Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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360
B.
G.
Konopelchenko
and
C.
Rogers
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1606, 1989.
Nonlinear
React
ion-Diff
usion
Systems
R.
H.
Martin,
Jr.
North
Carolina
State
University
M.
Pierre
Universith
de
Nancy
I
1.1
Introduction
Suppose
m
is a positive integer,
R+
=
[0,00),
and
F
=
(F;)?
is a locally
Lipschitz function from
RT
into
Rm
that is quasi-positive:
Under these circumstances the ordinary differential equation
has
a
unique, noncontinuable, nonnegative solution
y(-
;
C)
on an interval of
the form
[0,
b()
where
0
<
b(
5
00
for each
c
E
q.
Furthermore, if
b(
<
00
then
ly(t;()I
-+
00
as
t
Suppose
now that
S2
is a bounded domain in
RN
with smooth boundary
60.
Let
6,
denote the outward normal derivative on
60
and let
A
denote the
Laplacian on
R.
In conjunction with
(ODE)
consider the reaction-diffusion
6~.
Nonlinear
Equations in the Applied Sciences
363
Copyright
0
1992
by
Academic Press, Inc.
All rights
of
reproduction
in
any
form
reserved.
ISBN
0-12-056752-0
364
R.
H.
Martin
and
M.
Pierre
system
ut
=
DAu
+
F(u)
on
Q
x
(0,
oo)
B[u]
=
0
on
8Q
x
(0,oo)
u=z
on
nx
(0)
(RDS)
where
D
=
diag
(dl
,
..
.
,
d,)
is
a diagonal matrix with
di
>
0
for each
i, u
=
(ui)?, Au
=
(Au;)?,
and
B[u]
=
(Bi[ui])y
is
a
(possibly nonhomogeneous)
boundary operator of the form
Bi[ui]
=
X;aVui
+
(1
-
Xi)(~i
-
~i).
with
O<Xi<l
and
aj10
for
i=l,
...,
rn.
(1.2)
{
(1.3)
{
The initial function
z
=
(z;)?
is assumed nonnegative and continuous
on
n.
The local Lipschitz assumption on
F
may be used to show that (RDS)
has a unique noncontinuable solution
u(.
,
z)
on
Ti
x
[O,T(z)),
where
0
<
T(z)
<
00.
Furthermore, if
T(z)
<
00
then it must be the case that
both
lim
t+T(t)-
where
11
-
11,
denotes the essential supremum over
Q,
max{llui(.,t)l(,;
i
=
I,.
.
.
,
rn)
=
00
and
for some
r
>
1.
Since (RDS) can be viewed
as
a
semilinear differential equation involving an
analytic semigroup and a relatively continuous ,perturbation in any of the
spaces
Lp(n)'",
1
5
p
<
00,
the assertions in
(1.3)
and
(1.4)
may be deduced
from Ball
[2,
Theorem
3.11,
for example. Note further that the quasi-positive
assumption
(1.1)
on
F
and the maximum principle imply that each solution
to (RDS) remains nonnegative
so
long
a8
it exists.
The fundamental consideration in this paper is the following:
under what circumstances does global existence
of each solution to (ODE) [i.e.,
b(
=
00
for all
C]
imply global existence of each solution to (RDS)
[i.e.,
T(z)
=
00
for all
z]?
(1.5)
Nonlinear Reaction-Diffusion Systems
365
For
m
=
1
global existence for
(ODE)
always implies the same for
(RDS).
This can easily be established using the maximum principle and differential
inequalities. It also carries over to the system case
m
1
2
whenever
F
is
quasimonotone:
vk
E
{
1,
.
. .
,
m},
Fk(<l,
. . . ,
<,,,)
is nondecreasing in
&,
i
#
k.
These techniques are indicated in
[16],
for example.
classes of systems. Our purpose is to give some results in that direction.
A
general answer to
(1.5)
is unknown even when restricted to some basic
1.2
A
Negative Result
It is possible to destroy global existence for solutions to an ordinary dif-
ferential equation system by the addition of diffusion in one component.
Following Morgan
[18],
the nonnegative solutions to
exist
for
all
t
2
0
(see
[IS]).
