Systems of Parabolic Differential Equations

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Systems of Parabolic Differential Equations
On the First Boundary Problem for Quasilinear
Systems of Parabolic Differential Equations
in Non-cylindrical Domains
By Takasi KUSANO
(Chuo University)
Introduction
This paper deals with second order quasilinear systems of partial differential
equations which constitute some special cases of the nonlinear parabolic system
$u_{t}^{k}=F^{k}(t,$
$X$
,
$u^{1}$
,
$¥cdots$
,
$u^{N}$
,
$u_{x_{1}}^{k}$
,
$¥cdots$
,
$u_{x_{n}}^{k}$
,
$u_{x_{1}x_{1}}^{k}$
,
$u_{x_{1}x_{2}}^{k}$
,
$¥cdots$
,
$u_{x_{n}x_{n}}^{k})$
(0. 1)
$(k=1, ¥cdots, N)$
considered by several Polish mathematicians. For typical problems and results
concerning the system (0. 1) and the related systems of parabolic differential
inequalities we refer the reader to the works of Szarski [15-17], Mlak [10-13],
Besala [1, 2] and Brzychczy [20].
We shall be concerned specifically with the non-local solvability of the first
boundary problem for systems of the forms
$u_{t}^{k}=¥sum_{i,j=1}^{n}a_{ij}^{k}(t, X)u_{x_{i}x_{j}}^{k}+¥sum_{i=1}^{n}b_{i}^{k}(t, X)u_{x_{i}}^{k}+f^{k}(t, X, u^{1_{ }},¥cdots, u^{N})$
$(k=1, ¥cdots, N)$
$u_{t}^{k}=¥sum_{i.j=1}^{n}a_{ij}^{k}(t, X)u_{x_{i}x_{i}}^{k}+¥sum_{i=1}^{n}b_{i}^{k}(t, X, u^{k})u_{x_{i}}^{k}+c^{k}(t,$
$+f^{k}(t, X, u^{1}, ¥cdots, u^{N})$
$X$
,
$u^{k}$
,
$u_{x_{1}}^{k}$
,
$¥cdots$
,
(0. 2)
,
$u_{x_{h}}^{k})$
(0. 3)
$(k=1, ¥cdots, N)$
in non-cylindrical domains, since the existence problem seems to have been only
superficially touched in the literature quoted above. The basis of our method is
simple; the existence proofs are based on the method of iterations with the aid
of the comparison theorem of Nagumo-Westphal type and the theorems concerning existence and a priori estimates of solutions of single linear and quasilinear
parabolic equations.
The body of the paper is divided into two parts.
In Part I we state and
prove several comparison theorems for general nonlinear systems of parabolic
partial differential inequalities associated with (0. 1).
The theorems obtained
extend the well known Nagumo-Westphal theorem (Nagumo [14] and Westphal
[19] and modify the results given by Mlak [11] and Szarski [17]. Part
concerns the existence of solutions to the first boundary problem for the quasilinear parabolic systems (0. 2) and (0. 3) in non-cylindrical domains. A natural
$)$
$¥mathrm{I}¥mathrm{I}$
104
T. KUSANO
iteration procedure will be adopted to yield sequences of approximate solutions
whose limits determine the required solutions. In order to show that the iteration scheme we adopt is a good one which means that the approximating sequences are monotonie, uniformly bounded and compact in appropriate norms, we
need some results pertaining to single parabolic differential equations due to
Friedman [3, 4] and Kamynin and Maslennikova [5]. Such preparatory results
will be brought together at the beginning of Part .
In connection with this paper we have to mention some previous investigations on the non-local solvability of the first boundary problem for quasilinear
parabolic systems of the second order. (See Ventsel’ [18], Ladyzhenskaia and
UraFtseva [6] and others.) Our treatment is elementary and rather similar to
the idea originated by McNabb [9] in the theory of multicomponent diffusion
systems. (See also Maslennikova [7, 8].)
Finally the author wishes to thank Professor Masuo Hukuhara and Professor John R. Cannon for many helpful suggestions.
$¥mathrm{I}¥mathrm{I}$
Comparison Theorems of Nagumo-Westphal Type
Part I.
Let
be {he
-dimensional
(a) Definitions and terminologies.
space-time of generic points
. By a non-cylindrical domain
(more precisely, a domain not necessarily cylindrical) we mean a bounded domain in
enclosed by two hyperplanes $t=0$ and $t=T(T>0)$ , and by a lateral surface ( is a closed set) lying between these hyperplanes. We denote by
the projection of the cross-section
onto the hyper$D$
$t=0$
plane
.
is then the base of the domain
is
and the set
called the normal (or parabolic) boundary of $D$. Let us set
.
In this Part the domain is assumed to have the following property : To
each
and
and to every sequence
such that $0<t_{v}<T$
sequence
points
corresponds
there
and
of
a
for which
and
$E^{n+1}$
$(¥mathrm{n}+1)$
$(t, X)¥equiv(t, x_{1^{ }},¥cdots, x_{n})$
$E^{n+1}$
$S$
$S$
$¥{x=x_{0}¥}¥cap¥overline{D}(0¥leqq t_{0}¥leqq T)$
$¥Omega_{t_{0}}$
$¥partial D=S¥cup¥Omega_{0}$
$¥Omega_{0}$
$D_{0}=¥overline{D}-¥theta D$
$t_{0}(0¥leqq t_{0}¥leqq T)$
$X_{0}¥in¥Omega_{t_{0}}$
$¥{t_{v}¥}$
$¥{X_{¥mathcal{V}}¥}$
$t_{¥gamma}¥rightarrow t_{0}$
$X_{¥nu}¥rightarrow X_{0}$
$X_{¥gamma}¥in¥Omega_{t_{¥nu}}$
.
Following Polish authors (Mlak and Szarski) we introduce several definitions which are needed in formulating comparison theorems.
Definition 1. We shall say that a vector function $U(t, X)=¥{u^{k}(t, X)¥}$
$(k=1, ¥cdots, N)$ is regular in $D$ if each of its components $u^{k}(t, X)$ is continuous in
and possesses continuous first , , and second -derivatives in
.
$
¥
{f^{k}(t,
X,
Z,
Q,
R)
¥
}$
$(k=1,$
Definition 2. Consider a system of functions
,
¥
$(t,
X)
¥
in
D_{0}$
$N)$ defined for
and arbitrary
,
, $R=(r_{11}$ ,
, ,
. The system of functions is said to be elliptic with respect to a vec$tor$
function $U(t, X)=$ $¥{u^{k}(t, X)¥}$ $(k=1, ¥cdots, N)$ if for any symmetric matrices
$¥overline{D}$
$x$
$t-$
$¥mathrm{x}$
$D_{0}$
$¥cdots$
$Z=(z^{1_{ }}, cdots, z^{N})$
$r_{12}$
$¥cdots$
$r_{nn})$
$Q=(q_{1^{ }},¥cdots, q_{n})$
and
such that
equalities are satisfied:
$R=(r_{ij})$
$¥tilde{R}=(¥tilde{r}_{ij})$
105
of Parabolic Differential Equations
Quasilinear Systems
$R-¥tilde{R}$
is positive semi-definite the following in-
$f^{k}(t, X, U(t, X), u_{x}^{k}(t, X), R)¥geqq f^{k}(t, X, U(t, X), u_{x}^{k}(t, X),¥tilde{R})$
$(k=1, ¥cdots, N)$
where we have set
$u_{x}^{k}(t, X)=$ $(u_{x_{1}}^{k}(t, X), ¥cdots, u_{x_{n}}^{k}(t, X))$
,
.
The system is called simply elliptic when the inequalities
$f^{k}(t, X, Z, Q, R)¥geqq f^{k}(t, X, Z, Q,¥tilde{R})$
hold for arbitrary
$Z$
,
and for all matrices
$Q$
$(k=1, ¥cdots, N)$
such that
$R,¥overline{R}$
$R-¥tilde{R}$
is positive
semi-definite.
