)
\end{pmatrix}
\label{e1.3}
\end{equation}
and a weak form of the Dirichlet boundary condition,
\begin{equation}
\begin{pmatrix}
u(0,t)\\
v(0,t)
\end{pmatrix}
= \begin{pmatrix}
u_b(t)\\
v_b(t)
\end{pmatrix}
\label{e1.4}
\end{equation}
where $u_0(x)$ is bounded measurable and $v_0(x)$ are Lipschitz continuous
functions of $x$ and
$u_b(t)$ and $v_b(t)$ are Lipschitz continuous functions of $t$. Indeed
with strong form of Dirichlet boundary conditions \eqref{e1.4}, there is
neither existence nor uniqueness as the speed of propagation
$\lambda =f'(u)$ depends on the unknown variable $u$ and does not have
a definite sign at the boundary $x=0$. We note that the speed is
completely determined by the first equation. We use the Bardos Leroux and
Nedelec
\cite{b1} formulation of the boundary condition for the $u$ component
which for our case is
equivalent to the following condition (see LeFloch \cite{le2}):
\begin{equation}
\begin{gathered}
\text{either } u(0+,t)= u_{b}^{+}(t)\\
\text{or } f'(u(0+,t))\leq 0 \text{ and }
f(u(0+,t))\geq f(u_{b}^{+}(t)).
\end{gathered}\label{e1.5}
\end{equation}
Here
$u_{b}^{+}(t)= \max\{u_b(t),\lambda\}$ where
$\lambda$ is the unique point where $f'(u)$ changes sign. Because
of convexity of $f$, $f(\lambda)=\inf{f(u)}$. There are explicit
representations of the entropy weak solution of of the
first component $u$ of \eqref{e1.1} with initial condition
$u(x,0)=u_0(x)$ and the
boundary condition \eqref{e1.5} by Joseph and Gowda \cite{j3} and LeFloch
\cite{le2}. We use the formula in \cite{j3} for $u$ which involve a
minimization of functionals on certain class of paths
and generalized characteristics. Once $u$ is
obtained, the equation for $v$ is linear equation with a discontinuous
coefficient $f'(u(x,t))$. Now $v(0+,t) =v_b(t)$ is prescribed only if
the characteristics at $(0,t)$ has positive speed, ie $f'(u(0+,t))>0$.
So the weak form of boundary conditions for $v$ component is
\begin{equation}
\text{if $f'(u(0+,t))>0$, then $v(0+,t)=v_b(t)$.}
\label{e1.6}
\end{equation}
The aim of this paper is to construct explicit formula for
\eqref{e1.1}, with initial condition \eqref{e1.3} and boundary
conditions
\eqref{e1.5} and \eqref{e1.6}. We also indicate some generalizations to
some other systems. The question of uniqueness is under investigation.
\section{A formula for the solution}
In this section,
using the explicit formula derived in \cite{j1,j3} for the scalar convex
conservation laws
with initial condition and Bardos Leroux and Nedelec boundary
condition \eqref{e1.6}, we construct a solution for the problem stated
in the introduction. To be more precise, We
assume $f(u)$ satisfies the following
conditions
\begin{equation}
f''(u)>0, \quad \lim_{u \to \infty}\frac{f(u)}{u} = \infty,
\label{e2.1}
\end{equation}
and let $f^{*}(u)$ be the convex dual of $f(u)$ namely,
$f^{*}(u)= \max_{\theta}[\theta u -f(\theta)]$.
