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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{\hfil H\"{O}LDER SOLUTIONS FOR THE AMORPHOUS SILICON SYSTEM
\hfil\folio}
\def\leftheadline{\folio\hfil Walter Allegretto, Yanping Lin, \& Aihui Zhou\hfil}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Differential Equations and Computational Simulations III\hfill\break
J. Graef, R. Shivaji, B. Soni, \& J. Zhu (Editors)\hfill\break
Electronic Journal of Differential Equations, Conference~01, 1997, pp. 1--9.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp 147.26.103.110 or 129.120.3.113 (login: ftp)\bigskip} }
\topmatter
\title
H\"{O}LDER SOLUTIONS FOR THE AMORPHOUS SILICON SYSTEM AND RELATED PROBLEMS
\endtitle
\thanks
{\it 1991 Mathematics Subject Classifications:} 35J60.\hfil\break\indent
{\it Key words and phrases:} Reaction, diffusion, semiconductor, Holder continuoussolutions.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and
University of North Texas. \hfil\break\indent
Published November 12, 1998.
\endthanks
\author Walter Allegretto, Yanping Lin, \& Aihui Zhou \endauthor
\address
Walter Allegretto \hfill\break
Department of Mathematical Sciences,
University of Alberta,
Edmonton, Alberta,
Canada T6G 2G1 \endaddress
\email
retl\@retl.math.ualberta.ca\endemail
\address Yanping Lin \hfill\break
Department of Mathematical Sciences,
University of Alberta,
Edmonton, Alberta,
Canada T6G 2G1\endaddress
\email ylin\@hilbert.math.ualberta.ca\endemail
\address Aihui Zhou\hfill\break Institute of Systems Science,
Academia Sinica,
Beijing 100080,
China\endaddress
\email azhou\@bamboo.iss.ac.cn\endemail
\abstract
We present existence of solutions and other results for the partial
differential equation system with memory which models
amorphous silicon devices and related problems in ${\Bbb R}^3$.
Our approach employs only classical estimates and
Degree Theory; it shows the existence of
$C^{\alpha,\alpha/2}$ solutions for some $\alpha>0$. In view
of the mixed boundary conditions, this is the maximum
regularity that can be expected.
\endabstract
\endtopmatter
\document
\def\pd#1#2{\frac{\partial#1}{\partial#2}}
\def\wone{w^{(1)}}
\def\zone{z^{(1)}}
\head 1. Introduction \endhead
In the past few years micro-electronic devices employing amorphous
silicon as the semiconductor material have shown promise in a
variety of applications such as liquid crystal displays, image
sensors and solar cells. The mathematical model usually
employed to simulate such devices involves drift-diffusion equations
as well as equations describing the density of trapped charges,
[3, 8]. The latter may be explicitly integrated in time, giving a
drift-diffusion system with integral (i.e. ``memory'') terms.
Specifically, if we assume only one trapped charge state and set all
mathematically irrelevant coefficients to unity, we obtain:
$$ \align
-\Delta \varphi &=\Big(p-n+C_1(x)+\int^t_0 (p-n)e^{-\int^t_\xi
(n+p+2)d\eta }d\xi \Big) \tag 1' \\
\pd nt &-\nabla [D_n(x,t,n,p,| \nabla\varphi | )\nabla n
- n\mu _n(x,n,p,| \nabla\varphi | )\nabla\varphi ] \tag 2'\\
&= 1 -\int^t_0 (p-n)e^{-\int^t_\xi (n+p+2)d\eta }d\xi
- n\Big[1+\int^t_0 (p-n)e^{-\int^t_\xi (n+p+2)d\eta }d\xi \Big] \\
\pd pt &- \nabla [D_p(x,n,p,| \nabla \varphi | )\nabla p
+ p\mu _p(x,n,p,| \nabla \varphi | )\nabla \varphi ] \tag 3' \\
&= 1+ \int^t_0 (p-n) e^{-\int^t_\xi (n+p+2)d\eta }d\xi
- p\Big[1-\int^t_0 (p-n)e^{-\int^t_\xi (n+p+2)d\eta }d\xi \Big]
\endalign
$$
to be satisfied in a smooth domain $\Omega \subset R^3$. We
observe that the factor \;``2''\; is present in the various
integrals in equations (1')-(3') to ensure charge conservation,
[8]. With equations (1')--(3') we associate initial/mixed
boundary conditions as follows: Let $\partial \Omega =\Gamma
_D\cup \Gamma _N$ with $\Gamma _D$ a smooth nonempty closed
sub-manifold in which Dirichlet conditions are to hold:
$$
\varphi (x,t) =\overline \varphi (x), \quad n(x,t) =\overline n(x), \quad p(x,t)
=\overline p(x)
$$
for all $t$, while Neumann conditions are to hold on $
\Gamma _N =\partial \Omega -\Gamma _D:$
$$
\pd {\varphi}{ \vec\nu } = \pd{n}{\vec\nu } =\pd{p}{\vec \nu }=0\,.
