\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 158, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/158\hfil Black-Scholes equations]
{Solution to a nonlinear Black-Scholes equation}

\author[M. C. Mariani, E. K.  Ncheuguim, I. SenGupta \hfil EJDE-2011/158\hfilneg]
{Maria Cristina Mariani, Emmanuel K.  Ncheuguim, Indranil SenGupta}
% in alphabetical order

\address{Maria Cristina Mariani \newline
Department of Mathematical Sciences,
The University of Texas at El Paso,
Bell Hall 124, El Paso, TX 79968-0514, USA}
\email{mcmariani@utep.edu}

\address{Emmanuel Kengni Ncheuguim \newline
Department of Mathematical Sciences,
New Mexico State University, 
Las Cruces, NM 88003-8001, USA}
\email{emmanou@nmsu.edu}

\address{Indranil SenGupta \newline
Department of Mathematical Sciences,
The University of Texas at El Paso,
Bell Hall 316, El Paso, TX 79968-0514, USA}
\email{isengupta@utep.edu}

\thanks{Submitted August 20, 2010. Published November 28, 2011.}
\subjclass[2000]{91G80, 35B45, 58J35, 35D30}
\keywords{Option pricing; Black-Scholes equation; Sobolev space;
\hfill\break\indent Schaefer's fixed point theorem}

\begin{abstract}
 Option pricing with transaction costs leads to a nonlinear
 Black-Scholes type equation where the nonlinear term reflects
 the presence of transaction costs. Under suitable conditions,
 we prove the existence of weak solutions in a bounded domain
 and we extend the results to the whole domain using a diagonal
 process.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In a complete financial market without transaction costs,  the
celebrated Black-Scholes model (1973) \cite{BS}  provides not only
a rational option pricing formula, but also a hedging portfolio
that replicates the contingent claim. In the Black-Scholes
analysis, it is assumed that hedging takes place continuously, and
therefore, in a market with proportional transaction costs, it
tends to be infinitely expensive. So the requirement of
replicating the value of the option continuously has to be
relaxed. The first model in that direction was presented by Leland
(1985) \cite{LE}. He assumes that the portfolio is rebalanced at a
discrete time $\delta t$ fixed and that the transaction costs are
proportional to the value of the underlying; that is the costs
incurred at each step is $\kappa |\nu|S$, where $\nu$ is the
number of shares of the underlying bought ($\nu > 0$) or sold
($\nu <0)$ at price $S$ and $\kappa$ is a constant depending on
individual investors. Leland derived an option price formula that
is the Black-Scholes formula with an adjusted volatility
$$
\hat {\sigma}=\sigma \Big( 1+\sqrt{\frac{2}{\pi}}
\frac{\kappa}{\sigma \sqrt{\delta t}}\Big)^{1/2}.
$$
Hoggard, Whalley and Wilmott \cite{HWW} derived a model for
portfolios of options in the presence of transaction costs in
1994. We will outline the steps that they followed.

Let $C(S,t)$ be the value of the option and $\Pi$ be the value of
the hedge portfolio. Assume that the value of the underlying
follows the random walk
$$
\delta S=\mu S \delta t +\sigma S \phi\delta t^{1/2},
$$
where $\phi$ is drawn from a normal
distribution, $\mu$ is a measure of the average rate of growth of
the asset price also known as the drift, and $\sigma$ is a measure
of the fluctuation (risk) in the asset prices, it corresponds to
the diffusion coefficient. Then the change in the value of the
portfolio over the time step $\delta t$ is given by
\begin{equation*} %\label{problem3}
\delta \Pi =\sigma S \big( \frac{\partial C}{\partial S}-\Delta \big)
\phi \delta t^{1/2}+\Big(\frac12 \sigma^2 S^2
 \frac{\partial^2 C}{\partial  S^2}\phi^2
 + \mu S \frac{\partial C}{\partial S}
 +\frac{\partial C}{\partial t}-\mu \Delta S\Big) \delta t
-\kappa S |\nu|
\end{equation*}
 Let us consider the delta hedging strategy;  that is choose the
number of assets held short at time $t$ to be
$\Delta = \frac{\partial C}{\partial S}(S,t)$.
Therefore the number of assets to be traded after $\delta t$
is given by
$$
\nu= \frac{\partial C}{\partial S}(S+\delta S,t+\delta t)
-\frac{\partial C}{\partial S}(S,t) \simeq
\frac{\partial^2 C}{\partial S^2}\sigma S \phi \delta t^{1/2}.
$$
So the expected transaction cost over a time step is
$$
E[\kappa S |\nu|]=\sqrt{\frac{2}{\pi}} \kappa \sigma S^2
\big|\frac{\partial^2 C}{\partial S^2}\big| \delta t^{1/2},
$$
and the expected change in the value of the portfolio is
$$
E(\delta \Pi)=\Big(  \frac{\partial C}{\partial t}
+ \frac 12 \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}
-\kappa \sigma S^2 \sqrt{\frac{2}{\pi \delta t} }
\big| \frac{\partial^2 C}{\partial S^2}\big| \Big)\delta t.
$$
If the holder of the option expects to make as much from his
portfolio as from a bank account at a riskless  interest rate $r$
(no arbitrage), then
$$
E(\delta \Pi )= r\big( C-S \frac{\partial C}{\partial S}\big) \delta t.
$$
Hence the Hoggard, Whalley and Wilmott model for option pricing
with transaction costs is given by
\begin{equation}\label{refprob}
\frac{\partial C}{\partial t} +\frac 12 \sigma^2 S^2
\frac{\partial^2 C}{\partial^2 S}
+ r S \frac{\partial C}{\partial S}-r C-\kappa \sigma S^2
\sqrt{\frac{2}{\pi \delta t}}
\big| \frac{\partial^2 C}{\partial S^2}\big|=0,
\end{equation}
for $(S,t)  \in (0, \infty) \times(0, T)$,
and the final condition
\begin{equation*}%\label{IniCondRefProb}
C(S,T)= \max(S-E,0),  \quad  S \in(0, \infty)
\end{equation*}
for European call options with strike price $E$.