For each fixed $(x,y,t), x> 0, y \geq 0, t>0$,
$C(x,y,t)$ denotes the following class of paths $\beta$ in
the quarter plane
$D=\{ (z,s) : z\geq 0, s \geq 0\}$. Each path is connected from the
initial point
$(y,0)$ to $(x,t)$ and is of the form $z=\beta(s)$, where $\beta$ is a
piecewise linear function of maximum three lines and always linear in
the interior of $D$. Thus for $x>0$ and $y>0$, the curves are
either a straight line or have exactly three straight lines with one
lying on the boundary $x=0$. For $y=0$ the curves are made up
of one straight line or two straight
lines with one piece lying on the
boundary $x=0$. Associated with the
flux $f(u)$ and boundary data $u_b(t)$, we define the functional
$J(\beta)$
on $C(x,y,t)$
\begin{equation}
J(\beta) = -\int_{\{s:\beta(s)=0\}}f(u_B(s)^{+})ds + \int_{\{s:\beta(s)
\neq 0\}}f^{*}\big(\frac{d\beta(s)}{ds}\big)ds.
\label{e2.2}
\end{equation}
We call $\beta_0$ is straight line path connecting $(y,0)$ and $(x,t)$
which does not touch the boundary $x=0$, $\{(0,t), t>0\}$, then let
\begin{equation}
A(x,y,t)= J(\beta_0) =t f*\big(\frac{x-y}{t}\big).
\label{e2.3}
\end{equation}
For any $\beta \in C^{*}(x,y,t) = C(x,y,t)-{\beta_0}$, that
is made up of three straight lines
connecting $(y,0)$ to $(0,t_1)$ in the interior and $(0,t_1)$ to
$(0,t_2)$ on the boundary and $(0,t_2)$ to $(x,t)$ in the interior,
it can be easily seen from \eqref{e2.2} that
\begin{equation}
J(\beta) = J(x,y,t,t_1,t_2) = -\int_{t_1}^{t_2}f(u_B(s)^{+})ds +
t_1 f^{*}(\frac{y}{-t_1})+(t-t_2) f^{*}\big(\frac{x}{t-t_2}\big).
\label{e2.4}
\end{equation}
For the curves made up two straight
lines with one piece lying on the
boundary $x=0$ which connects $(0,0)$ and $(0,t_2)$ and the other
connecting $(0,t_2)$ to $(x,t)$.
\[
J(\beta) = J(x,y,t,t_1=0,t_2) = -\int_{0}^{t_2}f(u_B(s)^{+})ds +
(t-t_2) f^{*}(\frac{x}{t-t_2}).
\]
It was proved in \cite{j1,j3}, that there exists a
$\beta^{*} \in C^{*}(x,y,t)$ or correspondingly $t_1(x,y,t)$,
$t_2(x,y,t)$ so that
\begin{equation}
\begin{aligned}
B(x,y,t)& =J(\beta^{*})\\
&=\min \{J(\beta) :\beta \in C^{*}(x,y,t)\}\\
&= \min \{J(x,y,t,t_1,t_2): 0\leq t_1 < t_2 < t\}\\
&= J(x,y,t,t_1(x,y,t),t_2(x,y,t))
\end{aligned}
\label{e2.5}
\end{equation}
is a Lipschitz continuous so that
\begin{equation}
\begin{aligned}
Q(x,y,t)&= \min\{J(\beta) : \beta \in C(x,y,t)\}\\
& = \min \{A(x,y,t),B(x,y,t)\},
\end{aligned}
\label{e2.6}
\end{equation}
and
\begin{equation}
U(x,t)= \min \{Q(x,y,t) + U_0(z), \,\, 0\leq y< \infty\}
\label{e2.7}
\end{equation}
are Lipschitz
continuous functions in their variables,
where $U_0(y)=\int_0^y u_0(z)dz$.
Further minimum in \eqref{e2.7} is attained at some value of $y\geq 0$
which
depends on $(x,t)$, we call it $y(x,t)$. If
$A(x,y(x,t),t)\leq B(x,y(x,t),t)$
\begin{equation}
U(x,t)=tf^{*}(\frac{x-y(x,t)}{t}) + U_0(y),
\label{e2.8}
\end{equation}
and if $A(x,y(x,t),t)>B(x,y(x,t),t)$
\begin{equation}
U(x,t)=J(x,y(x,t),t,t_1(x,y(x,t),t),t_2(x,y(x,t),t))
+ U_0(y).