Definition 3. Consider a system of functions
, where
¥
and
stands for a collection of variables different from . We
shall say that this system satisfies the condition (W) with respect to
if for
each $k=1$ , , $N$, and for every couple of vectors $Z=(z^{1}, ¥cdots, z^{N})$ ,
,
such that
,
we have
$¥{g^{k}(Z_{ },¥underline{=})¥}(k=1, ¥cdots, N)$
$Z=(z^{1_{ }}, cdots, z^{N})$
$Z$
$¥Xi$
$Z$
$¥overline{Z}=¥tilde{(z}^{1}$
$¥cdots$
$z^{k}=¥tilde{z}^{k}$
$¥cdots,¥tilde{z}^{N})$
$z^{j}¥geqq¥tilde{z}^{j}(j¥neq k)$
$g^{k}(Z,¥underline{=})¥geqq g^{k}(¥tilde{Z}_{ },¥underline{=})$
.
(b) Comparison theorems. Let us now extend the Nagumo-Westphal
theorem ([14], [19]) to the case of systems of parabolic differential inequalities.
Although the comparison theorems to be obtained are the modified versions of
those of Mlak [11] and Szarski [17], it will be of interest to observe that they
are rather similar in outlook to the results recently found in the theory of diffusion systems (McNabb [9] and Maslennikova [7, 8]).
Theorem 1. 1. Suppose that:
I. $U(t, X)=$ $¥{u^{k}(t, X)¥}$ and $V(t, X)=$ $¥{v^{k}(t, X)¥}(k=1, ¥cdots, N)$ are regular
in
$D$
.
. The systems of functions $¥{f^{k}(t, X, Z, Q, R)¥}$ and
, $N)$ , defined for $(t, X)¥in D_{0}$ and arbitrary
$¥mathrm{I}¥mathrm{I}$
$¥ldots$
$r_{11}$
$Z=(z^{1}, ¥cdots, z^{N})$
,
$r_{12}$
,
$¥cdots$
,
$r_{nn})$
, have the following property :
every pair of vectors
$(j¥neq k)$ there holds the inequality
$Z=(z^{1_{ }},¥cdots, z^{N}),¥overline{Z}=$
For each
$(¥tilde{z}^{1}, ¥cdots,¥tilde{z}^{N})$
.
$U(t, X)=$
$¥mathrm{I}¥mathrm{V}$
.
The system
$¥{f^{k}(t, X, Z, Q, R)¥}$
$¥{u^{k}(t, X)¥}(k=1, ¥cdots, N)$
(,
$t$
$X$
,
$U$
,
$u_{x}^{k}$
,
$Q=(q_{1}, ¥cdots, q_{n})$
$k=1$ ,
$¥cdots$
$z^{k}=¥tilde{z}^{k}$
,
$z^{j}¥geqq¥tilde{z}^{j}$
.
is elliptic with respect to
$(k=1, ¥cdots, N)$
.
satisfied:
The following inequalities are
$u_{t}^{k}-f^{k}$
,
such that
$f^{k}(t, X, Z, Q, R)¥geqq g^{k}(t, X,¥overline{Z}, Q, R)$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
,
, $R=$
, $N$, and for
$¥{g^{k}(t, X, ¥mathrm{Z} , R)¥}(k=1$
$u_{xx}^{k})>v_{t}^{k}-g^{k}(t,$
$u^{k}(t, X)>v^{k}(t, X)$
on
$¥partial D$
$X$
,
,
$V$
$v_{x}^{k}$
,
$v_{xx}^{k})$
in
$(k=1, ¥cdots, N)$
.
$D_{0}$
,
(1. 1)
(1. 2)
Under these assumptions we have
$u^{k}(t, X)>v^{k}(t, X)$
in
$¥overline{D}$
$(k=1, ¥cdots, N)$
.
(1. 3)
106
T. KUSANO
Proof. We
set
$M^{k}(t)=¥min_{X¥in¥Omega_{l}}$
and
$¥{u^{k}(t, X) -v^{k}(t, X)¥}$
$M(t)=¥min_{1¥leqq k¥leqq N}M^{k}(t)$
.
Our theorem then asserts that $M(t)>0$ for all $t¥in[0, T]$ . To begin, note that
the function $M(t)$ is continuous in $[0, T]$ (Szarski [16, Lemme 1. 1]). Supposing, contrary to the statement of the theorem, there is a $t¥in(0,$ $T$ ] where
$M(t)¥leqq 0$ , we shall derive a contradiction.
To do this, put $ t_{0}=¥inf¥{t;M(t)¥leqq$
$T$
, $t¥in(0,$ ]}. Then, clearly $t_{0}>0$ and we have
$M(t_{0})=0$
and $M(t)>0$ for $0¥leqq t<t_{0}$ ,
whence we can find an index and a point
such that
$(j¥neq k)$ .
and
(1. 4)
The condition (1. 2) implies that the point
.
lies in
From the determination of
it follows readily that
$0$
$k$
$X_{0}¥in¥Omega_{t_{0}}$
$u^{j}(t_{0}, X_{0})¥geqq v^{j}(t_{0}, X_{0})$
$u^{k}(t_{0}, X_{0})=v^{k}(t_{0}, X_{0})$
$D_{0}=¥overline{D}-¥partial D$
$(t_{0}, X_{0})$
$(t_{0}, X_{0})$
$u_{t}^{k}(t_{0}, X_{0})¥leqq v_{t}^{k}(t_{0}, X_{0})$
.
On the other hand, observing that
$u_{x}^{k}(t_{0}, X_{0})=v_{x}^{k}(t_{0}, X_{0})$
$¥sum_{i.j=1}^{n}$
(1. 5)
,
$[u_{x_{i}x_{j}}^{k}(t_{0}, X_{0})-v_{x_{i}x_{j}}^{k}(t_{0}, X_{0})]¥xi_{i}¥xi j¥geqq 0$
(1. 6)
,
we derive from (1. 1)
$u_{t}^{k}(t_{0}, X_{0})-v_{t}^{k}(t_{0}, X_{0})>f^{k}(t_{0},X_{0},$
,
$U$
$u_{x}^{k}$
,
$u_{xx}^{h})-g^{k}(t_{0}$
,
$X_{0}$
,
,
$V$
$v_{x}^{k}$
,
$v_{xx}^{k})$
$=[f^{k}(t_{0}, X_{0}, U, u_{x}^{h}, u_{xx}^{k})-f^{k}(t_{0}, X_{0}, U, u_{x}^{k}, v_{xx}^{k})]$
$+[f^{k}(t_{0}, X_{0}, U, u_{x}^{h}, v_{xx}^{k})-g^{k}(t_{0}, X_{0}, V, v_{x}^{k}, v_{xx}^{k})]$
,
which is easily found non-negative. Indeed, the first difference in the last expression is non-negative on account of the ellipticity
(Assumption
)
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$¥mathrm{a}¥mathrm{s}¥mathrm{s}¥mathrm{u}¥mathrm{m}¥mathrm{p}¥mathrm{t}¥check{¥mathrm{i}}¥mathrm{o}¥mathrm{n}$
and the semi-definiteness (1. 6) of the matrix
$(u_{x_{i}x_{j}}^{k}-v_{x_{i}x_{j}}^{k})$
at
$(t_{0}, X_{0})$
, and so is
tegether with the relations (1. 4)
the second term by virtue of Assumption
and (1. 5). But this is absurd and the theorem is proved.
Theorem 1. 2. Suppose that:
I. $U(t, X)=$ $¥{u^{k}(t, X)¥}$ and $V(t, X)=¥{v^{k}(t, X)¥}$ $(k=1, ¥cdots, N)$ are regular
$¥mathrm{I}¥mathrm{I}$
$¥dot{z}nD$
.
The system of functions $¥{f^{k}(t, X, Z, Q, R)¥}$ $(k=1, ¥cdots, N)$ , defined for
and arbitrary $Z$ , $Q$ , $R$ , is elliptic with respect to $U(t, X)=¥{u^{k}(t, X)¥}$
and satisfies the condition (W) with respect to $Z$.
. The following inequalities are fuffilled:
$¥mathrm{I}¥mathrm{I}$
.
$(t, X)¥in D_{0}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$u_{t}^{k}-f^{k}$
(,
$t$
$X$
,
,
$U$
$u_{x}^{k}$
,
$u_{xx}^{k})>v_{t}^{k}-f^{k}(t,$
$u^{k}(t, X)>v^{k}(t, X)$
on
$¥partial D$
$X$
,
,
$V$
$v_{x}^{k}$
,
$v_{xx}^{k})$
in
$(k=1, ¥cdots, N)$ ,
$D_{0}$
,
(1. 7)
(1. 8)
Quasilinear Systems
107
of Parabolic Differential Equations
Under these assumptions we have
(1. 9)
This is a trivial consequence of Theorem 1. 1. We shall show that the
above theorems may be sharpened when the functions $f^{k}(t, X, Z, Q, R)$ satisfy
a uniform Lipschitz condition with respect to .