However, the mixed diffusion and ordinary
system
(2.2)
where
uo
is
Ut
=
lr-ZU,,
+
u
u,
=
0
on
{0,11
on
~t
=
U’
-
13
-
UIU~
U(E,
0)
=
6cos(lrz),
u(z,
0)
=
uo(t)
0
<
E
<
1
smooth and
u,(
1/3)
>
0,
has finite blow-up time. For note that
U(E,
t)
6
cos
AE,
and hence
u(
1/3,
t)
3
and
ut(1/3,
t)
=
[41/3,
t)lZ
Thus
u(1/3,
t)
blows up
as
t
T
1/u0(1/3).
A
similar example involving an
important scientific model
is
given in the next section.
Remark
2.1
It is interesting to note that if the associated vector field in
366
R.
H.
Martin
and
M. Pierre
then the resulting system has finite time blow-up for all
6
2
0
and all
u,,
u,
with
u,
=
3
and
u,
>
0.
We now give an example of
a
system where both of the diffusion ce
efficients are positive and
a
solution blows up in finite time even though
the associated ordinary equation has the global existence property for all
nonnegative initial data.
Consider the reaction-diffusion system
ut
=
aAu
+
2aN
ut
=
Au
+
2(N
-
t)+u2
-
811
-
u1u3
(2,
t)
E
x
(-1,oo)
(2,
t)
E
aR
x
(-1,oo)
(2.3)
I
u=o
,
u
=
(1+tZ)-'
where
R
is
the unit ball in
RN
and
(r)+
=
r
if
r
2
0,
(r)+
=
0
if
r
<
0.
Also,
consider the associated ordinary differential system
For convenience we use the time interval
(-1,oo)
instead of
(0,~).
Proposition
2.1
Assume that
(2.5)
a
>
(N
+
1)'/4N.
Then the solution to
(2.4)
exists on
[-1,oo)
for
all
initial data
C1,
c2
2
0.
However,
u(r,t)
=
1
-
r
2
,
u(r,t)
=
(t2
+
r2)-l
(2.6)
is
a
nonnegative solution
to
(2.3)
on
II
x
(-1,
0)
which blows up in
Lm(R)
as
t
0.
More precisely
if
N
>
4,
then
(u,
u)
is
a
weak solution
to
(2.3)
on the whole set
R
x
(-1,
N)
but blows up in
LN12(R)
at
t
=
0.
if
1
5
N
<
4,
then
(u,
u)
cannot be extended into
a
weak solution
to
(2.3)
around
t
=
0.
Remarks.
distributions.
(i)
(ii)
-
By weak solution, we mean that
(2.3)
holds in the sense of
Above example shows
at
the same time that
Nonlinear Reaction-Diffusion Systems
367
(a)
(b)
(c)
global existence for
all
nonnegative initial data in
O.D.E.
(2.4)
does
weak solutions of
R.D.S.
can be bounded (and even
Coo)
at
two
solutions
of
R.D.S.
might exist on
x
(tl, t2)
\
{(lo,
to)}
and blow
not imply global existence in associated
R.D.S.
(2.3).
different times
tl,
t2,
but not in
(tl
,
t2).
up
at
(lo,
to)
E
R
x
(t~,
t2)
only.
The latter pathological situations (b) and (c) can
also
happen for equations
(rn
=
1).
Indeed
v
as
defined in
(2.6)
satisfies
Vt
=
AV
+
2(N
-
t)v2
-
8r2v3
21
=
(1
+
t2)-'
(x,
t)
E
R
x
(-1,
N)
(x,
t)
E
8R
x
(-1,
N).
{
Proof of Proposition
2.1
note that
To check that
(2.6)
defines
a
solution of
(2.3),
ut
=
0,
AU
=
-2N
Vt
=
-
2t -2xt -2
SX?
(t2
+
r2)2
1
Vz*
=
(t2
+
r2)2'
Vz:z*
=
(t2
+
r2)2
+
(t2
+
r2)3
Hence
-2t
2N
-
8r2
=
2(N-t)v2-8(1-u)v3.
(t2
+
,212
+
(t2
+
r2)2
(t2
+
r2)3
(2.7)
v~-Av
=
This computation is valid for
t
E
(-1,
0).
It is
also
valid in the sense of
distributions on
R
x
(-1,
m)
if
N
>
4,
since
all
quantities in
(2.7)
are then
locally integrable on
R
x
(-1,m).
