Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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390
R.
H.
Martin
and
M. Pierre
Proof.
We have
vn(t)
=
S2(t)v,"
+
S2(t
-
S)gn
(un(S),
vn(s))
ds
It
(8.27)
where
(u:,
v,")
converges to
(u,,
v,)
in
L'(i-2)
and where
fn(un,
vn),
gn(un,
vn)
are bounded in
L1(Q~)
for
all
T
independently of
n
by Lemma 8.3. There-
fore, by Lemma 8.4
(un,
v,)
is precompact in
L'(&)
for
all
T.
Using a
sequence
Tk
with limk,,
Tk
=
00
and a diagonal process, we obtain (8.25).
Comments
are
very
general.
They are based only on the facts that
(8.28)
It
is
important to notice here that the arguments given above
the semigroups
Sr(t)
are nonexpansive and compact in
L'(i-2)
As
a
consequence the conclusion (8.25) (and therefore the existence of the
limit
(u,
v))
is
valid under assumptions (8.28), (8.29) even if the semigroups
are nonlinear. Indeed, Lemma 8.4
is
valid for compact nonlinear semigroups
of contractions if
w
is
defined
as
the "weak solution" (in the sense of the
nonlinear semigroup theory) of
(8.30)
12-
Aw3g
on
(0,
a)
w(0)
=
w,
where
A
is
the corresponding nonlinear dissipative operator (see Baras
[3]
for
a
proof of this compactness result).
All
the details of this generalization can be found in
[13].
At this point we have reached the main difficulty of the problem. We
need to prove that
(u,
v)
is
a
solution to our problem (8.6). Since
(unk,
vn,)
converges in
L1(Q~)
to
(u,
v),
up to extracting a new subsequence, we can
assume that the convergence also holds almost everywhere. Since
f
and
g
are continuous with respect to
u
and
v
and by construction of
fn,
gn
(see
(8.9), (8.10)), we can assume
Nonlinear Reaction-Diffusion Systems
39
1
However this
is
not sufficient to pass to the limit in
(8.26), (8.27).
We need
that the convergence in
(8.31)
holds in
L’(QT)
for
all
T
(recall the continuity
result in Lemma
8.4).
The gap can be filled in by the next lemma.
Lemma
8.6
(8.32)
u,
-+
u
a.e.
on
QT
(8.33)
Then
u,
converges
to
u
in
L’(QT).
Let
an
be
a
sequence in
L’(QT)
and
u
in
L1(Q~)
such that
u,
is uniformly integrable in
L’(QT).
Proof
This
is
a
classical result (see a proof in
[21]).
Recall that
(8.33)
means that for
all
E
>
0,
there exists
6
>
0
such that
(I<
c
QT
measurable, meas
I<
<
6)
j
J,
lunl
<
6
Vn.
Our
goal
is
now to show that
f(u,,
v,),
g(un,
u,)
are not only bounded
in
L1(Q~)
but uniformly integrable. We cannot accomplish this without an
extra structure assumption on
f,
g,
namely
(8.34)
%,
M2
2
0
This
is
a
nothing but assuming a “triangular” structure
for
the nonlinear
part
as
in Sections
4-5
(in other words we
ask
that
a
=
1,
/3
=
0
in
(8.16).
Theorem
8.7
Assume moreover
(8.34).
solution
(u,
v)
to
(8.6).
f(u,
V)
I
Lz(u
+
v)
+
Mz.
Let us make the same assumptions as in Proposition
8.1.
Then,
for
all
u,,
v,
as in
(8.7),
there exists
a
The proof of this theorem requires also the following result
Lemma
8.8
in
L’(U).
Then the following statements are equivalent:
(8.35)
u,
is
uniformly integrable in
L’(U)
Let
U
be
a
bounded open set in
Rd
and let
a,
be
a
sequence
There exists
J
:
(0,~)
--*
(0,~)
with
J(Ot)
=
0
and
(a)
J
is
convex,
J’
is
concave,
J’
2
0
I
I
392
R.
H.
Martin
and
M. Pierre
Proof
This
lemma
is
classical
except perhaps for the possibility of choosing
J'
concave
(see [15] for a proof).
Proof
of
Theorem
8.7
By Proposition 8.1,
we
have
existence
of
the
solution
(u,,,
vn)
to the approximate problem (8.12)
(or
(8.26), (8.27)). By
Proposition 8.5, up to extracting
a
subsequence,
we
can assume that
(8.37)
lim(un,vn)
=
(u,v)
(8.38)
(8.39)
a.e.
z,t
E
R
x
(0,oo)
and in
L1(Q~)
for all
T
n-o3
lim
fn(un,Vn)
=
f(u,~)
a.e.
~,t
E
R
x
(0,~)
lim
gn(unrvn)
=
g(u,v)
a.e.
z,t
E
R
x
(0,~).
n+
03
n-oo
We
now
fix
T
>
0
arbitrary and
J
as
in
(8.36) with (8.36~) replaced by
This
is
possible by Lemma 8.8 since
un+vn
converges
in
L1(Q~)
and 2uE+vF
converges in
L'(R).
