Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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378 15. Nonlinear Parabolic Equations
where A
jk
"
D A
jk
.t;x;D
1
x
J
"
u
"
/ and v D D
˛
J
"
u
"
. Note that, by the strong
ellipticity hypothesis (7.2), we have
(7.7)
X
.A
jk
"
@
j
v; @
k
v/ C
0
krvk
2
L
2
:
The commutator ŒD
˛
;L
"
D
P
ŒD
˛
;A
jk
"
@
j
@
k
can be treated using the Moser
estimates established in Chap. 13. From Proposition 3.7 of Chap. 13 we deduce
that
(7.8)
kŒD
˛
;L
"
wk
L
2
C
X
j;k
kA
jk
"
k
H
`
k@
j
@
k
wk
L
1
CkrA
jk
"
k
L
1
k@
j
@
k
wk
H
`1
;
provided j˛j`.SinceŒA
jk
"
;@
k
w D
P
.@
k
A
jk
"
/w, we have the elementary
estimate
(7.9) kŒA
jk
"
;@
k
@
j
vk
L
2
C krA
jk
"
k
L
1
k@
j
vk
L
2
:
Furthermore, Proposition 3.9 of Chap.13 implies
(7.10) kA
jk
"
k
H
`
C
`
kJ
"
u
"
k
C
1
1 CkJ
"
u
"
k
H
`C1
;
and we have the elementary estimate krA
jk
"
k
L
1
C
kJ
"
u
"
k
C
1
kJ
"
u
"
k
C
2
.
Hence (7.5) is less than or equal to
(7.11)
2C
0
krD
˛
J
"
u
"
.t/k
2
L
2
C C
kJ
"
u
"
k
C
1
kJ
"
u
"
k
C
2
1 CkJ
"
u
"
k
H
`C1
kD
˛
J
"
u
"
k
L
2
:
Furthermore, we have a bound
(7.12) 2.D
˛
B
"
;D
˛
J
"
u
"
/ C
kJ
"
u
"
k
C
1
1 CkJ
"
u
"
k
H
`C1
kD
˛
J
"
u
"
k
L
2
;
by the analogue of (7.10)forkB.t; x; D
1
x
J
"
u
"
/k
H
`
. Consequently, we have an
upper bound for (7.4). Summing over j˛j`, we obtain
(7.13)
d
dt
ku
"
.t/k
2
H
`
2C
0
kJ
"
u
"
k
2
H
`C1
C C
1
kJ
"
u
"
k
C
2
1 CkJ
"
u
"
k
H
`C1
kJ
"
u
"
k
H
`
:
Using AB C
0
A
2
C .1=4C
0
/B
2
, with A DkJ
"
u
"
k
H
`C1
, we obtain
(7.14)
d
dt
ku
"
.t/k
2
H
`
C
0
kJ
"
u
"
k
2
H
`C1
C C
2
kJ
"
u
"
k
C
2
kJ
"
u
"
k
2
H
`
C 1
:
7. Quasi-linear parabolic equations I 379
In particular, since fJ
"
W 0<" 1g is uniformly bounded on each space C
j
.M /
and H
`
.M /,wehaveanestimate
(7.15)
d
dt
ku
"
.t/k
2
H
`
C
`
ku
"
k
C
2
ku
"
.t/k
2
H
`
C 1
:
This estimate permits the following analysis of the evolution equations (7.3).
Lemma 7.1. Given f 2 H
`
;`>n=2C2, the solution to (7.3)existsfort in an
interval I D Œ0; A/, independent of ", and satisfies an estimate
(7.16) ku
"
.t/k
H
`
K.t/; t 2 I;
independent of " 2 .0; 1.
Proof. Using the Sobolev imbedding theorem, we can dominate the right side of
(7.15)byE
ku
"
.t/k
2
H
`
,soku
"
.t/k
2
H
`
D y.t/ satisfies the differential inequality
(7.17)
dy
dt
E.y/; y.0/ Dkf k
2
H
`
:
Gronwall’s inequality then yields a function K.t/, finite on some interval
I D Œ0; A/, giving an upper bound for all y.t/ satisfying (7.17). This I and
K.t/ work for (7.16).