However, note that the right-hand side
of
(2.7)
is not locally integrable if
1
5
N
5
3
which proves (ii).
Now, suppose for contradiction that
a
solution
(yl,
y2)
to
(2.4)
blows up
at
a
finite time
t*
>
-1.
Since
(2.8)
yl(t)
=
2~N(t
+
1)
+
C1
on
(-1,
t*),
it
is
the case that
y2(t)
-+
00
as
t
T
t*.
Furthermore
it
is
clear from the
form
of the vector field in
(2.4)
that
-1<t*<N
and
l-yl(t)--*O
as
tTt'.
Hence, using
(2.8)
(2.9) 2~N(t*
+
1)
+
=
1.
368
R.
H.
Martin
and
M. Pierre
It follows from
(2.4), (2.8),
(2.9)
that
(2.10)
yk
=
2(N
-
t)yz
-
16aN(t*
-
t)yz.
In particular
y;
5
2(N
+
l)yi
so
that for all
to
<
t
<
t*
y2(to)-l
-
yz(t)-l
I
2(N
+
l)(t
-
to)
and by letting
t
T
t*
On the other hand,
y’(to)
>
0
for
to
close enough to
t*
so
that
(2.10)
implies
(2.12)
2(N
-
to)
2
16aN(t*
-
to)y2(to).
Combining
(2.11)
and
(2.12)
leads to
2(N
-
to)
2
8aN/(N
+
1)
==+
(N
+
1)’
2
4aN.
This contradicts
(2.5)
and completes the proof.
Remark.
has the solution
y(t)
=
(t*
-
t)-’.
This justifies the need of above proof.
Note that equation
y’
=
2y2
-
(t’
-t)y3
which
is
similar to
(2.10)
1.3
A
Typical
Example
We now consider a very simple
2x 2
system of
P.D.E.
for which the associated
O.D.E.
has the global existence and positivity properties but where usual
arguments like maximum principle
or
invariant regions techniques do not
apply:
~t
=
dlAu
-
~h(v)
on
52
x
(0,~)
~lt
=
dzA~
+
~h(~l)
(3.1)
subject to the boundary conditions of the form
X1a”U
+
(1
-
X1)(U
-
a)
=
0
xzavv
+
(1
-
X,)(v
-
p)
=
0
on
852
x
(0,~)
(3.2)
Nonlinear Reaction-Diffusion
Systems
369
where
a,
p
2
0,
0
5
XI,
Xz
5
1
and
h
:
R+
-+
R+
is
C’
with
h(0)
=
0.
To
minimize subscripts, we use
(u,
v)
instead
of
(u1,
u~).
Here global nonnegative solutions exist
for
the corresponding
O.D.E.
(3.3)
Y:
=
-Ylh(Yz), Yl(0)
=
Cl
10
1
Yi
=
Ylh(Y2)r YZ(0)
=
cz
2
0.
However, global existence in (3.1),
(3.2)
is
not yet completely understood.
Let
us
first indicate some easy facts.
Note that condition (1.1) holds
for
the nonlinear terms
so
that local
existence of nonnegative solutions to (3.1),
(3.2)
is
assured
as
explained in
the introduction.
If
dl
=
dz
and
X1
=
Xz,
then
u
+
v
satisfies the heat equation with
In
all
cases, since
-uh(v)
5
0,
the maximum principle implies
u
is
uni-
standard boundary conditions. Therefore global existence
is
obvious.
formly bounded by
11u(.,
O)llm.
As
a
consequence if
(3.4)
h(v)
5
Lv
+
M
vv
2
0
for some
L,
M
>
0,
then from Gronwall’s inequality we get
a
uniform bound
on
v(z,
t)
for
(2,
t)
E
Rx
(0,
T)
and
for
all
T.
Therefore nonnegative solutions
exist globally.
Outside these situations, the question becomes more complicated even
when we have obvious uniform
a
priori
L’(R)-estimates. For example when
Let
us
summarize some recent results.
R
1.
Assume polynomial growth for
h:
(3.5)
3p
>
1,
3L,
M
>
0
h(v)
5
LvP
+
M
vv
>
0
and
Then (3.1),
(3.2)
has global nonnegative classical solutions.
When
p
is
small enough this can be proved using “bootstrap? arguments
deducing Lm-estimates from L’-estimates (see [l]).
A
proof
for
all
p
was