Note that equation \eqref{refprob} contents the usual
Black-Scholes terms with an additional nonlinear term modelling
the presence of transaction costs.
Setting
$$
x=\log(S/E), \quad t=T-\tau/\frac12 \sigma^2 ,\quad
C = E V(X,\tau),
$$
equation \eqref{refprob} becomes
\begin{equation} \label{eq2}
-\frac{\partial V}{\partial \tau}
+\frac{\partial^2 V}{\partial x^2}
+(k-1)\frac{\partial V}{\partial x}-kV
=\kappa^*\big| \frac{\partial^2 V}{\partial x^2}
-\frac{\partial V}{\partial x} \big|,
\end{equation}
for $(x,\tau) \in \mathbb{R} \times (0, T^*)$,
with the initial condition
$$
V(x,0)= \max (e^{x}-1,0), \quad  x \in \mathbb{R},
$$
where $k=r/(\sigma^2/2) $,
$\kappa^* = \kappa \sqrt{8/(\pi \sigma^2 \delta t)}$ and
$T^*=\sigma^2 T/2$.
Next set
$$
 V(x,\tau) = e^x U(x ,\tau).
$$
Then  \eqref{eq2} yields
\begin{equation}
-\frac{\partial U}{\partial \tau}
+\frac{\partial^2 U}{\partial x^2}
+(k+1)\frac{\partial U}{\partial x}=\kappa^*
\big| \frac{\partial^2 U}{\partial x^2}+\frac{\partial U}{\partial x}
\big|,  \quad  (x,\tau) \in \mathbb{R} \times (0, T^*)
\end{equation}
with the initial condition
$$
U(x,0)= \max(1-e^{-x}, 0) .
$$

The previous discussion motivates us to consider the following
problem that includes cost structures that go beyond proportional
transaction costs
\begin{equation}\label{TheoPb}
-\frac{\partial U}{\partial t}+\frac{\partial^2 U}{\partial x^2}
+\alpha \frac{\partial U}{\partial x}
=\beta F\big( \frac{\partial U}{\partial x} ,
\frac{\partial^2 U}{\partial x^2} \big),  \quad
 (x,t) \in \mathbb{R} \times (0,T)
\end{equation}
and
\begin {equation} \label{ICTheoPb}
U(x,0)=U_0(x) ,\quad  x \in \mathbb{R},
\end{equation}
where  $\alpha$ and $\beta$ are nonnegative constants.

Publications related to the above problem can be found in
\cite{M,MF,EMF}. It is also worth noting that
such problems can be solved using the techniques used
in \cite{ISG1,ISG2}. The goal of this paper is to show
that the theoretical problem \eqref{TheoPb}-\eqref{ICTheoPb}
has a strong solution where the derivatives are understood
in the distribution sense.
We use the following assumptions:
\begin{itemize}
\item[(H1)] $ F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}^+  $ is a continuous function,
\item[(H2)] $ F(p,q) \leq |p|+|q|$,
\item[(H3)] For $U\in H^2_{\rm loc}(\mathbb{R})$,
$\frac{\partial}{\partial x}
F(U, \frac{\partial U}{\partial x}) \in L^2(0,T;L^2_{\rm loc}(\mathbb{R}))$.
Let $B_R= \{ x\in \mathbb{R} : |x| <R\}$. Then if $w_k \to w$ in $L^2(0,T; H_0^1(B_R))$, then $\frac{\partial}{\partial x} F(w_k, \frac{\partial w_k}{\partial x}) \to \frac{\partial}{\partial x} F(w, \frac{\partial w}{\partial x})$ in $L^2(0,T; L^2(B_R))$.
\item[(H4)] $U_0 \in H^1_{\rm loc}(\mathbb{R})$,
\item[(H5)]  $\beta <  1$.
\end{itemize}

This work is organized as follows:  In section 2 we review some
notions of  functional analysis that will be used later, then in
section 3 we solve a similar problem in a ball and finally in
section 4  we construct a solution in the whole domain using a
diagonal process.