We are now prepared to establish the following existence result:
Theorem 7.2. If (7.1) satisfies the parabolicity hypothesis (7.2), and if f 2
H
`
.M /, with `>n=2C 2, then there is a solution u, on an interval I D Œ0; T /,
such that
(7.18) u 2 L
1
I;H
`
.M /
\ Lip
I;H
`2
.M /
:
Proof. Take the I above and shrink it slightly. The bounded family
u
"
2 C.I;H
`
/ \ C
1
.I; H
`2
/
will have a weak limit point u satisfying (7.18). Furthermore, by Ascoli’s theorem
(in the form given in Exercise 5,in 6 of Appendix A), there is a sequence
(7.19) u
"
! u in C
I;H
`2
.M /
;
since the inclusion H
`
,! H
`2
is compact. In addition, interpolation inequal-
ities imply that fu
"
W 0<" 1g is bounded in C
I;H
`2
.M /
for each
380 15. Nonlinear Parabolic Equations
2 .0; 1/. Since the inclusion H
`2
,! C
2
.M / is compact for small >0if
`>n=2C 2, we can arrange that
(7.20) u
"
! u in C
I;C
2
.M /
:
Consequently, with " D "
,
(7.21)
J
"
X
A
jk
.t;x;D
1
x
J
"
u
"
/@
j
@
k
J
"
u
"
!
X
A
jk
.t;x;D
1
x
u/@
j
@
k
u;
J
"
B.t; x; D
1
x
J
"
u
"
/ ! B.t; x; D
1
x
u/
in C.I M/, while clearly @u
"
=@t ! @u=@t weakly. Thus (7.1) follows in the
limit from (7.3), and the theorem is proved.
We turn now to questions of the uniqueness, stability, and rate of convergence
of u
"
to u; we can treat these questions simultaneously. Thus, with " 2 Œ0; 1,we
compare a solution u to (7.1) with a solution u
"
to
(7.22)
@u
"
@t
D J
"
X
A
jk
.t;x;D
1
x
J
"
u
"
/@
j
@
k
J
"
u
"
C J
"
B.t; x; D
1
x
J
"
u
"
/;
u
"
.0/ D h:
For brevity, we suppress the .t; x/-dependence and write
(7.23)
@u
@t
D L.D
1
x
u;D/u C B.D
1
x
u/;
@u
"
@t
D J
"
L.D
1
x
J
"
u
"
;D/J
"
u
"
C J
"
B.D
1
x
J
"
u
"
/:
Let v D u u
"
. Subtracting the two equations in (7.23), we have
(7.24)
@v
@t
D L.D
1
x
u;D/vC L.D
1
x
u;D/u
"
J
"
L.D
1
x
J
"
u
"
;D/J
"
u
"
C B.D
1
x
u/ J
"
B.D
1
x
J
"
u
"
/:
Write
(7.25)
L.D
1
x
u;D/u
"
J
"
L.D
1
x
J
"
u
"
;D/J
"
u
"
D
L.D
1
x
u;D/ L.D
1
x
u
"
;D/
u
"
C .1 J
"
/L.D
1
x
u
"
;D/u
"
CJ
"
L.D
1
x
u
"
; D/.1 J
"
/u
"
C J
"
L.D
1
x
u
"
;D/ L.D
1
x
J
"
u
"
;D/
J
"
u
"
and
(7.26)
B.D
1
x
u/ J
"
B.D
1
x
J
"
u
"
/ D
B.D
1
x
u/ B.D
1
x
u
"
/
C .1 J
"
/B.D
1
x
u
"
/
C J
"
B.D
1
x
u
"
/ B.D
1
x
J
"
u
"
/
:
7. Quasi-linear parabolic equations I 381
Now write
(7.27)
B.D
1
x
u/ B.D
1
x
w/ D G.D
1
x
u;D
1
x
w/.D
1
x
u D
1
x
w/;
G.D
1
x
u;D
1
x
w/ D
Z
1
0
B
0
D
1
x
u C .1 /D
1
x
w
d;
and similarly
(7.28) L.D
1
x
u;D/ L.D
1
x
w; D/ D .D
1
x
u D
1
x
w/ M.D
1
x
u;D
1
x
w; D/:
Then (7.24) yields
(7.29)
@v
@t
D L.D
1
x
u;D/vC A.D
1
x
u;D
2
x
u
"
/D
1
x
v C R
"
;
where
(7.30)
A.D
1
x
u;D
2
x
u
"
/D
1
x
v
D D
1
x
v M.D
1
x
u;D
1
x
u
"
;D/u
"
C G.D
1
x
u;D
1
x
u
"
/D
1
x
v
incorporates the first terms on the right sides of (7.25)and(7.26), and R
"
is the
sum of the rest of the terms in (7.25)and(7.26). Note that each term making up
R
"
hasasafactorI J
"
, acting on either D
1
x
u
"
;B.D
1
x
u
"
/,orL.D
1
x
u
"
;D/u
"
.
