Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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388 15. Nonlinear Parabolic Equations

We will obtain a solution to (8.1) as a limit of solutions u

(8.9)

D J

F.t;x;D

/; u

.0/ D f:

Thus we need to show that u

.t; x/ exists on an interval t 2 Œ0; T / independent of

" 2 .0; 1 and has a limit as " ! 0 solving (8.1). As before, all this follows from

an estimate on the H

-norm, and we begin with

(8.10)

kƒ

.t/k

D 2.ƒ

F.t;x;D

/; ƒ

D 2.ƒ

;ƒ

/ C 2.ƒ

;ƒ

The last term is easily bounded by

C.ku

.t/k



.t/k

C 1



Here M

D M.J

It;x;D/. Writing M

D M

C M

as in (8.6), we see that

(8.11)

.ƒ

;ƒ

D .ƒ

s1

;ƒ

sC1

 C.kJ

2Cr

/kJ

sC1rı

sC1

;

for s>1,sinceby(8.7), M

W H

sC1rı

! H

s1

. We next estimate

(8.12)

.ƒ

;ƒ

D .M

;ƒ

/ C .Œƒ

J

;ƒ

By (8.7), plus (10.99) of Chap. 13, we have Œƒ

 2 OPS

sC2r

1;ı

if 0<r<1,

so the last term in (8.12) is bounded by

(8.13)

.ƒ

1

Œƒ

J

;ƒ

sC1

 C.ku

2Cr

/kJ

sC1r

kJ

sC1

Finally, G˚arding’s inequality (Theorem 6.1 of Chap. 7) applies to M

(8.14) .M

w; w/ C

kwk

C C

.ku

2Cr

/kwk

Putting together the previous estimates, we obtain

(8.15)

.t/k



sC1

C C.ku

2Cr

/kJ

sC1rı

;

8. Quasi-linear parabolic equations II (sharper estimates) 389

and using Poincar´e’s inequality, we can replace C

=2 by C

=4 and the

sC1rı

-norm by the H

-norm, getting

(8.16)

.t/k



.t/k

sC1

C C

.ku

.t/k

2Cr

/kJ

.t/k

From here, the arguments used to establish Theorem 7.2 through Proposition 7.6

yield the following result:

Proposition 8.1. If (8.1) is strongly parabolic and f 2 H

.M /, with s>n=2C

2, then there is a unique solution

(8.17) u 2 C.Œ0;T /;H

.M // \ C

..0; T / M/;

which persists as long as ku.t/k

2Cr

is bounded, given r>0.

Note that if the method of quasi-linearization were applied to (8.1) in concert

with the results of  7, we would require s>n=2C 3 and for persistence of the

solution would need a bound on ku.t/k

We now specialize to the quasi-linear case (7.1), that is,

(8.18)

j;k

.t;x;D

u/@

u C B.t; x; D

u/; u.0/ D f:

This is the special case of (8.1)inwhich

(8.19) F.t;x;D

u/ D

.t;x;D

u/@

u C B.t; x; D

u/:

We form M.vIt;x;D/ as before, by (8.3). In this case, we can replace (8.4)by

(8.20) v 2 C

1Cr

H) M.vIt; x;/ 2 A

1;1

C S

2r

1;1

Thus we can produce a decomposition (8.6) such that (8.7) holds for v 2 C

1Cr

Hence the estimates (8.11)–(8.16) all hold with constants depending on the

1Cr

-norm of u

.t/, rather than the C

2Cr

-norm, and we have the following im-

provement of Theorem 7.2 and Proposition 7.6:

Proposition 8.2. If the quasi-linear system (8.18) is strongly parabolic and f 2

.M /, s>n=2C 1, then there is a unique solution satisfying (8.17), which

persists as long as ku.t/k

1Cr

is bounded, given r>0.

We look at the parabolic equation

(8.21)

.t; x; u/@

u C B.t; x; u/;

390 15. Nonlinear Parabolic Equations

which is a special case of (8.1), with

(8.22) F.t;x;D

u/ D

.t; x; u/@

u C B.t; x; u/:

In this case, if r>0,wehave

(8.23) v 2 C

H) M.vIt;x;/ 2 A

1;1

C S

2r

1;1

;

and the following results:

Proposition 8.3. Assume the system (8.21) is strongly parabolic. If f 2 H

.M /,

s>n=2C 1, then there is a unique solution satisfying (8.17), which persists as

long as ku.t/k

is bounded, given r>0.

It is also of interest to consider the case

(8.24)

.t; x; u/@

u; u.0/ D f:

Arguments similar to those done above yield the following.

Proposition 8.4. If the system (8.24) is strongly parabolic, and if f 2 H

.M /;

s>n=2C 1, then there is a unique solution to (8.24), satisfying (8.17), which

persists as long as ku.t/k

is bounded, for some r>0.