$$
With $n,p$ we also associate initial conditions. Both to avoid
technical difficulties and in keeping with the situation in the
physical problems we also ask that $n(x,0) =\overline n(x)$,
$p(x,0) =\overline p(x)$,
with $\overline n(x),\;\overline p(x)\in C^1(\overline \Omega)$ and
$\overline n,\,\overline p\ge 0$.
These may be weakened in what follows without essential proof
changes, but they do simplify the presentation.
For the same reason, we assume throughout the paper that all
equation coefficients are smooth in
their variables. Note that since $n,p$ are densities we shall only
seek solutions with $n,p\ge0$. The behavior of the equation
coefficients in $(n,p)$ for $n,p<0$ can thus be chosen for
convenience. Some regularity is also needed for $\partial (\Gamma
_N)\cap \Gamma _D$. Intuitively if $x\in\partial (\Gamma
_N)\cap\Gamma _D$ and $\Cal N$ is a small neighborhood of $x$, we
require that regularity considerations for $\Cal N\cap \partial
\Omega $ be reduced via bi-Lipschitz (resp. smooth)
coordinate maps to similar
problems on quarter-spheres (resp. hemispheres).
The reader interested in the explicit
formulations of such conditions may find them for example in
[7, 11, 13, 16].
In the next sections we introduce and analyze a system of equations
which contains as special cases not only (1')--(3') but
also the standard drift-diffusion equations. While we are not aware
of an earlier study of such a system with ``memory terms'', we point
out there have been numerous results on the non-augmented system in recent
years. It is not possible to present a detailed analysis of
previous results here, but we refer the interested reader in
particular to the papers [2, 4, 5], to the books
[9, 10, 12] and the
references therein. It is the paper by Fang and Ito [2], and
da~Veiga, [15] which
furnished
in part the motivation for this work, and
which are closest to the assumptions made.
Indeed the regularity of $n,p$ shown here is conjectured in [15].
In general terms, the
approaches usually employed in the past
are based on time discretization, on semigroup analysis,
on fixed point theorems and weak solutions are found in suitable
spaces. Often techniques involving maximum principles, the Einstein
relations and the introduction of quasi Fermi variables, and
coefficient truncation were employed. In this paper we use none of
these tools, and it is not clear how useful many of these
would be in our situation, given the memory term present in $(1').$
Instead we employ Degree Theory and work directly with
$C^{\alpha ,\alpha /2}$ spaces. Not only does this simplify
considerably the presentation but the solutions we find are of the
regularity one would expect from the physical point of view. More
global regularity cannot be realized in general due to the mixed boundary
conditions for $n,p,\varphi $.
Our procedures are based on simple
arguments involving classical results ([6, 17]) which are
well-known although in themselves far from simple. As presented,
the results are given for $\Omega \subset R^3$ -- the physically
interesting case. We conjecture that similar results hold for
$\Omega \subset R^N,\; N\ne 3$.
As a final observation, note that as mentioned above, we do not make
use of the Einstein Relations connecting $D_n,\,D_p$ and $\mu
_n,\,\mu _p$ as we shall have no need of them. It follows that the
results also hold if the system does not admit `` quasi-Fermi " variables.