Thus there is an estimate
(7.31) kR
"
.t/k
2
L
2
C
`
ku
"
.t/k
C
2
1 Cku
"
.t/k
2
H
`
r
`
."/
2
;
where
(7.32) r
`
."/ DkI J
"
k
L.H
`2
;L
2
/
kI J
"
k
L.H
`
;H
2
/
:
Now, estimating .d=dt/kv.t/k
2
L
2
via techniques parallel to those used for
(7.4)–(7.15) yields
(7.33)
d
dt
kv.t/k
2
L
2
C.t/kv.t/k
2
L
2
C S.t/;
with
(7.34) C.t/ D C
ku
"
.t/k
C
2
; ku.t/k
C
2
;S.t/DkR
"
.t/k
2
L
2
:
Consequently, by Gronwall’s inequality, with K.t/ D
R
t
0
C./d,
(7.35) kv.t/k
2
L
2
e
K.t/
kf hk
2
L
2
C
Z
t
0
S./e
K./
d
;
for t 2 Œ0; T /. Thus we have
382 15. Nonlinear Parabolic Equations
Proposition 7.3. Fo r `>n=2C2, solutions to (7.1) satisfying (7.18) are unique.
They are limits of soutions u
"
to (7.3), and, for t 2 I ,
(7.36) ku.t/ u
"
.t/k
L
2
K
1
.t/kI J
"
k
L.H
`2
;L
2
/
:
Note that if J
"
D '."
p
/ and ' 2 S.R/ satisfies './ D 1 for jj1,we
have the operator norm estimate
(7.37) kI J
"
k
L.H
`2
;L
2
/
C"
`2
:
We next establish smoothness of the solution u given by Theorem 7.2,away
from t D 0.
Proposition 7.4. The solution u of Theorem 7.2 has the property that
(7.38) u 2 C
1
.0; T / M
:
Proof. Fix any S<Tand take J D Œ0; S /.Ifweintegrate(7.14) over J ,we
obtain a bound on
R
J
kJ
"
u
"
.t/k
2
H
`C1
dt, provided we assume `>n=2C2,sothat
we can appeal to a bound on C
2
kJ
"
u
"
.t/k
C
2
and on kJ
"
u
"
.t/k
2
H
`
,fort 2 J .
Thus
(7.39) u 2 L
2
J;H
`C1
.M /
:
Recall that we know u 2 Lip
I;H
`2
.M /
. It follows that there is a subset E of
I such that
(7.40) meas.E/ D 0; t
0
2 I n E H) u.t
0
/ 2 H
`C1
.M /:
Given t
0
2 I nE, consider the initial-value problem
(7.41)
@U
@t
D
X
A
jk
.t;x;D
1
x
U/@
j
@
k
U C B.t; x; D
1
x
U/; U.t
0
/ D u.t
0
/:
By the uniqueness result of Proposition 7.3, U.t/ D u.t/ for t
0
t<T.
Now, the proof of Theorem 7.2 gives a length L>0, independent of t
0
2 J ,
such that the approximation U
"
defined by the obvious analogue of (7.3)converges
to U weakly in L
1
.Œt
0
;t
0
CL; H
`
.M //. In particular, kU
"
.t/k
C
2
is bounded on
Œt
0
;t
0
CL. On the other hand, there is also an analogue of (7.15), with ` replaced
by ` C 1:
(7.42)
d
dt
kU
"
.t/k
2
H
`C1
C
`C1
kU
"
.t/k
C
2
kU
"
.t/k
2
H
`C1
C 1
:
7. Quasi-linear parabolic equations I 383
Consequently, U
"
is bounded in C.Œt
0
;t
0
C L; H
`C1
.M //, and we obtain
(7.43) u 2 L
1
Œt
0
;t
0
C L; H
`C1
.M /
\ Lip
Œt
0
;t
0
C L; H
`1
.M /
:
Since the exceptional set E has measure 0, this is enough to guarantee that
(7.44) u 2 L
1
loc
.J; H
`C1
.M // \ Lip
loc
.J; H
`1
.M //;
and since J is obtained by shrinking I as little as one likes, we have
(7.44) with J replaced by I . Now we can iterate this argument, obtaining
u 2 L
1
loc
.I; H
`Cj
.M // for each j 2 Z
C
, from which (7.38) is easily deduced.
We can now sharpen the description (7.18) of the solution u in another fashion:
Proposition 7.5. The solution u of Theorem 7.2 has the property that
(7.45) u 2 C
I;H
`
.M /
\ C
1
I;H
`2
.M /
:
Proof. It suffices to show that u.t/ is continuous at t D 0, with values in H
`
.M /.