Next we set
(8.41)
where
J'
is
the conjugate function
of
J.
We easily check that
j
satisfies
(8.36a), (8.36b) and
j(r)
=
lr
min
(J'(s),
(J*)-'(s))
ds
(8.42)
Vr
2
0
j(r)
5
J(r) J*(j'(r))
5
r.
Our
goal
is
to show that
(8.43)
suPJ,,j1(2un
+
vn)(Ifn(Un, vn)l+ lgn(un, vn)l)
<
00.
Let
us
first indicate how the proof of Theorem 8.7 can then be finished.
on
QT.
Indeed let
E
>
0
and
QT
be measurable. Then
Estimate (8.43) implies the uniform integrability
of
fn(un, Vn), gn(un, Vn)
SUP
Ifn(z,tl
r,
s)l
dz
di
Kn[2un+vn<~]
o<r,s<p
(fn(un,.vn)I
=
11
+
12
J,
lfn(un, vn)l<
J
Nonlinear Reaction-Diffusion
Systems
393
The second integral
12
is estimated by
Thanks to
(8.43)
this can be made
less
than
$
by choosing
p
=
p(~)
large
enough depending only on
E.
For
I1,
we notice that by the assumptions
on
f,
g
(see
(4.8),
(4.9))
and the choice of
fnl
gn
(see
(8.9), (8.10)),
SUP
Ifn(t,t,r,s)I
I
M
+~PZ~(P)
olr,s<cr
As
a
consequence
11
5
[M
+
2p(E)~(p(E))]
meas
I<
<
5
if meas
I<
<
c/2[~
+
2p(E)~(p(c))].
This proves uniform integrability of
fn(un,
vn)
and similarly of
gn(unl
vn).
Using now
(8.38), (8.39)
and Lemma
8.6,
we obtain that
fn(un,
vn), gn(unj
Vn)
converge in
L1(Q~)
to
fn(u,
v),
gn(u,
v).
Then
the
proof
of
Theorem
8.7
is
finished by passing to the limit in the approximate equations
(8.26), (8.27).
2
Let
us
now prove
(8.43).
We set
Rn
=
M2
+
&(un
+
vn)
-
fn(un,
vn)
2
0
by
(8.34)
Sn
=
M
+
L(un
+
vn)
-
fn(uni
vn)
-
gn(unr
vn)
2
0
by
(4.9).
Combining the two first equations in
(8.12),
we obtain
(8.45) (2un+vn)t-( 2A’un
+A2vn)+R,
+Sn
=
M+M2+(
L1
+L~)(un
+vn).
We multiply by
j’(2u,
+
v,)
and integrate over
QT
to obtain
J,
j(2un
+-
vn)(T)
-
~1
+
j’(2Un
+
vn)(Rn
+
sn)
(8.46)
LT
=
J2
+
1
j(24
+
v,”)
where
j’(2~n
+
vn)(2A1un
+
A2vn)
J1
=
LT
J2
=
JQT
j’(2un
+
vn)
[M
+
MZ
+
(L
+
h)(un
+
vn)].
To bound
J2
we use that (see
(8.41), (8.42))
(8.47)
j’(r)
*
s
5
~(s)
+
J*(j’(r))
I
J(s)
+
r.
394
R. H.
Martin
and
M.
Pierre
Therefore
52
5
LT
J(M
+
~2)
+
2un
+
vn
+
(~1+
~2)
[J(un
+
vn)
+
2un
+
On]
which
is
bounded independently of
n
by the choice
of
J
(see
(8.40))
and by
Lemma
8.2.
If we can control
J1,
then from
(8.46),
the above estimate of
J2
and
(8.40)
again, we obtain
our
estimate
(8.43)
since, by definition
of
Rn,
Sn
lfnI+
Ignl
I
Rn
+
25,
+
M
+
2M2
+
(L
+
2L~)(un
+
vn)
and by
(8.47)
(8.48)
j’(2un
+
vn)(un
+
vn)
5
J(un
+
vn)
+
21-4,
+
vn.
Let
us
estimate
J1.
We have, after integration by parts
According to assumptions
(8.2)
and the boundary conditions in
(8.12),
the
first integral
is
nonnegative.
So
are the integrals involving
sib!
combined
with the last integral. Now, we use ellipticity of
a:j
in the second integral to
get
j/’(2un
+
21,)
[-a
I
v
(2un
+
Un)
+
(a:j
-
a:j)
aj
(221,
)ai
(221,
+
vn
)]
J1
J,T
Now use Young’s inequality and absorb the terms in
j”(
with
-a
to obtain
a
constant
C
depending on
a,
a!j,
bf
such that
)IV(2un
+
vn)I2
(8.49)
~15
cJgTj”(2un
+
vn)
[Ivun12
+
u:
+
v:)]
.