\head{II. Analysis}\endhead
Based on the model system considered in the Introduction, we
introduce the following equations:
$$
\gather
-\Delta \varphi = f\big(x,t,p-n,h(p,n)\big) \tag 1\\
\pd{n}{t} -\nabla [D_n(x,t,n,p,| \nabla\varphi | )\nabla
n - n\mu _n\big(x,t,n,p,| \nabla \varphi | \big)
\nabla\varphi ]\\ = 1-h(p,n) - n[1+h(p,n)] + R_n(x,t,n,p) \tag 2\\
\pd pt - \nabla \big[D_p(x,t,n,p,| \nabla\varphi | )\nabla p
+ p\mu _p \big(x,t,n,p,| \nabla \varphi |\big)
\nabla\varphi \big] \\ = 1+h(p,n) -p[1-h(p,n)] + R_p(x,t,n,p) \tag 3
\endgather
$$
with
$$
h(p,n) =\int^t_0 (p-n) e^{-\int^t_\xi (n+p+2)d\eta } d\xi \,.
$$
We keep the initial/boundary conditions given on $(n,p,\varphi )$
in the introduction as well as the requirement that $D_n,D_p,\mu
_n,\mu _p,R_n,R_p$ be smooth functions of their respective
arguments (at least for $n,\,p\ge 0)$. We now introduce the
following growth conditions on $\Omega \times (0,T)$, which may
depend on $T$.
\item {(A)} There exist positive constants $\alpha ,\beta $ such
that
$$
\alpha \le D_n(x,t,n,p,| \nabla\varphi | ), \quad
D_p(x,t,n,p,| \nabla\varphi | )\le \beta
$$
\item {(B)} $R_n(x,t,n,p) =
R_{n,1}(x,t,n,p) - n R_{n,2}(x,t,n,p)$
with $R_{n,1},\;R_{n,2}$ nonnegative,
bounded, smooth if $n,p\ge 0$. We
assume that $R_p$ admits a similar decomposition into
$R_{p,1}- p R_{p,2}$ with and $R_{p,1},\; R_{p,2}$
nonnegative, bounded, smooth.
\item {(C)} $\mu _n = \mu _{n,1} +\mu _{n,2}, $ with $\mu
_{n,1} $ a positive constant and $\mu _{n,2} =\mu
_{n,2}\big(x,t,n,p,| \nabla \varphi | \big)$ such
that $| \mu _{n,2}\nabla\varphi | \le a_n$ for some
positive constants $a_n$ if $\big(x,t\big)$ are bounded.
Similarly, $\mu _p =\mu _{p,1} +\mu _{p,2}$ with $0<\mu
_{p,1}$ constant
and $| \mu _{p,2}\nabla \varphi | \le
a_p.$
\item {(D)} There exist positive smooth functions $M_1,M_2$ of $(x,t)$
such that
$$
\big| f(x,t,\xi _1,\xi _2) - M_1| \xi _1|
^{\alpha_2}
\;\text{sign}\;\xi _1\big| \le M_2
$$
for some $\alpha_2 \ge 1$ and all $(x,t)\in \Omega \times [0,T], \quad
0\le \xi _2\le 1$.
Observe that system (1)--(3) with conditions (A -- D) includes both
the standard Drift-Diffusion model and the amorphous silicon model.
We choose and fix a parameter $\tau $ with $3<\tau <4, $ set
$Q_T=\Omega \times (0,T)$ and recall $\Omega \subset R^3$. We
observe the following results
\proclaim {Lemma 0}
\item{\rm (a)} Let $-\Delta u(x) =f_1(x)$ in
$\Omega $, with $f_1\in L^\tau(\Omega )$. If $u=\overline
u(x)\in C^1$ in $\Gamma _D, \quad \pd{u}{n} =0$ on $\Gamma _N$ then
$u\in H^{1,\tau}(\Omega )$ and
$$
\| \nabla u\| _{L^\tau(\Omega )} \le C\Big[\| f_1\|
_{L^\tau(\Omega )} +\| \overline u\| _{C^1(\Omega )}\Big]
$$
\item{\rm (b)} Let $v$ be a generalized solution of
$$
v_t -\nabla [w\nabla v +\vec\delta v] +m v = f_2 \tag 4
$$
with $0<\alpha 0$
(independent of $v)$ such that $v\in C^{\alpha _0,\alpha _0/2}
(\overline Q_T)$.
\item{\rm (c)} If $v$ solves {\rm (4)} with the given initial/boundary
conditions and $\| v\| _{L^2(\Omega )}(t)$ is bounded,
then $v$ is globally bounded in $L^\infty$.
\endproclaim
\demo{Proof} Part (a) is immediate from the results of Shamir,
[13].