\label{e2.9}
\end{equation}
Here and hence forth $y(x,t)$ is a
minimizer in \eqref{e2.7} and in the case of
\eqref{e2.9}, $t_2(x,t)=t_2(x,y(x,t),t)$ and $t_1(x,t)=t_1(x,y(x,t),t)$.
\begin{theorem}\label{thm2.1}
For every $(x,t)$ minimum in \eqref{e2.7}
is achieved by some $y(x,t)$, and $U(x,t)$ is a
Lipschitz continuous and for almost every $(x,t)$ there is only one
minimizer $y(x,t)$.
For every points $(x,t)$ satisfying
$U(x,t)=A(x,y(x,t),t)\leq B(x,y(x,t),t)$, define
\begin{equation}
\begin{gathered}
u(x,t)= (f^{*})'(\frac{x-y(x,t)}{t})\\
v(x,t)= v_0(y(x,t)).
\end{gathered}
\label{e2.10}
\end{equation}
and for the points $(x,t)$ where $B(x,y(x,t),t)0$, they are nondecreasing
function of
$x$ and hence except for a countable number of points they
are equal. Corresponding $t_2^{-}(x,t)$ and $t_2^{+}(x,t)$ have the
following properties. They are nondecreasing function of
$x$, for each fixed $t$ and except for a countable number of
points $x$ they are equal and nondecreasing function of
$t$, for each fixed $x$ and except for a countable number of
points $t$ they are equal.
Further if
$A(x,y(x,t),t)**B(x,y(x,t),t)$, for some
$x=x_0$ then this continues to be so for all $x>x_0$.
It was proved in \cite{j3}, that
$u(x,t)=Q_1(x,y(x,t))=\partial_{x}U(x,t)$ where
$Q_1(x,y,t)= \partial_x Q(x,y,t)$, is the weak solution of
\begin{equation}
u_t+f(u)_x=0
\label{e2.12}
\end{equation}
satisfying the initial condition $u(x,0)=u_0(x)$ and weak form of
boundary condition \eqref{e1.5}.
To show that $v$ satisfies the second equation, we follow LeFloch
\cite{le1} and use the
nonconservative product of Volpert \cite{v1} in sense of measures.
Since $u$ is a function of bounded variation, we write
\[
[0,\infty)\times[0,\infty) =S_c \cup S_j \cup S_n
\]
where $S_c$ and $S_j$ are points of approximate continuity of $u$ and
points of approximate jump of $u$ and $S_n$ is a set of one dimensional
Hausdorff-measure zero. At any point $(x,t) \in S_j$, $u(x-0,t)$ and
$u(x+0,t)$ denote the left and right values of $u(x,t)$. For any
continuous function $g :R^1 \to R^1$, the Volpert product
$g(u)v_x$ is defined as a Borel measure in the following manner.
Consider the averaged superposition of $g(u)$ (see Volpert \cite{v1})
\begin{equation}
\overline{g(u)}(x,t) = \begin{cases}
g(u(x,t)),&\text{if } (x,t) \in S_c,\\
\int_0^1 g(1-\alpha)(u(x-,t)+\alpha u(x+,t))d\alpha,
&\text{if }(x,t) \in S_j
\end{cases}
\label{e2.13}
\end{equation}
and the associated measure
\begin{equation}
[g(u)v_x](A)=\int_{A}\overline{g(u)}(x,t)v_x
\label{e2.14}
\end{equation}
where $A$ is a Borel measurable subset of $S_c$ and
\begin{equation}
[g(u)v_x](\{(x,t)\})=\overline{g(u)}(x,t)(v(x+0,t)-v(x-0,t))
\label{e2.15}
\end{equation}
provided $(x,t) \in S_j$.
The second equation in \eqref{e1.1} is understood as
\begin{equation}
\mu = v_t + \overline{f'(u)}(u)v_x=0
\label{e2.16}
\end{equation}
in the sense of measures.