Theorem 1. 3. Retaining Assumptions I and
of Theorem 1. 1, we suppose
$u^{k}(t, X)>v^{k}(t, X)$
in
$(k=1, ¥cdots, N)$
$¥overline{D}$
.
$Z$
$¥mathrm{I}¥mathrm{I}$
further that:
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
.
uniform
The system $¥{f^{k}(t,X, ¥mathrm{Z} , R)¥}(k=1, ¥cdots, N)$ is elliptic and
Lipschitz condition with respect to $z$ :
$|f^{k}(t, X, Z, Q, R)$
$¥mathrm{I}¥mathrm{V}$
.
$-f^{k}(t, X,¥overline{Z}, Q, R)|¥leqq M¥sum_{j=1}^{n}|z^{j}-¥tilde{z}^{j}|$
(,
$t$
$X$
,
$U$
,
$u_{x}^{k}$
,
$u_{xx}^{k})¥geqq v_{t}^{k}-g^{k}(t,$
on
$u^{k}(t, X)¥geqq v^{k}(t,X)$
$a$
(1. 10)
.
satisfied:
The following inequalities are
$u_{t}^{k}-f^{k}$
satisfies
$X$
,
,
$V$
$¥partial D$
$v_{x}^{k}$
,
in
$ v_{xx}^{k}¥rangle$
(1. 11)
,
$D_{0}$
$(k=1, ¥cdots,N)$
.
(1. 12)
$(k=1, ¥cdots, N)$
.
(1. 13)
Then, we have
$u^{k}(t, X)¥geqq v^{k}(t,X)$
Proof. Setting
ter and
$(t, X)¥}$
$M$
with
in
$¥overline{D}$
$u_{¥alpha}^{k}(t, X)=u^{k}(t,X)+¥mathrm{a}$ $e^{2MNt}$
, where
is a positive parame-
$¥alpha$
is the Lipschitz constant in (1. 10), we first compare $U(t,X)=$
$U_{a}(t,X)=$
$¥{u_{¥alpha}^{k}(t,X)¥}$
$¥{u^{k}$
. We have
$[u_{at}^{k}-f^{k}(t,X, U_{¥alpha}, u_{ax}^{k},¥mathrm{u}_{¥alpha xx}^{k})]-[u_{t}^{k}-f^{k}(t,X, U,u_{x}^{k},u_{xx}^{k})]$
and
(
$=2$
a
$¥geqq 2$
a $MNe^{MNt}-M¥sum_{j=1}^{N}|u^{j}-u_{a}^{j}|=$ a
$MNe^{¥mathit{2}MNt}+f^{k}$
$u_{a}^{k}(t, X)>u^{k}(t,X)$
$u_{at}^{k}-f^{k}$
(,
$t$
$X$
,
on
$U_{a}$
,
$¥partial D$
$u_{ax}^{k}$
.
,
$t,X$
,
$U,u_{x}^{k}$
,
$u_{xx}^{k})-f^{k}(t,$
$X$
,
$U_{¥mathrm{a}}$
,
$u_{¥alpha x}^{k}$
$MNe^{¥mathit{2}MNt}>¥mathit{0}$
,
$u_{¥alpha xx}^{k})$
in
$D_{0}$
Hence it follows from (1. 11) and (1. 12) that
$u_{¥alpha xx}^{k})>v_{t}^{k}-g^{k}(t,$
$X$
,
$V,v_{x}^{k}$
,
$v_{xx}^{k})$
in
$D_{0}$
and
$u_{a}^{k}(t, X)>v^{k}(t, X)$
on
$¥theta D$
$(k=1, ¥cdots,N)$
.
An application of Theorem 1. 1 then yields
$u_{¥alpha}^{k}(t, X)=u^{k}(t, X)+¥mathrm{a}e^{2MNt}>v^{k}(t, X)$
in
$¥overline{D}$
$(k=1, ¥cdots, N)$
,
whence, passing to the limit a
, we obtain the desired inequalities (1. 13).
Assume
that:
Theorem 1. 4.
$U(t,X)=$ $¥{u^{k}(t, X)¥}$ and $V(t, X)=¥{v^{k}(t, X)¥}(k=1, ¥cdots, N)$ are regular
I.
$¥rightarrow 0$
in
$D$
.
. The system of functions $¥{f^{k}(t, X, ¥mathrm{Z} , R)¥}(k=1, ¥cdots, N)$ is elliptic and
satisfies the condition (W) as $we/Z$ as a uniform Lipschitz condition with res$¥mathrm{I}¥mathrm{I}$
T. KUSANO
108
pect to
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$Z$
.
.
There hold the following inequalities
$u_{t}^{k}-f^{k}$
(
$t,X$
,
$U$
,
$u_{x}^{k}$
,
$u_{xx}^{k})¥geqq v_{t}^{k}-f^{h}(t,$
$u^{k}(t, X)¥geqq v^{k}(t, X)$
on
$X$
, $V$,
$v_{x}^{k}$
,
in
$v_{xx}^{k})$
$(k=1, ¥cdots, N)$
$¥partial D$
$D_{0}$
,
(1. 14)
.
(1. 15)
Then, we have
.
(1. 16)
This follows immediately from Theorem 1. 3.
(c) Comparison theorems {continued). In this paragraph we shall briefly
discuss some variants of the above comparison theorems, though not directly
relevant to the considerations in the next Part . Only the statements corresponding
Theorems 1. 2 and 1. 4 will be presented.
Theorem 1. 5. Let the following conditions be satisfied:
I. $U(t, X)=¥{u^{k}(t, X)¥}$ and $V(t, X)=¥{v^{k}(t,X)¥}$ $(k=1, ¥cdots, N)$ are regular
in D. In addition, the $u^{k}(t, X)$ and $v^{k}(t, X)$ admit on
an oblique derivative
$u^{k}(t,X)¥geqq v^{k}(t, X)$
in
$¥overline{D}$
$(k=1, ¥cdots, N)$
$¥mathrm{I}¥mathrm{I}$
$¥dot{¥mathrm{t}}¥mathrm{o}$
$S$
of the form
,
$¥frac{dw^{k}}{df_{k}}=X¥rightarrow X!.,¥mathrm{m}¥frac{w^{k}(i,¥dot{X})-w^{k}(t,X)}{|¥dot{X}-X|}X¥in l_{¥mathrm{k}}1$
or
$w^{k}=u^{k}$
$v^{k}$
,
(1. 17)
orthogonal to
is an assigned direction through
and
where
the -axis and penetrating into the domain $D$.
. The system of functions $¥{f^{k}(t, X, Z, Q, R)¥}(k=1, ¥cdots,N)$ is elliptic with
respect to $U(t, X)$ and satisfies the condition (W) with respect to $Z$ .
. The system of functions $¥{¥Psi^{k}(t, X, Z)¥}$ $(k=1, ¥cdots, N)$ is defined for
$(t, X)¥in S$ and arbitrary $Z=(z^{1_{ }},¥cdots, z^{N})$ and
the condition (W) with res$(¥dot{t},¥dot{X})¥in S$
$(t.,¥dot{X})$
$f_{h}$
$t$
$¥mathrm{I}¥mathrm{I}$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$sat¥dot{s}sfies$
pect to
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$Z$
.
.
The foffowing inequdities are
$u_{t}^{k}-f^{k}$
(,
$t$
$X$
,
,
$U$
$u_{x}^{k}$
,
satisfied:
$u_{xx}^{k})>v_{t}^{k}-f^{¥mathit{1}¥epsilon}(t,$
$u^{k}(0, X)>v^{k}(0, X)$
$¥frac{du^{k}}{df_{k}}-¥Psi^{k}(t, X, U)>¥frac{dv^{k}}{df_{k}}-¥Psi^{k}(t, X, V)$
$X$
,
,
$V$
on
$v_{x}^{k}$
$¥Omega_{0}$
on
,
$v_{xx}^{k})$
in
$D_{0}$
,
(1. 19)
,
$(k=1, ¥cdots, N)$
$S$
(1. 18)
. (1. 20)
Then, we have
.