We know that as t & 0; u.t/ is bounded in H
`
.M / and converges to u.0/ D f
in H
`2
.M /; hence u.t/ ! f weakly in H
`
as t & 0. To deduce that u.t/ ! f
in H
`
-norm, it suffices to show that
(7.46) lim sup
t&0
ku.t/k
H
`
kf k
H
`
:
However, the bounds on ku
"
.t/k
H
`
implied by (7.15) easily yield this result.
Now that we have smoothness, (7.38), an argument parallel to but a bit simpler
than that used to produce (7.4)–(7.15)gives
(7.47)
d
dt
ku.t/k
2
H
`
C
`
ku.t/k
C
2
ku.t/k
2
H
`
C 1
;
for a solution u 2 C
1
.0; T /M
to (7.1). This implies the following persistence
result:
Proposition 7.6. Suppose u 2 C
1
.0; T / M
is a solution to (7.1). Assume
also that
(7.48) ku.t/k
C
2
K<1;
for t 2 .0; T /. Then there exists T
1
>T such that u extends to a solution to (7.1),
belonging to C
1
.0; T
1
/ M
.
384 15. Nonlinear Parabolic Equations
A special case of (7.1) is the class of systems of the form
(7.49)
@u
@t
D
X
j;k
A
jk
.t; x; u/@
j
@
k
u C B.t; x; D
1
x
u/; u.0/ D f:
We retain the strong parabolicity hypothesis (7.2). In this case, when one does
estimates of the form (7.5)–(7.15), and so forth, C
2
-norms can be systematically
replaced by C
1
-norms. In particular, for a local smooth solution to (7.49), we have
the following improvement of (7.47):
(7.50)
d
dt
ku.t/k
2
H
`
C
`
ku.t/k
C
1
ku.t/k
2
H
`
C 1
:
Thus we have the following:
Proposition 7.7. If (7.49) is strongly parabolic and f 2 H
`
.M / with `>
n=2 C 1, then there is a solution u, on an interval I D Œ0; T /, such that
(7.51) u 2 C
Œ0; T /; H
`
.M /
\ C
1
.0; T / M
:
Furthermore, if
(7.52) ku.t/k
C
1
K<1;
for t 2 Œ0; T /, then there exists T
1
>T such that u extends to a solution to (7.49),
belonging to C
1
.0; T
1
/ M
.
We apply this to obtain a global existence result for a scalar parabolic equation,
in one space variable, of the form
(7.53)
@u
@t
D A.u/@
2
x
u C g.u; u
x
/; u.0/ D f:
Take M D S
1
. We assume g.u;p/ is smooth in its arguments. We will exploit
the maximum principle to obtain a bound on u
x
, which satisfies the equation
(7.54)
@
@t
u
x
D A.u/@
2
x
.u
x
/ C A
0
.u/u
x
@
x
.u
x
/
C g
p
.u; u
x
/@
x
.u
x
/ C g
u
.u; u
x
/u
x
:
The only restriction on applying the maximum principle to estimate ju
x
j is that
we need g
u
.u; u
x
/ 0. We can fix this by considering e
tK
u
x
, which satisfies
(7.55)
@
@t
e
tK
u
x
D e
tK
R Ku
x
;
7. Quasi-linear parabolic equations I 385
where R is the right side of (7.54). The maximum principle yields
(7.56) ku
x
k
C
0
e
tK
k@
x
f k
C
0
;
provided
(7.57) g
u
.u; u
x
/ K:
We have the following result:
Proposition 7.8. Given f 2 H
2
.S
1
/, suppose you have a global a priori bound
(7.58) ku.t/k
L
1
K
0
;
for a solution to the scalar equation (7.53). If there is also an a priori bound
(7.57)(forjujK
0
), which follows automatically in case g D g.u/ is a smooth
function of u alone, then (7.53) has a solution for all t 2 Œ0; 1/.