\section{Definitions and notion}

\subsection*{Function Spaces}
Let $\Omega \in \mathbb{R}^p$, $p\in \mathbb{N} $, be an open subset.

\begin{definition} \label{def2.1} \rm
Suppose $u$ and $v$ $\in L^1_{\rm loc}(\Omega)$, and $\alpha$ is a
multi-index constant.  $v$ is said to be the $\alpha^{th}$-weak
partial derivative of $u$, denoted $D^{\alpha} u= v$, if
$$
\int_{\Omega}u D^{\alpha} \phi \,dx
= (-1)^{|\alpha|} \int_{\Omega} v \phi \,dx
$$
for any test function $\phi \in C^{\infty}_c$.
\end{definition}

Note that a weak partial derivative of $u$, when it exists,
 is unique up to a set of measure zero.
The Sobolev space
$$
H^m({\Omega}) = \{ u \in L^2(\Omega) :
D^{\alpha}u \in L^2(\Omega),  \text{ for any multiindex
$\alpha$  with } |\alpha| \leq m \},
$$
where the derivatives are taken in the weak sense,
is a Hilbert space when endowed with the inner product
$$
(u,v)_{H^m(\Omega)}
=\sum_{|\alpha|\leq m} (D^{\alpha}u, D^{\alpha}v)_{L^2(\Omega)}.
$$
Let $H^1_0(\Omega)=\{ u  \in H^1
\text{ such that $u=0$ on }\partial \Omega \}$ be the closure of
$C^{\infty}_c$ in $H^1(\Omega)$. Let the space $H^{-1} (\Omega)$
is the topological  dual of $H^1_0(\Omega)$.

Let $X$ be a Banach space and let $T$ be a nonnegative integer.
The space $L^2(0,T;X)$  consists of all measurable functions
$u:(0,T)\to X$  with
$$
\|u\|_{L^2(0,T;X)}:= \Big(  \int_0^T \|u(t)\|^2_{X} dt\Big)^{1/2}
< \infty.
$$
Note that $L^2(0,T;X)$ is a Banach space endowed with the
norm $\|u\|_{L^2(0,T;X)}$.

The space $C([0,T]; X) $ consists of all continuous functions
$u:[0,T] \to X $ with
$$
\|u\|_{C([0,T]; X) } := \max_{0 \leq t \leq T} \|u(t)\|_{X} < \infty.
$$
Note that $C([0,T]; X)$ is a Banach space  endowed with the norm
$\|u\|_{C([0,T]; X)}$.


\subsection*{Schaefer's fixed point theorem}

Let $X$ be a real Banach space.

\begin{definition} \label{def2.3} \rm
A nonlinear mapping $A: X \to X$ is said to be compact if
for each bounded sequence $\{u_k\}_{k=1}^{\infty}$, the sequence
$\{A[u_k]\}_{k=1}^{\infty}$ is precompact; that is,
there exists a subsequence  $\{u_{k_j}\}_{j=1}^{\infty}$ such
that $\{A[u_{k_j}]\}_{j=1}^{\infty}$  converges in $X$.
\end{definition}

\begin{theorem}[Schaefer's fixed point Theorem]
Suppose  $A: X \to X$ is a continuous and compact mapping.
 Assume further that the set
\[
\{ u \in X \text{ such that $u= \lambda A[u]$
 for some }  0\leq \lambda \leq 1 \}\]
is bounded.
Then $A$ has a fixed point.
\end{theorem}

We will use the Schaefer's fixed point theorem in order to show the existence of a solution in a ball.