(1. 21)
Proof. Arguing in entirely the same way as in the proof of Theorem 1. 1,
in
we can find a point
and an index such that $M^{k}(t_{0})=u^{k}(t_{0}, X_{0})-$
is on , then
and the relation (1. 4) is valid.
If the point
we have
$u^{k}(t, X)>v^{k}(t, X)$
$(t_{0}, X_{0})$
$¥overline{D}$
in
$¥overline{D}$
$(k=1, ¥cdots, N)$
$k$
$v^{k}(t_{0}, X_{0})$
$(t_{0}, X_{0})$
$¥frac{du^{k}}{dl_{k}}(t_{0}, X_{0})¥leqq¥frac{dv^{k}}{dl_{k}}(t_{0}, X_{0})$
But this is impossible, since by (1. 20)
.
$S$
of Parabolic Differential Equations
Quasilinear Systems
109
$¥frac{du^{k}}{df^{k}}(t_{0}, X_{0})-¥frac{dv^{kk}}{df}(t_{0}, X_{0})>¥Psi^{k}(t_{0}, X_{0}, U(t_{0}, X_{0}))-¥Psi^{k}(t_{0}, X_{0}, V(t_{0}, X_{0}))$
,
which is non-negative by virtuve of the condition (W) assumed for the the sys¥
of functions ¥ ¥
and the relation (1. 4). Thus, it is shown that
the point
cannot lie on
and hence by (1. 19) it must belong to
.
The rest of the proof is exactly the same as in Theorem 1. 1.
Theorem 1. 6. Let the following conditions be satisfied:
I. $U(t, X)=$ $¥{u^{k}(t, X)¥}$ and $V(t, X)=¥{v^{k}(t, X)¥}(k=1, ¥cdots, N)$ are functions
$ { Psi^{k}(t, X, Z) }$
$¥mathrm{t}¥mathrm{e}¥mathrm{m}$
$S$
$(t_{0}, X_{0})$
$D_{0}$
which are regular in $D$ and whose components have the oblique derivatives of
the form (1. 17) on .
. The system of functions $¥{f^{k}(t, X, Z, Q, R)¥}(k=1, ¥cdots, N)$ is elliptic and
satisfies the condition (W) as well as a uniform Lipschitz condition with res$S$
$¥mathrm{I}¥mathrm{I}$
pect to
$Z$
.
. Each of the function $s¥Psi^{k}(t, X, z^{k})(k=1, ¥cdots, N)$ is defined for
and arbitrary
an is strictly decreasing in
.
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$d$
$z^{k}$
$¥mathrm{I}¥mathrm{V}$
.
The following inequalities are
$u_{t}^{k}-f^{k}$
(,
$t$
$(t, X)¥in S$
$z^{k}$
,
$X$ $U$
,
$u_{x}^{k}$
,
satisfied:
$u_{xx}^{k})¥geqq v_{t}^{k}-f^{k}(t,$
$u^{k}(0, X)¥geqq v^{k}(0, X)$
$¥frac{du^{k}}{df_{k}}-¥Psi^{k}(t, X, u^{k})¥geqq¥frac{dv^{k}}{dJ_{k}}-¥Psi^{k}(t, X_{r}v^{k})$
$X$
,
,
$V$
on
$v_{x}^{k}$
$¥Omega_{0}$
on
,
$v_{xx}^{k})$
in
$D_{0}$
,
(1. 23)
,
$S$
(1. 22)
$(k=1, ¥cdots, N)$
. (1. 24)
Then, we have
$e$
$(k=1, ¥cdots, N)$ .
in
(1. 25)
The proof proceeds in the same way as Theorem 1. 3, except that we use
Theorem 1. 5 in place of Theorem 1. 1.
(d) Uniqueness of solutions. Consider the system of nonlinear partial differential equations
$u^{k}(t, X)¥geqq v^{k}(t, X)$
$u_{t}^{k}=f^{k}$
(,
$t$
,
$X$
,
$U$
$¥overline{D}$
$u_{x}^{k}$
,
$u_{xx}^{k}$
)
$(k=1, ¥cdots, N)$
(1. 26)
is elliptic in the
which we call parabolic when the system
sense defined in Definition 2. Given functions
$(k=1, ¥cdots, N)$ prescribed
on the normal boundary
of the domain $D$ , the problem of finding solutions
$U(t, X)=¥{u^{k}(t, X)¥}$ of the parabolic system (1. 26) satisfying the boundary condition
$u^{k}(t, X)=¥varphi^{h}(t, X)$
$(k=1, ¥cdots, N)$
on
(1. 27)
is called the first boundary problem for the system (1. 26). If we require the
solution $U(t, X)=$ $¥{u^{k}(t, X)¥}$ to satisfy instead of (1. 27) the following condition
$¥{f^{k}(t, X, Z, Q, R)¥}$
$¥varphi^{k}(t, X)$
$¥partial D$
$¥partial D$
$u^{k}(0, X)=¥varphi^{k}(X)$
where
$¥varphi^{k}(X)$
and
on
$¥Omega_{0}$
$¥Psi^{k}(t, X, z^{k})$
and
$¥frac{du}{d}f_{k}-k=¥Psi^{k}(t, X, u^{k})$
on
$S$
$(k=1, ¥cdots N)(1.28)$
are given functions, then the problem is referred
T. KUSANO
110
to as the third boundary problem for the system.
Theorems 1.4 and 1.6 lead to the uniqueness theorems for the first and the
third boundary problems, respectively, concerning the parabolic system (1. 1).
These theorems are contained in Szarski’s general unicity theorems (Szarski
[15-17] .
$)$
Part II.
Existence Theorems
the introduction this Part is devoted
(a) Preliminaries. As mentioned
to the construction of solutions to the first boundary problem for some quasilinear parabolic systems in non-cylindrical domains. The existence proof is based
on the simple method of successive approximations and at the same time makes
strong use of, together with the comparison theorem proven in Part , several
existence and estimation theorems concerning single linear and quasilinear parabolic equations. Such basic results will be grouped here.
we define
in
For points $P(t, X)$ ,
’
$¥ln$
$¥mathrm{I}$
$E^{n+1}$
$P^{¥prime}(t^{¥prime}, X^{¥prime})$
$¥rho(P, P^{¥prime})=$ $(|t-t^{¥prime}|^{2}+|X-X^{¥prime}|^{2})^{1/2}$
,
.
is said to be the parabolic distance between $P$ and .
Following Friedman we introduce the following norms for functions defined
:
in some set $B$ in
$d(P, P^{¥prime})=$ $(|t-t^{¥prime}|+|X-X^{J}|^{2})^{1/2}$
$d(P, P^{¥prime})$
$P^{¥prime}$
$E^{n+1}$
$|u|_{0}^{B}=¥sup_{P¥in B}|u(P)|$
,
$Pu_{¥frac{(P^{¥prime})|}{)^{¥alpha}}},$
$|u¥left¥{¥begin{array}{l}B=¥¥a¥end{array}¥right¥}u|_{¥mathrm{o}}^{B}+¥sup_{P.P^{¥prime}¥in B_{¥sim}P¥neq P^{¥prime}}¥underline{|u(}d¥overline{(}¥overline{P}P),--$
$|u¥left¥{¥begin{array}{l}B=¥¥1+a¥end{array}¥right¥}u|_{¥alpha}^{B}+¥sum_{t=1}^{n}|u_{x_{i}}|_{a}^{B}$
,
,
$|u|_{2+a}^{B}=|u|_{1+¥alpha}^{B}+l¥sum_{=1}^{n}|u_{x_{i}}|_{1+a}^{B}+|u_{t}|_{¥alpha}^{B}$
,
( $0<$ a $<1$ ).
is finite $(q=0,$ , $1+$
We shall say that $u(t, X)$ is in $C^{q}(B)(u¥in C^{q}(B))$ if
.