Part (b) follows from e.g. [6, Theorem~10.1, p.~205] (see
also [1]) and a reflection process to establish
the needed regularity on $\overline \Gamma _N\cap \Gamma _D,$
[7], [16], and
Part (c) follows from [6, p.~192]. More explicitly, let
$v$ satisfy (4) and suppose first that
$\Omega_0\subset\subset \Omega$. Then for $\Omega_0$ Parts~(b), (c) are
found explicitly in [6]. Next,
if $P\in\Gamma _N$ then we map a neighborhood $\Cal N$
of $P$ by a bi-Lipschitz map $L$ to a sphere $S$ with
$L(\Gamma _N\cap\Cal N)\subset\{x \mid x_3=0\}$,
$L(P)=0$ and $L(\Omega\cap\Cal N)\subset \{x\mid x_3>0\}.$
We extend $v$ as an even function to the whole of $S$
and the coefficients as in [14], [16] so that the extended
function $\widehat v$ satisfies (the extended) (4) in
$S$. We can now use the interior/initial results to conclude
first that $\widehat v$ is bounded in $L^\infty$ and then
that $\widehat v\in C^{\alpha_0,\alpha_0/2}$ in a
neighborhood of $0\times[0,T]$. Applying $L^{-1}$ then
shows the result for $\Cal N$. If $P\in\overline{\Gamma }
_N\cap \Gamma _D$ then the process is the
same except now $L(\Omega\cap\Cal N)\subset \{x\mid x_2>0,\
x_3>0\}$, $L(\Gamma _N\cap\Cal N\}\subset\{x\mid
x_2=0\}$, $L(\Gamma _D\cap\Cal N)\subset\{x\mid
x_3=0\}$. We first extend $v$ as an even function to the
upper hemisphere and then apply the Dirichlet problem results.
The results for $P$ on the Dirichlet Boundary are in [6].
In summary, for each $P\in\overline\Omega$, there exists a neighborhood
$\Cal M$ such that $u\in C^{\alpha_0,\alpha_0/2} (\Cal
M\times[0,T])$ and thus $u\in C^{\alpha_0,\alpha_0/2}(\overline Q_T)$
by boundary regularity. The same arguments also show
Part~(c).
\enddemo
\proclaim {Theorem 1} There exist $\alpha _1>0$ and $K>0$ such
that all solutions of {\rm (1~--~3)} in $C^{\alpha,\alpha
/2}(\overline Q_T)$ with $0<\alpha <\alpha _1$ and $n,p\ge 0$ actually
satisfy
$$
\| n\| _{C^{\alpha _1,\alpha _1/2}} + \| p\|
_{C^{\alpha _1,\alpha _1/2}} +\| \varphi \| _{C^{\alpha
_1,\alpha _1/2}} \le K.
$$
\endproclaim
\demo {Proof} Let $(n,p,\varphi )$ represent a solution in
$C^{\alpha,\alpha /2}$ for some $\alpha >0$. First note
that
$$
\frac{\partial h}{\partial t} + (n+p+2)h = p-n.
$$
Since $p,n$ are assumed nonnegative and $h(x,0) =0$, we immediately
conclude that $| h| \le 1$. We next show that $p,n$ are
bounded in $L^\xi(L^\xi)$ for some large $\xi $.
Assume without loss of generality that $\mu _{n,1} = \mu _{p,1}
=\mu _1$. Otherwise we multiply the $\text{``}n$ equation'' in
procedures that follow by $\frac{\mu _{p,1}}{\mu _{n,1}}$ and
repeat.
Put $E=\max\,[ \| \overline n
+\overline p\| _{L^\infty }, 1]$ and let $n=Ew, $ $p=Ez$ in equations
(1~--~3). We then have $01$. Since $n,\,p$ are assumed of class
$C^{\alpha,\alpha/2}$, these are suitable test functions.
We find from assumptions~(A), (B), (C) that:
$$
\align
&\frac1{(\theta+1)}\int_\Omega
[w^{(1)}]^{\theta+1}\bigg|_{t_1}^{t_2}+\frac{4\theta}{(\theta+1)^2}\int
_{t_1}^{t_2}\int_\Omega
\alpha\Big|\nabla\left([w^{(1)}]^{\frac{\theta+1}2}\right)\Big|^2
\\ &- \int_{t_1}^{t_2}\int_\Omega
a_n\theta\{[w^{(1)}]^\theta+[w^{(1)}]^{\theta -1}\}
|\nabla w^{(1)}| \\
&-\int_{t_1}^{t_2}\int_\Omega
\mu_1\theta[(w^{(1)})^\theta+(w^{(1)})^{\theta-1}]\nabla\varphi\nabla
w^{(1)}\\ &\le \int_{t_1}^{t_2}\int_\Omega \frac CE(w^{(1)})^\theta.