Nonlinear Reaction-Diffusion Systems
395
Since
j‘
is
concave,
j”(r)
5
j’(r)/r
and we have
j’‘(2un
+
vn)(ui
+
v:)
5
J,,
j‘(2un
+
vn)(un
+
On)
J,,
which
is
treated
as
in
(8.48)
to get a uniform bound. The last term in
(8.49)
is
estimated by using again the equation in
u
and the assumption
(8.34)
on
f.
We have (using ellipticity of
a:j)
Adding the information that
j”(un)
2
jN(2u,,
+
vn),
we now easily deduce
that
JQTj”(2un
+
v,)~VU,~~
is
uniformly bounded. By
(8.49),
so
is
J1,
and
the proof
is
complete.
Remarks
The method described
is
a straightforward extension to general
linear diffusions of the method introduced in
[21].
As
we mentioned before,
applications to nonlinear diffusions like
A,,
have been made in
[13].
Obviously, the most difficult part in above proof
is
the last one where
uniform integrability of the nonlinear terms
f(un,
v,,),
g(u,,,
vn)
is
estab-
lished. It
is
in that proof that the stronger assumption
(8.34)
has to be
made. Moreover, we also use the specific structure of the diffusions.
For
in-
stance, this proof does not seem to generalize
as
such to nonlinear diffusions
of the form
(-A4(.),
-A$(.))
where
4,
I)
are different increasing functions
(see
[13]
for some positive results in that direction).
Similarly, we do not
know how to extend it to systems like
(6.1)
where
(8.36)
does not hold.
Let
us
point out that there are “good” reasons for which some important
work has to be made in the last part. Indeed,
all
the estimates up to that
point and, in particular, Lemmas
8.2,
8.3
and Proposition
8.5
remain valid
if
u:,
v,“
converge to finite measures
uo,
vo
rather than in
L’.
But, one can
find cases for which uniform integrability fails and for which solutions (at
least in usual sense) do not exist.
For
example consider
ut
-
AU
=
-UV’
on
B(0,
1)
x
(0,
T)
vt
-
AV
=
UV’
u=v=o
on
aB(0,
1)
x
(0,
T)
?A(.,
0)
=
v(.,
0)
=
6
Dirac mass at the origin
396
R.
H.
Martin
and
M.
Pierre
where
y
>
N/(N
-
2).
In this case
f(u,,
v,)
4
0
a.e. but not in
L’(&T).
Note that here the diffusions are the same.
This emphasizes the fact that
even when the diffusions are the same, the case
of
L’-initial data
is
not
completely obvious.
Finally, an interesting question
is
to know whether the solutions obtained
are in
Loo
for
t
>
0
(and therefore classical) although the initial data are only
L’.
In most cases one can show that this
is
indeed the case if again
f
and
g
have polynomial growth. The proof strongly relies on the “LP-techniques”
of
Section
4
coupled with well-known regularizing effects
of
semigroups
Sr(t)
having the property
ll~r(~)~olla3
5
~~-all~olll.
This
will
be studied in detail elsewhere.
Nonlinear Reaction-Diffusion
Systems
397
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BrCzis and
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A. Strauss,
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Hollis and
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Morgan,
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Hollis and J. Morgan,
Interior estimates
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class
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reaction-
diffusion system from L'-a priori estimates,
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I.
Kanel,
Cauchy's problem for semilinear parabolic equations with
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Dig. Uravneniya,
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E. Laarnri,
Existence globale pour des systkmes de re'action-diffusion
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Riccati-type Pseudopotentials
and
Their
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M.C.
Nucci
Dipartimento di Matematica, Universita di Perugia,
06100
Perugia, Italy
and
School
of
Mathematics, Georgia Institute
of
Technology,
Atlanta,
GA
30332,
U.S.A.
1
Introduction
Innumerable papers are dedicated to the study of nonlinear evolution
(NLE) equations. These equations can be divided into
3
groups
[6]:
0
S-integrable equations which are solvable by
a
linear spectral
problem i.e. the inverse scattering transform
[7], [2].
0
C-integrable equations which are linearizable by an appropriate
change of variables.
0
Non-integrable equations.
S-integrable equations possess
a
rich structure and have wide appli-
cability to many different physical phenomena. The most famous is
the Korteweg-de Vries equation
[22].
Other examples of S-integrable
equations are the sine-Gordon, Harry Dym, nonlinear Schrodinger,
Boussinesq,
and
Kadomtsev-Petviashvili equations
[2].
If
a
NLE
equation is S-integrable then the two coupled equations of its spec-
tral problem are called Lax equations
[21].
Another approach to the
spectral problem
is
given in
[l].
It gives rise to the scrcalled AKNS
system. A
NLE
equation is generally believed to be S-integrable
if an auto-Backlund transformation
[35]
is found. Another test for
S-integrability is the PainlevQ property
[44],
which
is
widely used
Nonlinear Equations in the Applied Sciences
399
Copyright
0
1992
by
Academic Press, Inc.
All
rights
of
reproduction
in
any
form
reserved.
ISBN
0-12-056752-0