,
We shall assume that the lateral surface of the non-cylindrical domain $D$
may
where the first boundary problem is studied has the following property:
in such a way that the portion
be covered by a finite number of spheres
admits (for some ) a parametric representation (caret
of cut off by each
omit)
,
x
with the properties:
(i) the function is in $C^{2+a}(T_{v})$ ,
satisfy on
a Lipschitz condition with respect
(ii) the derivatives
.
to the usual metric
prescribed on the normal bounFinally we shall say that a function
$|u|_{q}^{B}$
$¥alpha$
$¥alpha$
$2+¥mathrm{a})$
$S$
$S$
$¥{¥Sigma_{v}¥}$
$S$
$i$
$¥Sigma_{¥gamma}$
$¥equiv$
$¥wedge$
$i=h(t, x_{1’ 1}¥ldots x_{i^{ }},¥cdots, x_{n})$
$(t, x_{1^{ }},¥cdots, x_{i^{ }},¥cdots, x_{n})¥in T_{¥gamma}$
$h$
$¥partial h/¥partial x_{k}$
$T_{¥gamma}$
$¥rho(P, P^{¥prime})$
$¥varphi(t, X)$
Quasilinear Systems
of Parabolic Differential Equations
111
satisfies the condition (F) if it may be prolonged over
function $¥Phi(t, X)$ whose norm
is finite. We set
dary
$¥partial D$
of
$D$
$¥overline{D}$
to a
$|¥Phi|_{2+¥alpha}^{D}$
$|¥varphi|_{2+¥alpha}^{3D}=¥inf|¥Phi|_{2+¥alpha}^{D}$
,
the infimum being taken over all possible prolongations of .
Lemma 1. (Friedman [3]) Consider the first boundary probfem for the
$¥varphi$
linear parabolic equation
in
$u_{t}-¥sum_{i.j=1}^{n}a_{ij}(t, X)u_{x_{i}x_{j}}+¥sum_{i=1}^{n}b_{i}(t, X)u_{x_{i}}+c(t, X)u=f(t, X)$
$D_{0}$
(2. 1)
with the boundary condition
on
D.
Let the following assumptions be made:
1) At $aff(t, X)¥in¥overline{D}$ and for all real -tuples
$u(t, X)=¥varphi(t, X)$
(2. 2)
$¥partial$
$(¥xi_{1^{ }},¥cdots, ¥xi_{n})$
$n$
.
$¥sum_{i.j=1}^{n}a_{ij}(t, X)¥xi_{i}¥xi j¥geqq A_{0}¥sum_{i=1}^{n}¥xi_{i}^{2}$
2)
$¥sum_{j=1}^{n}|a_{ij}|_{a}^{D}+¥sum_{i=1}^{n}|b_{i}|_{¥alpha}^{D}+|c|_{¥mathrm{a}}^{D}¥leqq A_{1}$
$i$
,
3)
,
$|f|_{a}^{D}<+¥infty$
.
The boundary condition ¥
satisfies the condition (F).
Then, there exists one and only one solution $u(t, X)$ to the problem
$(2.2)$ satisfying the folfowing Schauder type
-estimate:
$ varphi(t, X)$
$(2.
1)-$
$(¥mathit{2}+a)$
$|u|_{2+¥alpha}^{D}¥leqq C(|f|_{a}^{D}+|¥varphi|_{2+¥alpha}^{¥partial D})$
where
$C$
(2. 3)
,
is a constant depending only on the constants
$A_{0}$
,
and the geometry
$A_{1}$
of .
$D$
Lemma 2.
(Friedman [4])
under the following hypotheses:
1) At $afl(t, X)¥in¥overline{D}$ and for
Consider the boundary problem
$aff$
real -tuples
$n$
$TAe$
coefficients
and the right member
1)-(2.2)$
$(¥xi_{1}, ¥cdots, ¥xi_{n})$
$¥sum_{i.j=1}^{n}a_{i}j(t, X)¥xi_{i}¥xi_{j}¥geqq H_{0}¥sum_{i=1}^{n}¥xi_{i}^{2}$
2)
$(2.
.
of (2. 1) are
continuous in
$¥overline{D}$
and
satisfy the inequalities
,
$¥sum_{i.j=1}^{n}|a_{ij}|_{¥alpha}^{D}+¥sum_{i=1}^{n}|b_{i}|_{0}^{D}+|c|_{0}^{D}¥leqq H_{1}$
$¥sum_{i,j=1}^{n}||a_{ij}||_{1}^{S}¥leqq H_{2}$
,
where
$||a_{ij}|_{11}^{1^{S}}=|a_{ij}|_{0}^{S}+¥sup_{P.P^{¥prime}¥in S,P¥neq P^{¥prime}}¥frac{|a_{ij}(P)-a_{ij}(P^{¥prime})|}{¥rho(P,P)},$
3)
that
The boundary
function
$¥varphi(t, X)$
is the trace on
$¥partial D$
of
.
a
function
$|¥Psi|_{2}^{D}¥equiv|¥Psi|_{0}^{D}+¥sum_{i=1}^{n}|¥Psi_{x_{i}}|_{0}^{D}+¥sum_{i,j=1}^{n}|¥Psi_{x_{i}x_{j}}|_{0}^{D}+|¥Psi_{t}|_{0}^{D}<+¥infty$
.
$¥backslash $
suck
112
T. KUSANO
In this case, if $u(t,
any $¥delta(0<¥delta<1)$ the
X)$
$(¥mathit{1}+¥delta)$
of
is a solution
the problem
$(2.
1)-(2.2)$
,
for
$the¥overline{n}$
-estimate
(2. 4)
$|u|_{1+¥delta}^{D}¥leqq C(|f_{1_{0}}^{D}|+|¥Psi|_{2}^{D})$
holds true, where $C$ is a constant which depends only on the constants ,
,
$D$
,
and the geometry of
and .
Lemma 3. (Kamynin and Maslennikova [5]) Let there be given the quasilinear parabolic equation
$¥delta$
$H_{1}$
$H_{0}$
$S$
$H_{2}$
(2. 5)
$u_{t}-¥sum_{i,j=1}^{n}a_{ij}(t, X)u_{x_{i}x_{j}}+¥sum_{i=1}^{n}b_{i}(t, X, u)u_{x_{i}}+f(t, X, u, u_{x})=0$
$u_{x}=(u_{x1}, ¥cdots, u_{x_{n}})$
together with the boundary condition (2. 2). We assume that:
1) At $afl(t, X)¥in¥overline{D}$ and for $aff$ reaZ -tuples
$(¥xi_{1^{ }},¥cdots, ¥xi_{n})$
$n$
$¥sum_{i,j=1}^{n}a_{ij}(t, X)¥geqq K_{0}¥sum_{¥iota=1}^{n}¥xi_{i}^{2}$
2)
The
usud metric
3) For
$b_{0}>0)$
$¥sum_{i.j=1}^{n}||a_{i¥dot{¥mathrm{J}}}||_{1}^{S}¥leqq K_{1}$
and any
,
$|u|<+¥infty$
,
with respect to the
and
$p=(p_{1^{ }},¥cdots,p_{n})$
$|¥frac{¥partial f}{¥partial p_{i}}(t, X, u,p)|¥leqq b_{1}$
.
$|u|¥leqq e^{¥gamma T}(¥frac{¥sup_{D}|f(t,X,0,0)|}{¥gamma+b_{0}}+¥sup_{¥partial D}|¥varphi(t, X)|)$
$p=(p_{1^{ }},¥cdots,p_{n})$
and
$S$
.
and
$(¥mathrm{t} , u, 0)¥geqq b_{0}$
$(t, X)¥in¥overline{D}$
$¥frac{¥partial f}{¥partial u}(t, X, u, 0)$
a
:
$aff(t, X)¥in¥overline{D}$
$¥frac{¥partial f}{¥partial u}$
4) For
satisfy a Lipschitz condition on
$a_{ij}(t, X)$
$¥rho(P, P^{f})$
.
, the
$¥frac{¥partial f}{¥partial p_{i}}(t, X, u,p)$
functions
$a_{ij}(t,X)$
,
$b_{i}(t,X, u)$
$(7^{¥prime}>0,$
,
$¥gamma+$
$f(t,X, 0, 0)$ ,
satisfy a Ho fder condition (exponent
$¥mathrm{a}$
, $0<$
with respect to $(t, X)$ and a Ho fder condition (exponent $¥beta,0<¥beta¥leqq 1$ )
with respect to
and .
5) T&e boundary function ¥
satisfies the condition (F).
Under the above assumptions the boundary probfem $(2. 5)-(2.2)$ has at least
one solution $u(t, X)$ which is of class
for any $¥delta(0<¥delta<1)$ and of class
for some $¥epsilon(0<¥epsilon<1)$ .