Let $(x,t)\in S_c$ and $u={f^{*}}'(\frac{x-y(x,t)}{t})$, since $u$
satisfies \eqref{e2.12}, we have
\[
\begin{aligned}
f''(u)\{-\frac{(x-y(x,t))}{t^2}-\frac{\partial_{t}y(x,t)}{t}+
f'(u)\frac{(1-\partial_{x}y(x,t))}{t}\}=0.
\end{aligned}
\]
This can be written as
\begin{equation}
f''(u)\{-\frac{1}{t}[(\frac{(x-y(x,t))}{t}-f'(u))]-\frac{1}{t}[\partial_{t}y(x,t)+
f'(u)\partial_{x}y(x,t)]\}=0.
\label{e2.17}
\end{equation}
Using $f''(u)>0$ and $f'(u)$ and $(f^{*}{'})(u)$ are inverses of each
other, it follows from \eqref{e2.17} that
\begin{equation}
\partial_{t}y(x,t) + f'(u)\partial_{x}y(x,t)=0.
\label{e2.18}
\end{equation}
Now
\[
\partial_{t}v(x,t) + f'(u)\partial_{x}v(x,t)=(\frac{dv_0}{dx})(y(x,t)
\{\partial_{t}y(x,t) + f'(u)\partial_{x}y(x,t)\}
\]
and from \eqref{e2.18}, we get
\begin{equation}
\partial_{t}v(x,t) + f'(u)\partial_{x}v(x,t)=0. \label{e2.19}
\end{equation}
Similarly if
$(x,t)\in S_c$ and $u(x,t)={f^{*}}'(\frac{x}{t-t_2(x,y(x,t),t)})$,
we can show that
\[
\partial_t(t_2(x,y(x,t),t)) + f'(u(x,t))\partial_x(t_2(x,y(x,t),t) =0
\]
and hence
\begin{equation}
\partial_{t}v(x,t) + f'(u)\partial_{x}v(x,t)=0, \label{e2.20}
\end{equation}
So from \eqref{e2.19} and \eqref{e2.20}, for any Borel subset $A$
of $S_c$
\begin{equation}
\mu(A)=0.
\label{e2.21}
\end{equation}
Now we consider a point $(s(t),t) \in S_j$, then
\[
\frac{ds(t)}{dt} =
\frac{f(u(s(t)+,t))-f(u(s(t)-,t))}{u(s(t)+,t)-u(s(t)-,t)}
\]
is the speed of propagation of the discontinuity at this point.
\begin{equation}
\begin{aligned}
&\mu\{(s(t),t)\}\\
&=-\frac{ds(t)}{dt}(v(s(t)+,t)-v(s(t)-,t))\\
&\quad +\int_0^1 f'(u(s(t)-,t)
+\alpha (u(s(t)+,t) - u(s(t)-,t))d\alpha (v(s(t)+,t)-v(s(t)-,t))\\
&=[-\frac{ds(t)}{dt} +
\frac{f(u(s(t)+,t))-f(u(s(t)-,t))}{u(s(t)+,t)-u(s(t)-,t)}]
(v(s(t)+,t)-v(s(t)-,t))\\
&=0.
\end{aligned}
\label{e2.22}
\end{equation}
Form \eqref{e2.21} and \eqref{e2.22}, \eqref{e2.16} follows.
To show that the solution satisfies the initial conditions, first
we observe that given $\epsilon>0$ there exists $\delta>0$ such that for
all $x\geq\epsilon$, $t\leq \delta$,
\[
u(x,t)=(f^{*})'(\frac{x-y(x,t)}{t})
\]
where $y(x,t)$ minimizes
$\min_{y\geq 0}[U_0(y)+t f^{*}(\frac{x-y}{t})]$ see \cite{j3}. So $u$
and $v$ are given by the formula \eqref{e2.10}.
Then
Lax's argument \cite{la1}, gives $\lim_{t\to 0}u(x,t)=u_0(x)$
a.e.