\endalign
$$
We repeat with equation (7) and add to obtain
$$
\align
&\frac1{(\theta+1)}\int_\Omega\big\{(\wone)^{\theta+1}+(\zone)^{\theta+1}\big\}
\Big|_{t_1}^{t_2}\\ & +\frac{4\theta}{(\theta+1)^2}\int_{t_1}^{t_2}
\int_\Omega
\alpha\left\{\big|\nabla([\wone]^{\frac{\theta+1}2})\big|^2+\big|\nabla([\zone]
^{\frac{\theta+1}2})\big|^2\right\}\\ & -\int_{t_1}^{t_2}
\int_\Omega\theta (a_n+b_n)
\big[\{[\wone]^\theta +[w^{(1)}]^{\theta -1}\}|\nabla
\wone|+\{[\zone]^\theta + [z^{(1)}]^{\theta -1}\}
|\nabla \zone|\big]\\ & -\int_{t_1}^{t_2}\int_\Omega\mu_1
\theta\nabla\varphi
\nabla\bigg[\frac{(\wone)^{\theta+1}}{\theta+1}+\frac{(\wone)^\theta}{\theta}-
\frac{(\zone)^{\theta+1}}{\theta+1}-\frac{(\zone)^\theta}{\theta}\bigg]\\
& \le \int_{t_1}^{t_2}\int_\Omega \frac CE
[(\wone)^\theta+(\zone)^\theta].\tag 8
\endalign
$$
Let $I_1,\ I_2,\ I_3,\ I_4$ denote the four integrals on
the left hand side of (8). $I_3,\ I_4$ can be estimated by
elementary means as follows. The first part of $I_3$ can
be estimated by
$$
\align
\theta a_n& \int_{t_1}^{t_2} \int_\Omega \{[\wone]^\theta +[w^{(1)}]^{\theta
-1}\} |\nabla \wone| \\
&\le \frac{2\theta a_n}{\theta+1}
\int_{t_1}^{t_2}\int_\Omega \{[\wone]^{\frac{\theta+1}2} +
[w^{(1)}]^{\frac{\theta -1}{2}}\}
\big|\nabla[(\wone)^{\frac{\theta+1}2}]\big|\\ &\le \frac{2\theta a_n}{\theta+1}
\bigg[\frac1{2\varepsilon}\int_{t_1}^{t_2}\int_\Omega
\{[\wone])^{\theta+1} + [w^{(1)}]^{\theta -1}\}
+\frac{\varepsilon}2\int_{t_1}^{t_2}\int_\Omega
\big|\nabla[(\wone)^{\frac{\theta+1}2}]\big|^2\bigg].
\endalign
$$
If we choose $\varepsilon$ small enough (depending on
$a_n,\ \alpha, \theta)$ then the second integral on the
right hand side has coefficient smaller than the
corresponding term in $I_2$. Observe that if
$\frac\alpha{a_n(\theta+1)}$ is big enough, then
we can also employ to advantage the estimate
$$\int_\Omega (\wone)^{\theta+1}\le\frac1{\rho_1} \int_\Omega
\big|\nabla[(\wone)^{\frac{\theta+1}2}]\big|^2$$
where $\rho_1$
denotes the least eigenvalue of $-\Delta$ with mixed
boundary conditions. While this comment is irrelevant here,
it is useful both for the existence of steady state
solutions and of an absorbing set. The second part of $I_3$
is treated identically, with $z$ replacing $w$. Next:
$$
-I_4 = \int_{t_1}^{t_2} \int_\Omega \mu_1
\theta\bigg\{-\frac{(\wone)^{\theta+1}}{\theta+1}-\frac{(\wone)^\theta}{\theta}+\frac
{(\zone)^{\theta+1}}{\theta+1}+\frac{(\zone)^\theta}{\theta}\bigg\}
f\big(x,t,E(z-w),h\big)
$$
Without loss of generality, at any given point $(x,t)$ we
may first assume $\zone>\wone$ with $\zone>0$ and also note that
$(z-w)^{\alpha_2}\ge (\zone-\wone)^{\alpha_2}$ and recall
$\alpha_2\ge 1$. We then have from (5)
$$
\align
&\frac1{\theta+1}[(\zone)^{\theta+1}-(\wone)^{\theta+1}][M_1E^{\alpha_2}(\zone-\wone)^{\alpha_2}
-M_2]\\ &\qquad \ge
\frac1{\theta+1}[(\zone)^{\theta+1}-(\wone)^{\theta+1}][M_1
E^{\alpha_2}(\zone-\wone)-(M_2+M_1 E^{\alpha_2})]\\ &\qquad \ge
-\frac1{\theta+1}\,\frac{[(\zone)^{\theta+1}-(\wone)^{\theta+1}]}{(\zone-\wone)}\,
\frac{(M_1 E^{\alpha_2}+M_2)^2}{4M_1 E^{\alpha_2}}\\ &\qquad \ge -\frac{(M_1E^{\alpha_2}+M_2)^2}{4M_1 E^{\alpha_2}}
\, [(\zone)^\theta+(\wone)^\theta].