Remarks. 1) It is easily verified by Lemmas 1 and 2 that the solution guaranteed by Lemma 3 is subject to the estimates:
$<1¥grave{)}$
$u$
$p$
$ varphi(t, X)$
$C^{1+¥delta}(¥overline{D})$
$C^{2+¥epsilon}(¥overline{D})$
$|u|_{1+¥delta}^{D}¥leqq C$
(
(
$|u|_{2+¥mathrm{S}}^{D}¥leqq C^{¥prime}$
where $C$ and
of the domain.
2) If ¥
$C^{¥prime}$
$ beta=1$
$|f(t,X, 0, 0)|_{0}^{D}+|¥Psi|_{2}^{D}$
),
),
$f(t, X, 0, 0)|_{a}^{D}+|¥varphi|_{2+¥alpha}^{¥partial D}$
depend only on the structure of the equation and the geometry
in Assumption 4), then the solution is unique in the class of
Quasilinear Systems
continuous functions in .
(b) An existence theorem.
of Parabolic Differential Equations
113
$D-$
We now turn to the study of the parabolic
system
$L^{k}(u^{k})¥equiv u_{t}^{k}-¥sum_{i,j=1}^{n}a_{ij}^{k}(t, X)u_{x_{i^{X}¥mathrm{j}}}^{k}+¥sum_{i=1}^{n}b_{i}^{k}(t, X)u_{x_{i}}^{k}=f^{k}(t, X, u^{1_{ }},¥cdots, u^{N})$
$(k=1, ¥cdots, N)$
.
The boundary condition is of the first kind:
$u^{k}(t, X)=¥varphi^{k}(t, X)$
$(k=1, ¥cdots, N)$ .
on
We shall list the restrictions placed on the problem $(2. 6)-(2.7)$ :
I. At all
and for all real -tuples
$¥partial D$
$(t, X)¥in¥overline{D}$
. The coefficients
$a_{ij}^{k}(t, X)$
,
$b_{i}^{k}(t, X)$
(2. 7)
$(¥xi_{1}, ¥cdots, ¥xi_{n})$
$¥mathrm{n}$
$¥sum_{i.j=1}^{n}a_{ij}^{k}(t, X)¥xi_{i}¥xi j¥geqq a_{0}¥sum_{i=1}^{n}¥xi_{i}^{2}$
$¥mathrm{I}¥mathrm{I}$
(2. 6)
.
are Holder continuous with exponent
satisfy a Lipschitz condition on
( $0<$ a $<1$ ) in . Moreover the
with respect to the usual metric
.
. The right members
satisfy the condition (W) as
well as a uniform Lipschitz condition with respect to $U=(u^{1}, ¥cdots, u^{N})$ . In addition, the
are Holder continuous with exponent a in
for
each fixed $U=(u^{1}, ¥cdots, u^{N})$ .
. The boundary functions
satisfy the condition (F).
V. There exist two systems of functions
and
$¥overline{D}$
$S$
$a_{ij}^{k}(t, X)$
$¥mathrm{a}$
$¥rho(P, P^{¥prime})$
$f^{k}(t, X, u^{1_{ }},¥cdots, u^{N})$
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$f^{k}(t, X, u^{1_{ }},¥cdots, ¥mathrm{u}^{N})$
$¥overline{D}$
$¥varphi^{k}(t, X)$
$¥mathrm{I}¥mathrm{V}$
$¥tilde{O^{¥mu}}(t, X)=$
$=$
$¥{¥tilde{¥theta}^{k}¥zeta t, X)¥}$
$¥sim O^{¥mu}(t, X)$
are HoIder continuous with exponent a in
and satisfy the inequalities
$¥overline{D}$
$¥{¥underline{¥theta}^{k}(t, X)¥}(k=1, ¥cdots, N)¥mathrm{w}¥mathrm{h}¥overline{¥mathrm{l}}¥mathrm{C}¥mathrm{h}$
are regular in
$D$
in
$L^{k}(¥tilde{¥theta}^{k})-f^{k}(t, X, ¥tilde{O^{¥mu}}(t, X))¥geqq 0¥geqq L^{k}(¥sim¥sim¥theta^{k})-f^{k}(t, X,O^{¥mu}(t, X))$
$¥tilde{¥theta}^{k}(t, X)¥geqq¥varphi^{k}(t, X)¥geqq¥sim¥theta^{h}(t, X)$
on
$¥partial D$
$D_{0}$
.
$(k=1, ¥cdots, N)$
,
(2. 8)
(2. 9)
The following is one of our main existence theorems.
Theorem 2. 1. Under Assumptions
the first boundary problem $(2. 6)-$
$(2.7)$ possesses a unique solution $U(t, X)=¥{u^{k}(t, X)¥}$ $(k=1, ¥cdots, N)$ which is in
¥
.
for any $¥delta(0<¥delta<1)$ and in
for one ¥
$¥mathrm{I}-¥mathrm{V}$
$C^{1+¥delta}(¥overline{D})$
$C^{2+¥mathrm{e}}(¥overline{D})$
Proof. We first
construct a sequence of vector functions
$(t, X)¥}(k=1, ¥cdots, N)$ $(m=0,1,2, ¥cdots)$
$L^{k}(u_{m}^{k})+Mu_{m}^{k}=f^{k}(t,$ $X$
$u_{0}^{k}(t, X)=¥tilde{¥theta}^{k}(t, X)$
$U_{m}(t, X)=$
$¥{u_{m}^{k}$
by solving successively the boundary problems
,
$u_{m}^{k}(t, X)=¥varphi^{k}(t, X)$
where
$ epsilon(0< epsilon<1)$
$u_{m-1}^{1}$
on
$(k=1, ¥cdots, N)$
,
$¥cdots$
,
$u_{m-1}^{N})+Mu_{m-1}^{k}$
$¥partial D$
and
$M$
in
$(k=1, ¥cdots, N)$
$D_{0}$
,
,
(2. 10)
(2. 11)
is the Lipschitz constant of
$f^{k}$
114
T. KUSANO
with respect to ,
Such a sequence is surely yielded by Friedman’s theory
(Lemma 1). It will now be shown that the sequence $¥{U_{m}(t, X)¥}$ thus obtained
is monotonically non-increasing and uniformly bounded. Indeed, noting that
$¥mathrm{U}$
$L^{k}(u_{1}^{k})+Mu_{1}^{k^{ }}=f^{k}(t, X,¥tilde{¥theta}^{1_{ }},¥cdots,¥overline{¥theta}^{N})+M¥tilde{¥theta}^{k}$
in
$¥leqq L^{k}(¥tilde{¥theta}^{k})+M¥overline{¥theta}^{k}=L^{k}(u_{0}^{k})+Mu_{0}^{k}$
on
$u_{1}^{k}(t, X)=¥varphi^{k}(t, X)¥leqq¥tilde{¥theta}^{k}(t, X)=u_{0}^{k}(t, X)$
$D_{0}$
,
$¥partial D$
and applying Theorem 1. 4, we see that
in
$u_{1}^{k}(t, X)¥leqq u_{0}^{k}(t, X)$
$(k=1, ¥cdots, N)$
$¥overline{D}$
.
(2. 12)
Next we obtain
$[L^{k}(u_{m+1}^{k})+Mu_{m+1}^{k}]-[L^{k}(u_{m}^{k})+Mu_{m}^{k}]$
$=f^{k}$
(,
$X$
$t$
,
$u_{m}^{1}$
,
$¥cdots$
,
$u_{m}^{N})-f^{k}(t,$
$X$
,
$u_{m-1}^{1}$
,
$¥cdots$
,
$u_{m-1}^{N})+M(u_{m}^{k}-u_{m-1}^{k})$
$=[f^{k}(t, X, u_{m}^{1}, ¥cdots, u_{m-1}^{k}, ¥cdots, u_{m}^{N})-f^{k}(t, X, u_{m-1}^{1}, ¥cdots, u_{m-1}^{k}, ¥cdots, u_{m-1}^{N})]$
$+[f^{k}$
(,
$t$
,
$X$
$u_{m}^{1}$
,
$¥cdots$
,
$u_{m}^{k}$
$u_{m-1}^{k}(t, X)-u_{m}^{k}(t, X)=0$
If
$¥cdots$
,
in
$+M(u_{m}^{k}-u_{m-1}^{k})]$
and
,
on
$D_{¥mathrm{Q}}$
$¥partial D$
in
$u_{m}^{k}(t, X)¥leqq u_{m-1}^{k}(t, X)$
$u_{m}^{N})-f^{k}(t,$
$X$
,
$u_{m}^{1}$
,
$¥cdots$
,
$u_{m-1}^{k}$
,
$¥cdots$
,
$u_{m}^{N})$
(2. 13)
,
.