The class of equations described by (7.53) includes those of the form
(7.59)
@u
@t
D @
x
A.u/@
x
u
C '.u/; u.0/ D f:
In fact, this is of the form (7.53), with
(7.60) g.u; u
x
/ D A
0
.u/u
2
x
C '.u/:
Thus
(7.61) g
u
.u; u
x
/ D A
00
.u/u
2
x
C '
0
.u/:
In such a case, (7.57) applies if and only if A
00
.u/ 0,thatis,A.u/ is concave
in u. For example, Proposition 7.8 applies to the equation
(7.62)
@u
@t
D @
x
u @
x
u
C '.u/; u.0/ D f;
in cases where it can be shown that, for some a; b 2 .0; 1/; a u.t; x/ b for
all t 0; x 2 S
1
. This in turn holds if f.x/ takes values in the interval Œa; b
with '.a/ > 0 and '.b/ < 0, by arguments similar to the proof of Proposition
4.3. A specific example is the equation
(7.63)
@u
@t
D
@
@x
u
@u
@x
C u.1 u/; u.0/ D f;
arising in models of population growth (see [Grin], p. 224, or [Mur], p. 289). This
is similar to reaction-diffusion equations studied in 4, but this time there is a
386 15. Nonlinear Parabolic Equations
nonlinear diffusion as well as a nonlinear reaction. In this case we see that (7.63)
has a global solution, given smooth f with values in a interval I D Œa; b, with
a>0I u.t; x/ 2 I for all .t; x/ 2 R
C
S
1
if also b>1. This existence result is
rather special; a much larger class of global existence results will be established
in 9.
Exercises
1. In the setting of Proposition 7.3,givenu.0/ D f 2 H
`
.M /, initial data for (7.1), work
out estimates for
ku.t/ u
"
.t/k
H
j
.M /
;1 j ` 1:
2. Establish global solvability on Œ0; 1/ S
1
for
(7.64)
@u
@t
D A.u/@
2
x
u C'.u/; u.0/ D f;
given f 2 H
2
.S
1
/, under the hypotheses
a f.x/ b; '.a/ 0; '.b/ 0;
and
A.u/ C>0; for a u b:
3. Establish global solvability on Œ0; 1/ S
1
for
(7.65)
@u
@t
D A.u
x
/u
xx
; u.0/ D f;
given f 2 H
3
.S
1
/, under the hypothesis
(7.66) A.p/ C>0and A
00
.p/ 0; for jpjsup jf
0
.x/j:
(Hint: The function v D u
x
satisfies
@v
@t
D @
x
A.v/ @
x
v
;v.0/D f
0
.x/:
Estimate v
x
D u
xx
:)
Consider the example
(7.67)
@u
@t
D
1 u
2
x
1=2
u
xx
;
assuming jf
0
.x/ja<1, or the example
(7.68)
@u
@t
D .1 C u
2
x
/
1
u
xx
;
assuming jf
0
.x/jb<
p
1=3.
A much more general global existence result is derived in 9. See Exercise 3 of 9 for
a better existence result for (7.68).
8. Quasi-linear parabolic equations II (sharper estimates) 387
8. Quasi-linear parabolic equations II (sharper estimates)
While most of the analysis in 7 was fairly straightforward, the results are not as
sharp as they can be, and we obtain sharper results here, making use of paradif-
ferential operator calculus. The improvements obtained here will be coupled with
Nash–Moser estimates and applied to global existence results in the next section.
Most of the material of this section follows the exposition in [Tay].
Though we intend to concentrate on the quasi-linear case, we begin with com-
pletely nonlinear equations:
(8.1)
@u
@t
D F.t;x;D
2
x
u/; u.0/ D f;
for u taking values in R
K
. We suppose F D F.t;x;/; D .
˛j
Wj˛j2; 1
j K/ is smooth in its arguments, and our strong parabolicity hypothesis is
(8.2) Re
X
j˛jD2
.@F=@
˛
/
˛
C jj
2
I;
for 2 R
n
,whereReA D .1=2/.A C A
/,foraK K matrix A.Usingthe
paradifferential operator calculus developed in Chap. 13, 10, we write
(8.3) F.t;x;D
2
x
v/ D M.vIt;x;D/v C R.v/:
By Proposition 10.7 of Chap. 13, we have, for r>0,
(8.4) v.t/ 2 C
2Cr
H) M.vIt;x;/ 2 A
r
0
S
2
1;1
C
r
S
2
1;0
\ S
2
1;1
;
where the symbol class A
r
0
S
m
1;ı
is defined by (10.31) of Chap.13. The hypothesis
(8.2) implies
(8.5) Re M.vIt;x;/ C jj
2
I>0;
for jj large. Note that symbol smoothing in x,asin(9.27) of Chap.13, gives
(8.6) M.vIt;x;/ D M
#
.t;x;/C M
b
.t;x;/;
and when (8.4) holds (for fixed t),
(8.7) M
#
.t;x;/ 2 A
r
0
S
2
1;ı
;M
b
.t;x;/ 2 S
2rı
1;1
:
We also have
(8.8) Re M
#
.t;x;/ C jj
2
I>0;
for jj large.