\endalign
$$
An identical estimate, with $\theta$ replaced by $\theta-1,$
holds for the other two terms in the integrand of $I_4$ and for the
points where $w^{(1)}>z^{(1)}.$
Thus:
$$
-I_4\ge -C\int_{t_1}^{t_2} \int_{\Omega}
\{(\zone)^\theta+(\wone)^\theta+(\zone)^{\theta-1}
+(\wone)^{\theta-1}\}
$$
with a calculable constant $C$. In summary, setting
$s=(\wone)^{\frac{\theta+1}2}$ and
$r=(\zone)^{\frac{\theta+1}2}$, we obtain from equation (8):
$$
\int_\Omega (s^2+r^2)\Big|_{t_1}^{t_2} +c_0\int_{t_1}^{t_2}
\int_\Omega [|\nabla s|^2+|\nabla r|^2] \le
c_1\int_{t_1}^{t_2} \int_\Omega (s^2+r^2)
+c_2
$$
with calculable positive constants $c_0,\ c_1,\ c_2$. We
thus have that $(\zone)^{(\theta+1)/2}$ and
$(\wone)^{(\theta+1)/2}$ are bounded in $C(L^2)\cap
L^2(H^{1,2})$ and thus, see e.g. [6], $w^{\theta+1},\
z^{\theta+1}$ are bounded in $L^{10/3}(L^{10/3})$, i.e.,
$n,\,p$ are bounded in $L^\xi(L^\xi)$ for any large chosen
$\xi$. In particular, $f$ is bounded in $L^\xi(L^\xi)$ and
thus $|\nabla\varphi|$ is bounded in $L^\xi(L^\tau)$ for
$\xi$ large, where we recall $3<\tau<4$. We now employ [6]
and Lemma~0 to conclude that $n,\,p$ (and thus $\varphi)$
are bounded in $C^{\alpha_1,\alpha_1/2}$ with $\alpha_1$ and bound
independent of $n,\,p$.
It is useful to embed (1 -- 3) and the associated boundary/initial
conditions in the following system:
$$
\gather
-\Delta \varphi =\lambda f(x,t,p-n,h(p^+,n^+)) \tag 9\\
\pd nt - \nabla [D_n\nabla n -n\mu _n\nabla \varphi ] =\lambda
\{1-h(p^+,n^+)-n^+ [1+h(p^+,n^+)]+\widetilde R_n\} \tag 10\\
\pd pt -\nabla [D_p \nabla p +p\mu _p \nabla \varphi ]=\lambda
\{1+h(p^+,n^+)-p^+[1-h(p^+,n^+)]+\widetilde R_p\} \tag 11
\endgather
$$
with boundary/initial Dirichlet conditions
$$
\varphi =\lambda\, \overline\varphi , \quad n=\lambda \,\overline n,
\quad p =
\lambda \,\overline p\quad \text{on} \quad \Gamma _D;\quad n=\lambda
\overline n,\quad
p=\lambda\overline p\quad\text{at}\quad t=0\tag 12
$$
and $\widehat R_n=R_n(x,t,n^+,p^+), \quad \widehat R_p = R_p(x,t,n^+,p^+)$.
Observe that for $\lambda =1$
and $n,p\ge 0$, this reduces to the original problem, and
the solutions $n,\,p$ must be nonnegative by the weak
maximum principle and equation~(9).
\enddemo
\proclaim {Theorem 2} There exists a $C^{\alpha ,\alpha /2}$
solution $(n,p,\varphi )$ of system {\rm (1,3)} with the associated
boundary/initial conditions for some $\alpha >0$ independent of
$(n,p,\varphi )$, with
$(n,p)$ nonnegative. If $D_n,\ D_p,\ \mu_n,\ \mu_p$ are only functions of
$(x,t)$, the solution is unique.