$¥overline{D}(k=1, ¥cdots, N)$
, then the last expression in
(2. 13) is non-positive, since the first bracket is non-positive according to the
condition (W) assumed for , and at the same time so is the second in virtue
in $U$ . Hence, by Theorem 1. 4 we have
of the Lipschitz continuity of
$f^{k}$
$f^{k}$
(2. 14)
In order to
proving the monotonicity of the sequence $¥{U_{m}(t, X)¥}$ .
verify that the sequence $¥{U_{m}(t, X)¥}$ is bounded uniformly assume that
in
$u_{m+1}^{k}(t, X)¥leqq u_{m}^{k}(t, X)$
,
$(k=1, ¥cdots, N)$
$¥overline{D}$
$¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{r}¥mathrm{e}¥dot{¥mathrm{b}¥mathrm{y}}$
$u_{m}^{k}(t, X)¥geqq¥underline{¥theta}^{k}(t, X)$
in
$¥overline{D}$
$(k=1, ¥cdots, N)$
.
(2. 15)
Since, arguing as in handling (2. 13),
$[L^{k}(u_{m+1}^{k})+Mu_{m+1}^{k}]-[L^{k}(¥sim¥sim¥theta^{k})+M¥theta^{k}]$
$¥geqq[f^{k}(t, X, u_{m}^{1}, ¥cdots, ¥theta^{k}¥sim’¥cdots, u_{m}^{N})-f^{k}(t, X_{¥sim¥sim},¥theta^{1}, ¥cdots, ¥theta^{k_{ _{ }}},¥cdots,¥underline{¥theta}^{N})]$
$+[f^{k}$
(,
$t$
$X$
,
$u_{m}^{1}$
,
$¥cdots$
,
$u_{m}^{k}$
$+M(u_{m}^{k}-¥sim¥theta^{k})]¥geqq 0$
and
,
$¥cdots$
in
,
$u_{m}^{N})-f^{k}(t,$
$D_{0}$
,
$X$
,
$u_{m}^{1}$
,
$¥cdots$
,
$¥theta^{k}¥sim’¥cdots$
,
$¥iota/_{m}^{N})$
Quasilinear
$Syste¥prime ns$
115
of Parabolic Differential Equations
on
$u_{m+1}^{k}(t, X)=¥varphi^{k}(t, X)¥geqq¥sim¥theta^{k}(t, X)$
$(k=1, ¥cdots, N)$ ,
$¥partial D$
another use of Theorem 1. 4 then yields
$u_{m+1}^{k}(t, X)¥geqq¥sim¥theta^{k}(t, X)$
in
.
$(k=1, ¥cdots, N)$
$¥overline{D}$
This establishes the uniform boundedness of the sequence $¥{U_{n}(t, X)¥}$
ximate solutions. (Note that (2. 15) holds for $m=0$ (Assumption
The proof of the theorem will be concluded if one verifies that
$¥mathrm{V}).$
$¥{u^{k}(t, X)¥}$
, where
$(t, X)=¥lim_{m¥rightarrow¥infty}u_{m}^{k}(t, X)$
$(k=1, ¥cdots, N)$
(2. 16)
of appro)
$U(t, X)=$
becomes the desired
,
solution of the problem in question. First we apply Lemma 2 to the problem
(2. 10)?(2. 11) to derive for any $¥delta(0<¥delta<1)$
,
(2. 17)
$|u_{m}^{k}|_{1+¥delta}^{D}¥leqq C(^{1}f_{m-1}^{k}|_{0_{1}}^{D}+M|u_{m-10}^{k1^{D}+}|¥Phi^{k^{¥mathrm{I}}}|_{2}^{D})$
where we have set
$f_{n-1}^{k}(t, X)=f^{k}(t, X, U_{m1}¥_(t, X))$
and
$¥Phi^{k}$
is a prolongation of
. The constant depends only on the structure of
with finite
In view of the uniform boundedness of
and the geometry of $D$ and .
$(t, X)¥}$ we infer from (2. 17) that the quantities
are bounded by a con-
$¥varphi^{k}$
over
$¥overline{D}$
$L^{k}$
$C$
$|¥Phi^{k}|_{2}^{D}$
$¥{U_{rn}$
$¥delta$
$|u_{m}^{k}|_{1+¥delta}^{D}$
stant independent of
$m$
. In particular,
$|u_{m}^{k}|_{¥alpha}^{D}$
are bounded independently of
Now we apply Lemma 1 to the problem (2. 10)?(2. 11), thus obtaining the
$¥mathrm{a})$
$m$
.
$(2+$
-estimates:
$|u_{m}^{k}|_{2+¥mathrm{a}}^{D}¥leqq C^{¥prime}(|f_{m-1}^{k}|_{¥alpha}^{D}+M^{¥mathrm{I}}|u_{m-1}^{k}|_{a|}^{D|}+¥varphi^{k}|_{2+¥alpha}^{3D})$
where
$C^{¥prime}$
depends only on the structure of
we can assert that
$|u_{m}^{k}|_{2+¥alpha}^{D}$
$L^{k}$
(2. 18)
,
and the geometry of
are bounded independently of
$m$
$D$
, whence
Consequently, we
.
together
can extract from $¥{U_{m}(t, X)¥}$ a subsequence converging uniformly in
necessarily
limit
function
which
with the first , and second -derivatives to a
Now it will be almost obvious that
agrees with $U(t, X)$ constructed above.
$U(t, X)$ becomes the solution to the boundary problem $(2. 6)-(2.7)$ and has the
required smoothness property. That the solution is unique follows from the
considerations in Part I. Thus the theorem is completely proved.
(c) Another existence theorem. In this paragraph we shall indicate a
slight generalization of the preceding Theorem 2. 1. Let us consider the quasilinear parabolic system of the form
$¥overline{D}$
$x$
$t-$
$¥mathrm{x}$
$¥Lambda^{k}(u^{k})¥equiv u_{t}^{k}-¥sum_{i.j=1}^{n}a_{ij}^{k}(t, X)u_{x_{i}x_{j}}^{k}+¥sum_{¥iota=1}^{n}b_{i}^{k}(t, X, u^{k})u_{x_{i}}^{k}+c^{k}(t,$
$=f^{k}(t, X, u^{1_{ }},¥cdots, u^{N})$
together with the boundary condition (2. 7).
Theorem 2. 2. Let the following conditions be
$X$
,
$u^{k}$
$(k=1, ¥cdots, N)$
satisfied.
,
$u_{x}^{k})$
(2. 19)
116
T. KUSANO
I.
At
$aff(t, X)¥in¥overline{D}$
for
and
all real -tuples
$n$
$¥sum_{i,j=1}^{n}a_{ij}^{k}(t, X)¥xi_{i}¥xi_{¥dot{f}}¥geqq a_{0}¥sum_{i=1}^{n}¥xi_{i}^{2}$
$¥mathrm{I}¥mathrm{I}$
.
.
$¥mathrm{I}¥mathrm{I}¥mathrm{I}$
$¥rho(P, P^{¥prime})$
For
functions
$(t, X)¥in¥overline{D}$
$a_{ij}^{k}(t, X)$
,
with respect to the
$S$
.
$(t, X)¥in¥overline{D}$
,
and any
$|u^{k}|<+¥infty$
and
$¥frac{¥partial c^{k}}{¥partial u^{k}}(t, X, u^{k}, 0)¥geqq b_{0}$
.
.
Th $ea_{ij}^{k}(t, X)$ satisfy a Lipschitz condition on
usual metric
. For
$¥mathrm{I}¥mathrm{V}$
$(¥xi_{1^{ }},¥cdots, ¥xi_{n})$
,
(
$|u^{k}|¥leqq¥gamma^{k}$
$b_{i}^{k}(t, X, u^{k})$
,
$¥gamma^{k}$
$p=(p_{1^{ }},¥cdots,p_{n})$
$|_{¥partial}^{¥partial}-¥frac{c^{k}}{p_{i}}(t, X, u^{k},p)|¥leqq b_{1}$
: arbitrary) and any
.