\section{Solutions in bounded domains}


Let $B_{R} =\{x \in \mathbb{R} : |x|< R\}$ be the open ball centered
at the origin with radius $R$. Assume that $U_0$ is suitable
cut into bounded functions defined on $B_R$  and such that
(H1)-(H5) are satisfied in $B_R \times [0,T]$.
Set $w= \frac{\partial U}{\partial x}$ and consider an analogous
problem in $B_R \times [0,T]$ with zero Dirichlet  condition on
the lateral boundary.
\begin{gather} \label{TheoPball}
-\frac{\partial w}{\partial t}+\frac{\partial^2 w}{\partial x^2}
+\alpha \frac{\partial w}{\partial x}
= \beta \frac{\partial}{\partial x}F
\big( w , \frac{\partial w}{\partial x} \big)   \quad
 (x,t) \in B_R \times (0, T)\\
\label{ICTheoPball}
w(x,0) = w_0(x) \quad  x \in B_R\\
\label{BCTheoPball}
w(x,t) = 0 ,  \quad ( x,t) \in  \partial B_R \times [0,T].
\end{gather}

\begin{definition} \label{def3.1} \rm
A function $w$ is said to be a weak solution of
\eqref{TheoPball}-\eqref{BCTheoPball} if
$w \in L^2(0,T;H^1_0(B_R))$,
$\frac{\partial w}{\partial t}\in L^2(0,T; H^{-1}(B_R))$
and
\begin{equation}\label{DefWeakSol}
\int_{B_R}\Big( \frac{\partial w}{\partial t} \phi
 + \frac{\partial w}{\partial x} \frac{\partial \phi}{\partial x}
+ \alpha  w  \frac{\partial \phi}{\partial x}\Big) \,dx
= -\beta \int_{B_R} F \big(w, \frac{\partial w}{\partial x}\big)
\frac{\partial \phi}{\partial x} \,dx
\end{equation}
for all $\phi \in H^1_0 (B_R)$.
\end{definition}

\begin{remark}[{\cite[Thm. 3, sec. 5.9.2]{Evans}}]
 \label{rmk3.2} \rm
If $w$ belongs to $L^2(0,T;H^1_0(B_R)) $ and
$\frac{\partial w}{\partial t}$ belongs to $L^2(0,T;H^{-1}(B_R))$,
then:
\begin{itemize}
\item[(i)]  $w \in C([0,T]; L^2(B_R))$;
\item[(ii)] the mapping $t \to \| w(t)\|^2_{L^2(B_R)}$ is
absolutely continuous with
\begin{equation}\label{eqn15}
\frac{d}{dt}\| w(t)\|^2_{L^2(B_R)}=2\int_{B_R}
\frac{\partial w}{\partial t} w dt \quad \text{a.e. } 0 \leq  t \leq T
\end{equation}
\end{itemize}
\end{remark}

\begin{theorem}[A priori estimate] \label{ape}
If $w$ is a weak solution of \eqref{TheoPball}-\eqref{BCTheoPball},
then there exists a positive constant $C$ independent of $w $
 such that
 \begin{equation*}
\max_{0\leq t \leq T} \|w(t)\|_{L^2(B_R)}
+ \| w\|_{ L^2(0,T; H^1_0(B_R))}
+ \| \frac{\partial w}{\partial t}\| _{L^2(0,T;H^{-1}(B_R))} \leq
 C \|w_0\|_{L^2(B_R)} .
\end{equation*}
\end{theorem}

\begin{proof}
Choosing $w(t) \in H^1_0(B_R) $ as the test function in
\eqref{DefWeakSol}, we obtain
$$
\int_{B_R}\Big( \frac{\partial w}{\partial t} w
+ \frac{\partial w}{\partial x} \frac{\partial w}{\partial x}
+ \alpha  w  \frac{\partial w}{\partial x}\Big) \,dx
= -\beta \int_{B_R} F \big(w, \frac{\partial w}{\partial x}\big)
\frac{\partial w}{\partial x} \,dx .
$$
Therefore,  by \eqref{eqn15},
$$
\frac 12 \frac{d}{dt}\| w(t)\|^2_{L^2(B_R)}
+\| \frac{\partial w}{\partial x} \|^2_{L^2(B_R)}
+ \frac 12  \alpha \int_{B_R}  \frac{\partial w^2}{\partial x}\,dx
=- \beta  \int_{B_R} F \big(w, \frac{\partial w}{\partial x}\big)
 \frac{\partial w}{\partial x} \,dx
 $$
and from \eqref{BCTheoPball},
$$
\frac 12 \frac{d}{dt}\| w(t)\|^2_{L^2(B_R)}
+\| \frac{\partial w}{\partial x}  \|^2_{L^2(B_R)}
\leq  \beta \int_{B_R} \big|F \big(w, \frac{\partial w}{\partial x}\big)
 \frac{\partial w}{\partial x}\big| \,dx.
 $$
Using (H2), we obtain
$$
\frac 12\frac{d}{dt}\| w(t)\|^2_{L^2(B_R)}
+\| \frac{\partial w}{\partial x}  \|^2_{L^2(B_R)}
\leq  \beta \int_{B_R} \Big( |w| \big| \frac{\partial w}{\partial x}\big|
+ \big| \frac{\partial w}{\partial x}\big|^2 \Big)  \,dx.
 $$
By the Cauchy-Schwartz  inequality with $\epsilon >0 $,
\begin{equation}
\begin{split}
&\frac 12 \frac{d}{dt}\| w(t)\|^2_{L^2(B_R)}
 +\| \frac{\partial w}{\partial x}  \|^2_{L^2(B_R)}\\
& \leq  \beta \Big(  \int_{B_R} | \frac{\partial w}{\partial x}|^2 \,dx
 +\epsilon \int_{B_R} | \frac{\partial w}{\partial x}|^2 \,dx
+\frac{1}{4\epsilon}  \int_{B_R} |w|^2 \,dx\Big).
\end{split}
\end{equation}
Since $\beta < 1$, choosing $\epsilon \ll 1$, yields
\begin{equation} \label{eq17}
 \frac{d}{dt}\| w(t)\|^2_{L^2(B_R)} +C_1 \|  w \|^2_{H^1_0(B_R)}
\leq C_2 \| w  \|^2_{L^2(B_R)}
\end{equation}
for a.e. $ 0\leq t\leq T$, and appropriate positive constants
$C_1$ and $C_2$.