$x\geq \epsilon$. Since $\epsilon>0$ is arbitrary,
\[
\lim_{t\to 0}u(x,t)=u_0(x),\,\,\, a.e.\,\,x.
\]
Since $f'$ and ${f^{*}}'$ are
inverses of each other $y(x,t)-x = -t f'(u(x,t))$, then it follows that
$y(x,t)\to x$ as $t\to 0$ a.e $x$. Since $v_0$ is
continuous we get
\[
\lim_{t\to 0}v(x,t)=\lim_{t\to
0}v_0(y(x,t))=v_0(x),\,\,\, a.e.\,\,x.
\]
Now we show the solution satisfies the boundary condition \eqref{e1.5}
and \eqref{e1.6}. That the $u$ component
satisfies the boundary condition \eqref{e1.5} is proved in \cite{j3}.
Further
if $f'(u(0+,t))>0$ then $f'(u(x,t))>0$ for $00$. So we have
\[
\lim_{x\to 0} v(x,t)=\lim_{x\to 0}v_b(t_2(x,t))=v_b(t).
\]
as $v_b$ is continuous. This proves $v$ satisfies the boundary condition
\eqref{e1.6}.
The proof of the theorem is complete.
\end{proof}
\section{Extensions to some other cases}
\subsection*{Generalized Lax equation}
The initial value problem for the system
\begin{equation}
\begin{gathered}
u_t +(\log(a e^u +b e^{-u}))_x =0\\
v_t + \frac{ae^u-be^{-u}}{ae^u+be^{-u}}v_x=0
\end{gathered}
\label{e3.1}
\end{equation}
was studied and explicit solution was constructed by Joseph and Gowda
\cite{j5} using a difference scheme of Lax \cite{la1}.
This system of equations is of the form \eqref{e1.1},with
\begin{equation}
f(u)=\log(ae^u + be^{-u})
\label{e3.2}
\end{equation}
For the case $f(u)$ satisfying \eqref{e2.1}, $f^{*}$ is defined
everywhere. The flux $f(u)$ given by \eqref{e3.2} is
convex but does not satisfies \eqref{e2.1} and $f^{*}$ is not defined
everywhere. Indeed $f^{*}$ is defined
only on $(-1,1)$ and is given by
\begin{equation}
f^{*}(u)=\frac{1}{2}\log\begin{pmatrix}(1+u)^{1+u}
(1-u)^{1-u}\end{pmatrix} -\frac{1}{2}\log\begin{pmatrix}4
a^{1+u} b^{1-u}\end{pmatrix}
\label{e3.3}
\end{equation}
and its derivative is
\begin{equation}
{f^{*}}'(u)=\frac{1}{2}\log(\frac{b}{a}\frac{1+u}{1-u}).
\label{e3.4}
\end{equation}
Explicit formula of the theorem (2.1) can be obtained for \eqref{e3.1}
on the domain $D = \{(x,t), x>0,t>0\}$
with initial condition \eqref{e1.3} and boundary conditions \eqref{e1.5}
and \eqref{e1.6} with minor modifications. Here we
define $C(x,y,t)$, the set of curves $\beta$ as in section 2,
but with a restriction on its slope $|\frac{d\beta(s)}{ds}|<1$. Using
the same notations as in theorem, and using the explicit form of
${f^{*}}'(u)$ given by \eqref{e3.4}, we have the following result.
\begin{theorem}\label{thm3.1}
For every $(x,t)$, $x>0$, $t>0$, let
$(u,v)$ be defined as follows:
When $A(x,y(x,t),t)\leq B(x,y(x,t),t)$, by
\[
u(x,t)=\frac{1}{2}\log[\frac{b}{a}\frac{t+x-y(x,t)}{t-x+y(x,t)}],\quad
v(x,t)=v_0(y(x,t));
\]
when $A(x,y(x,t),t)>B(x,y(x,t),t)$, by
\[
u(x,t)=\frac{1}{2}\log[\frac{b}{a}\frac{t+x+t_2(x,t)}{t-x+t_2(x,t)}],\quad
v(x,t)=v_b(t_2(x,t)).