\endproclaim
\demo {Proof} We transform (9)--(12) into an operator
equation in the usual way. Let $(\lambda_0,n_0,p_0)$ be
given with $(n_0,p_0)\in C^{\alpha,\alpha/2}$ with $\alpha>0$ chosen,
evaluate $f$ at this point and calculate $\varphi_0$ from
(9). Evaluate the coefficients $D_n,\ D_p,\ \mu_n,\ \mu_p,\
h$, boundary/initial conditions and the right hand sides of
(10), (11) at $(\lambda_0,n_0,p_0,\varphi_0)$ and solve the
now linear equations to obtain the new $(n,p)$. We may
express this process in the form:
$$
\gather
\lambda =\lambda_0\\
\varphi =\lambda T_0(n_0,p_0)\\
(n,p)=T_1(n_0,p_0,T_0(n_0,p_0),\lambda).
\endgather
$$
Observe that $T_1:C^{\alpha,\alpha/2}\times [0,1]\to
C^{\alpha_1,\alpha_1/2}$ whence if we choose some $\alpha<\alpha_1$ we have
compactness, since $C^{\alpha_1,\alpha_1/2}\subset\subset
C^{\alpha,\alpha/2}$ and the
earlier estimates of
Theorem~1 still hold (indeed the presence of $\lambda $ helps as
$0\le \lambda \le1)$. Finally, that $T_1$ is continuous can be seen from
lengthy but routine arguments. Note in particular that
the compactness of $T_1$ implies that continuity need only be
shown $C^{\alpha,\alpha/2}\times [0,1]\to L^2$ and that the
coefficients $D_n,\,\mu _n,\,D_p,\,\mu _p$ are assumed bounded and
thus the Lebesgue Convergence Theorem can be applied, much as for
example, was done in [2]. Once again the framework of
$C^{\alpha ,\alpha /2}$ spaces makes this process easier.
The existence of a solution is then immediate by the
Leray-Schauder Degree using $\lambda$ as a homotopy
parameter, [17].
The uniqueness of
$(n,p,\varphi )$ under the extra assumption on $D_n,\ D_p,
\ \mu_n,\ \mu_p$, is immediate from Gronwall's Lemma (see
once again, e.g., [2]), since if we let $(n,p,\varphi)$ and
$(\widehat n, \widehat p,\widehat\varphi)$ denote two solutions we then observe the estimate:
$$
\align
|h(n,p)-h(\widehat n,\widehat p)|&=
\bigg|\int_0^t ((p-\widehat p)-(n-\widehat n)) e^{-\int_\xi^t (p+n+2)d\xi}\\
&\quad +\int_0^t (\widehat p-\widehat n) \bigg[ e^{-\int_\xi^t (p+n+2)}-
e^{-\int_\xi^t (\widehat p+\widehat n+2)}\bigg]\bigg|\\
&\le [C_0+C_1(n,p,\widehat n,\widehat p)t]\int_0^t |p-\widehat p|+|n-\widehat n|
\endalign
$$
for some constant $C_0$. Choose $T$ and let $0\le t\le t_10$. Obviously a similar estimate
holds for $\| n\| _{L^\infty } +\| p\| _{L^\infty}$. Furthermore a similar
proof shows that in this case there exists at least
one steady state solution $\widehat n, \, \widehat p,\, \widehat \varphi $ in
$C^{\alpha _1,\alpha _1/2}(\Omega )$, with $h=\frac{p-n}{n+p+2}$,
Absorbing set considerations can also be based in this case directly
on the proof of Theorem~1. Indeed, choose $E=\sup\big\{\|\overline n+\overline p\|
_{L^\infty(\partial \Omega_D)},\;1\big\}$. We then repeat and find
that there exist a $K,\ t_0$ such that for $t\ge t_0$ we have
$\|n+p\|_{L^\infty}\le K$, where $K$ depends only on $\|\overline n+\overline p\|
_{L^\infty(\partial\Omega_D)}$ and $t_0$ on $\|\overline n+\overline p\|_{L^\infty
(\Omega)}$. Some idea of the precise nature of the bounds $K$ and $t_0$
can be obtained by following the various proofs in [6], [13] and Theorem~1.
In general, however, precise estimates seem extremely difficult to
obtain due to the difficulty in estimating the various constants.
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\enddocument
\bye