Next we write
$ \eta (t):= \| w(t)\|^2_{L^2(B_R)}$ , then by \eqref{eq17},
$$
\eta' (t) \leq  C_2 \eta (t), \text{ for a.e.}  0\leq t\leq T.
$$
The differential form of Gronwall inequality implies
$$
\eta(t)\leq e^{C_2t}\eta(0) \quad \text{a.e.}  0\leq t\leq T.
$$
Since $\eta(0)= \| w(0)\|^2_{L^2(B_R)} =\| w_0\|^2_{L^2(B_R)}$,
$$
\| w(t)\|^2_{L^2(B_R)} \leq e^{C_2t}\| w_0\|^2_{L^2(B_R)}.
$$
Hence
\begin{equation}\label{eq18}
\max_{0\leq t \leq T} \|w(t)\|_{L^2(B_R)}
\leq  C_{11} \| w_0\|^2_{L^2(B_R)},
\end{equation}
where $C_{11}$ is some constant.
To obtain a  bound for the second term, we consider \eqref{eq17},
and integrate from $0$ to $T$, obtaining
$$
\|w(T)\|^2_{L^2(B_R)} - \|w_0\|^2_{L^2(B_R)}
+  C_1 \int^T_0 \|  w  \|^2_{H^1_0(B_R)} dt
\leq C_2 \int_0^T\| w  \|^2_{L^2(B_R)} dt.
$$
Therefore,
$$
\|w(T)\|^2_{L^2(B_R)} +  C_1 \int^T_0 \|  w  \|^2_{H^1_0(B_R)} dt
\leq C_2 \int_0^T\| w  \|^2_{L^2(B_R)} dt + \|w_0\|^2_{L^2(B_R)}.
$$
Hence
$$
C_1 \int^T_0 \|  w  \|^2_{H^1_0(B_R)} dt
\leq C_2 \int_0^T\| w  \|^2_{L^2(B_R)} dt + \|w_0\|^2_{L^2(B_R)}.
$$
Using  \eqref{eq18} we conclude that
\begin{equation}
\label{eq19}
\|w\|_{L^2 (0,T;H^1_0(B_R))}\leq  C_{22} \|w_0\|_{L^2(B_R)},
\end{equation}
where $C_{22}$ is a constant.
Finally, to obtain a bound for the third term, by  fixing
 $v \in H^1_0(B_R)$ with $\| v\|_{H^1_0(B_R)} \leq 1$.
By \eqref{DefWeakSol}, we have
$$ \int_{B_R}\frac{\partial w}{\partial t} v \,dx
+ \int_{B_R} \Big(\frac{\partial w}{\partial x}
\frac{\partial v}{\partial x}  + \alpha  w  \frac{\partial v}{\partial x}
\Big) \,dx
= -\beta \int_{B_R} F \big(w, \frac{\partial w}{\partial x}\big)
\frac{\partial v}{\partial x} \,dx.
$$
Thus
$$
\big|  \int_{B_R} \frac{\partial w}{\partial t} v \,dx \big|
\leq \big| \int_{B_R} \Big(\frac{\partial w}{\partial x}
\frac{\partial v}{\partial x}  + \alpha  w  \frac{\partial v}{\partial x}
\Big) \,dx \big|
+ \beta \big| \int_{B_R} F \big(w, \frac{\partial w}{\partial x}\big)
 \frac{\partial v}{\partial x} \,dx.  \big|
$$
By H\"older inequality,
\begin{align*}
  \big|  \int_{B_R} \frac{\partial w}{\partial t} v \,dx \big|
&\leq \Big(  \int _{B_R} \big| \frac{\partial w }{\partial x} \big|^2
 \,dx \Big)^{1/2}
  \Big(  \int _{B_R} \big| \frac{\partial v }{\partial x}\big|^2 \,dx
  \Big)^{1/2}\\
&\quad + \alpha \Big(  \int _{B_R} |w|^2 \,dx \Big)^{1/2}
 \Big( \int _{B_R} \big| \frac{\partial v }{\partial x} \big|^2 \,dx
 \Big)^{1/2}\\
&\quad + \Big(  \int _{B_R} \big| F\big(w,
\frac{\partial w }{\partial x} \big) \big|^2 \,dx \Big)^{1/2}
\Big(  \int _{B_R} \big| \frac{\partial v }{\partial x} \big|^2 \,dx
\Big)^{1/2}.
\end{align*}
Since  $\| v\|_{H^1_0(B_R)} \leq 1$, using (H2) and the Poincar\'e
inequality we deduce
$$
\big| \int_{B_R} \frac{\partial w}{\partial t} v \,dx \big|
 \leq C_{33} \| w(t)\|_{H^1_0(B_R)},
$$
where $C_{33}$ is some constant.
So
$$
\| \frac{\partial w}{\partial t} (t)  \|_{H^{-1}(B_R)}
\leq  C_{33} \| w(t)\|_{H^1_0(B_R)}.
 $$
Therefore,
\[
\int_0^T \| \frac{\partial w}{\partial t} (t)  \|^2_{H^{-1}(B_R)}dt
\leq  C_{33} \int_0^T  \| w(t)\|^2_{H^1_0(B_R)} dt
 = C_{33} \| w\|^2_{L^2(0,T;H^1_0(B_R))}.
\]
Then \eqref{eq19} implies
\begin{equation}
\label{eq20}
\| \frac{\partial w}{\partial t} \|_{L^2(0,T;H^{-1}(B_R))}
\leq C_{33}  \| w_0\|^2_{L^2(B_R)}.
\end{equation}
Take $C= \max \{C_{11}, C_{22}, C_{33} \}$ to obtain the required
result in the Theorem.
\end{proof}