\]
Then $(u,v)$ solves \eqref{e3.1}, satisfies the initial
conditions \eqref{e1.3} and the boundary conditions \eqref{e1.5} and
\eqref{e1.6}.
\end{theorem}
\subsection*{Generalized Hopf equation}
Solution for the initial-value problem for the nonconservative system
for $u_j, j=1,2,\dots, n$
\begin{equation}
(u_j)_t + (\sum_{k=1}^n c_k u_k)(u_j)_x =0, \quad j=1,2,\dots ,n
\label{e3.5}
\end{equation}
was constructed by Joseph \cite{j2,j4} by a vanishing viscosity method
and a generalization
of Hopf-Cole transformation. Here we assume that at least one
$k$, $c_k\neq 0$. When $n=1,c_1=1$, \eqref{e3.5} is the inviscid Burgers
equation or the Hopf equation and Hopf \cite{h1} derived a formula for
the entropy weak solution for the initial value problem and boundary case
was treated in \cite{j1}. In the present discussion we consider
\eqref{e3.5} in $D=\{(x,t): x>0, t>0\}$
with initial condition
\begin{equation}
u_j(x,0)=u_{0j}(x),\quad x>0,\quad j=1,2,\dots ,n
\label{e3.6}
\end{equation}
and boundary conditions
\begin{equation}
u_j(0,t)=u_{bj}(t),\quad t>0\quad j=1,2,\dots ,n.
\label{e3.7}
\end{equation}
Here again a weak form of the boundary condition is required as
characteristic speed of the system,
$\sigma =\sum_{k=1}^n c_k u_k$ need not be positive at the boundary
$x=0$. First we note
from \eqref{e3.5} that $u_j$ satisfies
\begin{equation}
(u_j)_t + \sigma (u_j)_x =0, \quad j=1,2,\dots ,n
\label{e3.8}
\end{equation}
where $\sigma$ satisfies
\begin{equation}
\sigma_t + (\frac{\sigma^2}{2})_x =0.
\label{e3.9}
\end{equation}
Now \eqref{e3.9} together with \eqref{e3.8} is exactly the
form of equation we have studied in section 1, with $f(u)={u^2/2}$.
Let $\sigma$ is the entropy weak solution of \eqref{e3.9}
with the initial condition
\begin{equation}
\sigma(x,0)=\sigma_0(x)
\label{e3.10}
\end{equation}
and weak form of boundary condition
\begin{equation}
\begin{gathered}
\text{either $\sigma(0+,t)= \sigma_{b}^{+}(t)$}\\
\text{or $\sigma(0+,t)\leq 0$ and $\frac{u(0+,t)}{2}\geq
\frac{u_{b}^{+}(t)}{2}$},
\end{gathered}
\label{e3.11}
\end{equation}
with $\sigma_0(x)=\sum_{k=1}^n c_k u_{0k}(x)$ and
$\sigma_b(t)=\sum_{k=1}^n c_k u_{bk}(t)$ constructed in \cite{j1,j3}.
The analysis of section 1 then shows that with the formulation of
boundary condition
\begin{equation}
\text{if $\sigma(0+,t)>0$, then $u_j(0+,t)=u_{bj}(t)$}.
\label{e3.12}
\end{equation}
for $u_j$, Theorem (1.1) applies to the present case with
$f(u)=\frac{u^2}{2}$. With the same notations as Theorem 1.1, we obtain
the following theorem.
\begin{theorem}\label{thm3.2}
For $x>0$, $t>0$, let $u_j$ be defined as follows:\\
For points $(x,t)$ where $U(x,t)=A(x,y(x,t),t)\leq B(x,y(x,t),t)$,
define
\[
u_j(x,t)= u_{0j}(y(x,t)),
\]
and for the points $(x,t)$ where $B(x,y(x,t),t)**