$p=(p_{1^{ }},¥cdots,p_{n})$
and
$c(t, X, 0, 0),-¥partial¥frac{¥partial c^{k}}{u^{k}}(t, X, u^{k}, 0)$
, the
$¥frac{¥partial c^{k}}{¥partial p_{i}}(t,X, u^{k},p)$
satisfy a Ho fder condition (exponent , $0<$ a $<1$ ) with respect to $(t, X)$ , a Lipschitz condition with respect to
an a Ho lder condition (exponent , $0<¥beta¥leqq 1$)
with respect to .
V. The functions $f^{k}(t, X, u^{1}, ¥cdots, u^{N})$ satisfy the condition (W) as all as
a uniform Lipschitz condition with respect to $U=(u^{1_{ }},¥cdots, u^{N})$ . Moreover
are
$U$ .
Ho fder continuous (exponent a) in
each
for
fixed
. The boundary functions
satisfy the condition (F).
I. There exist a pair of functions
$¥sim OH(t, X)=$
$¥mathrm{a}$
$d$
$u^{k}$
$¥beta$
$p$
$f^{k}$
$¥overline{D}$
$¥varphi^{k}(t, X)$
$¥mathrm{V}¥mathrm{I}$
$¥tilde{O^{¥mu}}(t, X)=$
$¥mathrm{V}$
$¥{¥tilde{¥theta}^{k}(t, X)¥},$
which are Ho fder continuous (exponent a) in
and satisfy the inequalities
$(k=1, ¥cdots, N)$
$¥{¥sim¥theta^{k}(t,X)¥}$
$¥overline{D}$
, are regular in
$¥Lambda^{k}(¥tilde{¥theta}^{k})-f^{k}(t, X,¥overline{O^{¥mu}}(t, X))¥geqq 0¥geqq¥Lambda^{k}(¥sim¥sim¥theta^{k})-f^{k}(t, X, O^{¥mu}(t, X))$
$¥tilde{¥theta}^{k}(t, X)¥geqq¥varphi^{k}(t, X)¥geqq¥sim¥theta^{k}(t, X)$
on
$¥partial D$
in
$D_{0}$
$(k=1, ¥cdots, N)$
.
$D$
(2. 20)
(2. 21)
Under these assumptions the boundary problem (2. 19)?(2. 7) possesses
unique solution $U(t, X)=¥{u^{k}(t, X)¥}$ $(k=1, ¥cdots, N)$ which is in
for any
$¥delta(0<¥delta<1)$ and is in
for some $¥epsilon(0<¥epsilon<1)$ .
Proof. This follows the same pattern as the proof of Theorem 2. 1 with
some necessary modifications. We define a sequence of approximate solutions
$a$
$C^{1+¥delta}(¥overline{D})$
$C^{2+¥mathrm{P}}(¥overline{D})$
$U_{m}(t, X)=$ $¥{u_{m}^{k}(t, X)¥}(k=1, ¥cdots, N)$
$¥Lambda^{k}(u_{m}^{k})+Mu_{m}^{k}=f^{k}(t,$
$X$
$u_{m}^{k}(t, X)=¥varphi^{k}(t, X)$
$(m=0,1,2, ¥cdots)$
,
$u_{m-1}^{1}$
on
,
$¥cdots$
,
$¥partial D$
by the iteration scheme:
$u_{m-1}^{N})+Mu_{m-1}^{k}$
in
$(k=1, ¥cdots, N)$
,
$D_{0}$
,
(2. 22)
(2. 23)
$(k=1, ¥cdots, N)$ and $M$ denotes the Lipschitz constant of
where
. Lemma 3 due to Kamynin and Maslennikova insures legitimacy of the above
iteration procedure.
It turns out exactly as in Theorem 2. 1 that the sequence $¥{U_{m}(t, X)¥}$ is
$u_{0}^{k}(t, X)=¥tilde{¥theta}^{h}(t, X)$
$f^{k}$
uniformly bounded and monotonically decreasing in
it remains only to show that the limit function
$(k=1, ¥cdots, N)$
$u^{k}(t, X)=¥lim_{m¥rightarrow¥infty}u_{m}^{k}(t, X)$
117
of Parabolic Differential Equations
Quasilinear Systems
$¥overline{D}$
.
$U(t,
To complete the proof
X)=¥{u^{k}(t, X)¥}$ , where
, is actually the solution of the problem
considered. But this follows readily by applying Friedman’ a priori estimates
to the problem (2. 22)?(2. 23). The details need not be described here.
Remark, We can exhibit another way of establishing that the limit function $U(t, X)=$ $¥{u^{k}(t, X)¥}$ $(k=1, ¥cdots, N)$ is the solution sought of the problem
.
(2. 19)?(2. 7). It resembles the method of Maslennikova
$U_{m}(t,
X)$
-norm by
uniformly
in
the
sequence
is
bounded
Since the
Lemma 2, there is a subsequence of $U_{m}(t, X)$ which we shall denote again by
$¥mathrm{s}$
$[7¥mathrm{j}$
$C^{1+¥delta}(¥overline{D})$
$U_{m}(t, X)$
such that
$u_{m}^{k}(t, X)¥rightarrow u^{k}(t, X)$
and
$¥partial u_{m}^{k}(t, X)/¥partial x_{j}¥rightarrow¥partial u^{k}(t, X)/¥partial Xj$
formly in $¥overline{D}(k=1, ¥cdots, N;j=1, ¥cdots, n)$ .
Now let $Z(t, X)=¥{z^{k}(t,
$N)$ be the solution of the system of linear parabolic equations
X)¥}(k=1,$
$M^{k}(z^{k})¥equiv z_{t}^{k}-¥sum_{i.j=1}^{n}a_{ij}^{k}(t, X)z_{x_{¥dot{¥mathrm{t}}}x_{j}}^{k}+Mz^{k}=-¥sum_{¥iota=1}^{n}b_{i}^{k}(t, X, u^{k})u_{x_{i}}^{k}$
$-c^{k}$
(,
$t$
$X$
,
$u^{k}$
, $u_{x}^{k})+f^{k}(t, X, u^{1_{ }},¥cdots, u^{N})+Mu^{k}$ in
uni$¥cdots$
,
(2. 24)
$D_{0}$
satisfying the boundary condition (2. 7).
Subtracting (2. 22) from (2. 24) we
obtain
$M^{k}(z^{k}-u_{m}^{k})=F_{m}^{k}(t, X)$
,
in
and $z^{k}-u_{m}^{k}=0$ on
$¥partial D$
$D_{0}$
where we have set
$F_{m}^{k}(t, X)=¥sum_{i=1}^{n}[b_{i}^{k}(t, X, u_{m-1}^{k})u_{(m-1)x_{i}}^{k}-b_{i}^{k}(t, X, u^{k})u_{x_{i}}^{k}]$
$+c^{k}$
(,
(,
$-f^{k}$
$t$
$t$
,
$X$
$X$
$u_{m-1}^{k}$
,
,
$u_{m-¥iota}^{1}$
$u_{(m-1)x}^{k})-c^{k}(t,$
,
$¥cdots$
,
$X$
$u^{k}$
,
$u_{x}^{k})+f^{k}(t, X, u^{1_{ }},¥cdots, u^{N})$
, $u_{m-1}^{N})+M(u^{k}-u_{m-1}^{k})$ .
By the standard maximum principle it follows that
$-u_{m}^{k}(t, X)|¥leqq$ const .
$¥mathrm{m}¥frac{¥mathrm{a}}{D}¥mathrm{x}|F_{m}^{k}(t, X)|$
$¥mathrm{m}_{¥frac{¥mathrm{a}}{D}}¥mathrm{x}|z^{k}(t, X)$
$(k=1, ¥cdots, N;m=1,2, ¥cdots)$
Since
$F_{m}^{k}(t, X)¥rightarrow 0$
uniformly in
$¥overline{D}$
as
(2. 25)
.
$m¥rightarrow¥infty(k=1, ¥cdots, N)$
, it follows that
$(k=1, ¥cdots, N)$
throughout
X)¥equiv Z(t, X)$ is indeed the solution of the problem
$z^{k}(t, X)¥equiv u^{k}(t, X)$
$¥overline{D}$
and consequently that $U(t,
(2. 19)?(2. 7) in view of (2. 24).
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T. KUSANO
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