Before proving the existence theorem in a ball,  we state the
following energy estimate from the theory of linear parabolic partial
differential equations.

\begin{lemma}[{\cite[Theorem 2 page 354]{Evans}}]  \label{LinProb}
Consider the problem
\begin{equation}\label{Problin}
\begin{gathered}
\frac{\partial w}{\partial t} -\frac{\partial^2 w}{\partial x^2}
-\alpha \frac{\partial w}{\partial x} = f(x,t) \quad \text{in }
  B_R\times (0,T)\\
w(x,0) = w_0(x) \quad\text{on }  B_R\times\{ 0\}\\
w(x,t) = 0 \quad\text{on }  \partial B_R\times [0,T]
\end{gathered}
\end{equation}
with $f\in L^2(0,T;L^2(B_R))$ and $w_0 \in L^2(B_R)$.

Then there exists a unique $u \in
L^2(0,T; H^1_0(B_R)) \cap C([0,T];L^2(B_R)) $
 solution of \eqref{Problin} that satisfies
\begin{equation} \label{EnerEst}
\begin{split}
&\max_{0\leq t \leq T} \|u(t)\|_{L^2(B_R)}
 + \| u \|_{ L^2(0,T; H^1_0(B_R))}
 +\| \frac{\partial u}{\partial t}\| _{L^2(0,T;H^{-1}(B_R))} \\
& \leq  C\Big(\|f\|_{L^2(0,T;L^2(B_R))}+ \|w_0\|_{L^2(B_R)} \Big),
\end{split}
\end{equation}
where $C$ is a positive constant depending only on $R$ and $T$.
\end{lemma}

We need another Lemma that follows directly from
\cite[Theorem 5, page 360]{Evans}.

\begin{lemma}[Improved regularity] \label{LinProb2}
Consider the problem
\begin{gather*}
\frac{\partial w}{\partial t} -\frac{\partial^2 w}{\partial x^2}
-\alpha \frac{\partial w}{\partial x}
= f(x,t)\quad \text{in }  B_R\times (0,T)\\
w(x,0) = w_0(x) \quad \text{on }  B_R\times\{ 0\}\\
w(x,t) = 0 \quad\text{on }  \partial B_R\times [0,T]
\end{gather*}
with $f\in L^2(0,T;L^2(B_R))$ and $w_0 \in H_0^1(B_R)$.
Then this problem has a unique weak solution
$u \in L^2(0,T; H^1_0(B_R)) \cap C([0,T];L^2(B_R))$, with
$\frac{\partial u}{\partial t} \in L^2(0,T; H^{-1}(B_R))$.
Moreover,
$$
u \in L^2(0,T; H^2(B_R)) \cap L^{\infty}(0,T; H_0^1(B_R)), \quad
\frac{\partial u}{\partial t} \in L^2(0,T; L^2(B_R)).
$$
We also have the estimate
\begin{equation} \label{EnerEst1}
\begin{split}
& \operatorname{ess\,sup}_{0\leq t \leq T} \|u(t)\|_{H_0^1(B_R)}
 + \| u \|_{ L^2(0,T; H^2(B_R))}
 +\| \frac{\partial u}{\partial t}\| _{L^2(0,T; L^{2}(B_R))}  \\
&\leq  C''\Big( \|f\|_{L^2(0,T;L^2(B_R))} + \|w_0\|_{H_0^1(B_R)}\Big),
\end{split}
\end{equation}
where $C''$ is a positive constant depending only on $B_R$, $T$
and the coefficients of the operator $L$.
\end{lemma}

Next we show the existence of a solution in a ball by using
the Schaefer's fixed point Theorem.

\begin{theorem} \label{ExistenceBall}
If {\rm (H1)--(H5)} are satisfied, then
\eqref{TheoPball}-\eqref{BCTheoPball} has a weak solution
$w \in L^2(0,T; H^1_0(B_R)) \cap C([0,T];L^2(B_R))$.
\end{theorem}

\begin{proof}
Given $w \in L^2(0,T; H^1_0(B_R))$, set
$f_w(x,t):=  \beta \frac{\partial}{\partial x} F
\big(w, \frac{\partial w}{\partial x}\big) $. By
(H3),  $f_w \in L^2(0,T;L^2(B_R))$.
From Lemma \ref{LinProb} there exists a unique
$v \in   L^2(0,T; H^1_0(B_R)) \cap C([0,T];L^2(B_R))$  solution of
\begin{equation}
\label{eq21}
\begin{gathered}
\frac{\partial v}{\partial t} -\frac{\partial^2 v}{\partial x^2}
-\alpha \frac{\partial v}{\partial x}
= f(x,t)\quad \text{in }  B_R\times (0,T)\\
v(x,0) = v_0(x) \quad\text{on }  B_R\times\{ 0\}\\
v(x,t) = 0 \quad\text{on }  \partial B_R\times [0,T]
\end{gathered}
\end{equation}
Define the mapping
$$
A: L^2(0,T;H^1_0(B_R)) \to L^2(0,T;H^1_0(B_R))
$$
by $w \mapsto A(w) = v$,
where $v$ is derived from $w$  via \eqref{eq21}.

We now show that the mapping $A$ is continuous and compact.
We first prove the continuity.
Let $\{ w_k\}_k  \subset L^2(0,T;H^1_0(B_R))$ be a sequence such that
\begin{equation}
\label{eq22}
w_k \to w \quad \text{in } L^2(0,T;H^1_0(B_R)).
\end{equation}
By the \emph{improved regularity} \eqref{EnerEst1},
there exists a constant $C''$, independent of $\{ w_k\}_k$ such that
\begin{equation}
\|v_k\| _{L^2(0,T;H^2(B_R))}
\leq C''\Big( \| f_{w_k}\|_{L^2(0,T;L^2(B_R))}
+ \|w_0\|_{H_0^1(B_R)}\Big),
\label{indra10}
\end{equation}
for $v_k=A[w_k]$, $k=1,2,\dots$.
By (H3) as $w_k \to w$ in $L^2(0,T;H^1_0(B_R))$, we must have
$f_{w_k}(x,t) \to f_w(x,t)$ in ${L^2(0,T;L^2(B_R))}$.
Therefore $\|f_{w_k}(x,t)\|_{L^2(0,T;L^2(B_R))} \to
\|f_w(x,t)\|_{L^2(0,T;L^2(B_R))}$.
Then the sequence
$\{ \|f_{w_k}\|_{L^2(0,T;L^2(B_R))} \}_k $ is bounded and
\begin{equation}
\sup_{k} \|f_{w_k}\|_{L^2(0,T;L^2(B_R))} \leq C''',
\label{indra11}
\end{equation}
for a constant $C'''$.
Thus by \eqref{indra10} and \eqref{indra11} the sequence
$\{v_k\}_k$ is bounded uniformly in $L^2(0,T;H^2(B_R))$.
Similarly it can be proved that
$\{\frac{\partial v_k}{\partial t}\}_k$ is uniformly bounded
in $L^2(0,T;H^{-1}(B_R))$. Thus by Rellich's Theorem
(see \cite{Folland2}) there exists a subsequence
$\{v_{k_j}\}_j \in L^2(0,T;H_0^1(B_R))$ and a function
$v\in L^2(0,T;H^1_0(B_R))$ with
\begin{equation}\label{eq23}
v_{k_j} \to v \quad \text{in }  L^2(0,T;H^1_0(B_R)), \text{ as  }
j \to \infty.
\end{equation}
Therefore,
$$
\int_{B_R}\Big( \frac{\partial v_{k_j}}{\partial t} \phi
+ \frac{\partial v_{k_j}}{\partial x} \frac{\partial \phi}{\partial x}
+ \alpha v_{k_j} \frac{\partial \phi}{\partial x}\Big) \,dx
= \int_{B_R} f_{w_{k_j}}(x,t)\phi  \,dx
$$
for each $\phi \in H^1_0(B_R)$.
Using \eqref{eq22} and \eqref{eq23} we see that
$$
\int_{B_R}\Big( \frac{\partial v}{\partial t} \phi
+ \frac{\partial v}{\partial x} \frac{\partial \phi}{\partial x}
+ \alpha v \frac{\partial \phi}{\partial x}\Big) \,dx
= \int_{B_R} f_w(x,t)\phi  \,dx.
$$
Thus $v=A[w]$. Therefore
$$
A[w_k] \to A[w] \text{ in }  L^2(0,T;H^1_0(B_R)).
$$
The compactness result follows from similar arguments.

To apply Schaefer's fixed point Theorem in $ L^2(0,T;H^1_0(B_R))$
we need to show that the set
$ \{ w \in  L^2(0,T;H^1_0(B_R)) : w=\lambda A[w]  \text{ for some } 0 \leq \lambda \leq 1 \}$ is bounded.
This follows directly from the a priori estimate
(Theorem \ref{ape}) with $\lambda=1$.
\end{proof}

\begin{remark} \label{rmk3.7} \rm
Theorem \ref{ExistenceBall} shows that
$w=\frac{\partial u}{\partial x} \in  L^2(0,T;H^1_0(B_R)) $
 solves problem \eqref{TheoPball}-\eqref{BCTheoPball}; so
$u \in L^2(0,T;H^1_0(B_R) \cap H^2(B_R)) $ and is a strong
solution of problem \eqref{TheoPb}-\eqref{ICTheoPb} in the
bounded domain $B_R\times [0,T]$ with zero Dirichlet condition
on the lateral boundary of the domain.
\end{remark}

\section{Construction of the solution in the whole domain}

The next step is to construct a solution of
 \eqref{TheoPball}-\eqref{BCTheoPball} in the whole real line.
To do that, we approximate the real line by
 $$
\mathbb{R}= \cup_{N\in \mathbb{N}}B_N= \lim_{N \to \infty}B_N
$$
where $B_N=\{ x \in \mathbb{R} : |x| < N\}$.
We also approximate $w_0$ by a sequence of bounded function
$w_{0N}$ defined in $B_N$ such that $|w_{0N}| \leq |w_0| $
and $w_{0N} \to  w_0$ in $L^2_{\rm loc}(\mathbb{R})$.
For $N \in \mathbb{N}$, there exists
$w_N \in  L^2(0,T;H^1_0(B_N)) \cap C([0,T];L^2(B_N)) $  with
$\frac{\partial w_N }{\partial t}  \in L^2(0,T;H^{-1}(B_N))$,
weak solution of
\begin{gather}
-\frac{\partial w}{\partial t}+\frac{\partial^2 w}{\partial x^2}
+\alpha \frac{\partial w}{\partial x}
= \beta \frac{\partial}{\partial x}F
\big( w , \frac{\partial w}{\partial x} \big)   \quad
(x,t) \in B_N \times (0, T)\\
\label{ICTheoPballN}
w(x,0) = w_{0N}(x) \quad  x \in B_N\\
\label{BCTheoPballN}
w(x,t) = 0 ,  \quad(x,t) \in  \partial B_N \times [0,T].
\end{gather}
For any given $\rho >0$, the following sequences are bounded
uniformly for $N>2 \rho$:
\begin{gather*}
\{ w_N\}_N \quad \text{ in } L^2(0,T;H^1_0(B_{\rho})),\\
\{\frac{\partial  w_N}{\partial t}\}_N \quad
\text{ in } L^2(0,T;H^{-1} (B_{\rho}))
\end{gather*}
Since these spaces are compactly embedded in
$L^2(( B_{\rho} \times (0,T))$
then  the sequence $\{ w_N\}_N$  is relatively compact in
$L^2(( B_{\rho} \times (0,T))$.
By a standard diagonal process we may select a subsequence
also denoted  $\{w_N\}_N$ so that
\begin{gather*}
w_N \to w \quad \text{a.e.  in } L^2(0,T;L^2_{\rm loc}(\mathbb{R})),\\
w_N \to w \quad \text{weakly  in } L^2(0,T;H^1_{\rm loc}(\mathbb{R})).
\end{gather*}
Since $F$ is continuous, passing to the limit in \eqref{DefWeakSol}
yields that $w$ is a weak solution of
 \eqref{TheoPball}-\eqref{BCTheoPball} in $\mathbb{R}$.

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\end{document}