We continue to study the quasi-linear system (8.18), but we replace the strong

parabolicity hypothesis (7.2) with the following more general hypothesis on

(8.25) L

.t;v;x;/ D

j;k

.t;x;v/



namely,

(8.26) spec L

.t;v;x;/ fz 2 C W Re z C

jj

for some C

>0. When this holds, we say that the system (8.18)isPetrowski-

parabolic. Again we will try to produce the solution to (8.18) as a limit of

solutions u

to (8.9). In order to get estimates, we construct a symmetrizer.

Lemma 8.5. Given (8.26), there exists P

.t;v;x;/, smooth in its arguments, for

 ¤ 0, homogeneous of degree 0 in , positive-deﬁnite (i.e., P

 cI > 0), such

that .P

C L



/ is also positive-deﬁnite, that is,

(8.27) .P

C L



/  C jj

I>0:

8. Quasi-linear parabolic equations II (sharper estimates) 391

Such a construction is done in Chap. 5. We brieﬂy recall the argument used

there, where this result is stated as Lemma 11.5. The symmetrizer P

,which

is not unique, is constructed by establishing ﬁrst that if L

is a ﬁxed K  K

matrix with spectrum in Re z <0, then there exists a K  K matrix P

such

that P

and .P

C L



/ are positive-deﬁnite. This is an exercise in linear

algebra. One then observes the following facts. One, for a given positive matrix

,thesetofL

such that .P

C L



/ is positive-deﬁnite is open. Next,

for given L

with spectrum in Re z <0,thesetfP

W P

>0;.P



/>0g is an open convex set of matrices, within the linear space of self

adjoint K  K matrices. Using this and a partition-of-unity argument, one can

establish the following, which then yields Lemma 8.5. (Compare with Lemma

11.4 in Chap.5. Also compare with the construction in  8 of Chap. 15.)

Lemma 8.6. If M



denotes the space of real K  K matrices with spectrum in

Re z <0and P

the space of positive-deﬁnite (complex) K  K matrices, there

is a smooth map

ˆ W M



! P

;

homogeneous of degree 0, such that if L 2 M



and P D ˆ.L/,then.PL C



P/ 2 P

Having constructed P

.t;v;x;/, note that, for ﬁxed t; r 2 R,

(8.28)

u 2 C

1Cr

H) L.t; D

u;x;/ 2 C



and

.t; D

u;x;/ 2 C



Now apply symbol smoothing in x to

.t;x;/ D P

.t; D

u;x;/, to obtain

(8.29) P.t/ 2 OP A

1;ı

I P.t/ 

P.t;D

u;x;D/ 2 OP C

rı

1;ı

Then set

(8.30) Q D

.P C P



/ C Kƒ

1

;

with K>0chosen so that Q is positive-deﬁnite on L

. Now, with u

deﬁned as

the solution to (8.9), u

.0/ D f , we estimate

(8.31)

.ƒ

/ D 2.ƒ

/ C .ƒ

where P

is obtained as in (8.29)–(8.30), from symbol smoothing of the family

of operators

D P

.t; D

;x;D/,andQ

comes from P

via (8.30). Note

that if u

.t/ is bounded in C

1Cr

.M /,thenP

.t/ is bounded in OP S

2r

1;ı

.M /,so

(8.32) j.ƒ

/jC



.t/k

1Cr



.t/k

sC1r=2

392 15. Nonlinear Parabolic Equations

We can write the ﬁrst term on the right side of (8.31)astwice

(8.33) .Q

;ƒ

/ C .Q

;ƒ

where M

is as in (8.10). The last term here is easily dominated by

(8.34) C.ku

.t/k

/kJ

.t/k

sC1

ku

.t/k

We write the ﬁrst term in (8.33)as

(8.35)

;ƒ

/ C .Q

Œƒ

J

;ƒ

C .ŒQ

M

;ƒ

We have Q

.t/ 2 OP A

1;ı

, by (10.100) of Chap.13, and hence, by (10.99) of

Chap. 13,

(8.36) ŒQ

 bounded in L.H

s1

with a bound given in terms of ku

1Cr

if r>1.Furthermore,wehave

(8.37) kM

s1

 C.ku

1Cr

/kJ

sC1

;

so we can dominate the last term in (8.35)by

(8.38) C.ku

.t/k

1Cr

/kJ

sC1

ku

;

provided r>1. Moving to the second term in (8.35), since

2 OP A

1;1

C OPS

2r

1;1

;

we have

(8.39) kŒƒ

vk

 C



1Cr



kvk

sC1

;

provided r>1. Hence the second term in (8.35) is also bounded by (8.38).

This brings us to the ﬁrst term in (8.35), and for this we apply the G˚arding

inequality to the main term arising from M

D M

C M

,toget

(8.40) .Q

v; v/ C

kvk

C C.ku

/kvk

Substituting v D ƒ

and using the other estimates on terms from (8.31), we

have

(8.41)

.ƒ

/ C

sC1

C C.ku

3Cı

/ku



sC1

Cku



Exercises 393

which we can further dominate as in (8.16). Note that (8.32)istheworstterm;we

need r>2for it to be useful.

From here, all the other arguments yielding Propositions 8.1 and 8.2 apply, and

we have the following:

Proposition 8.7. Given the Petrowski-parabolicity hypothesis (8.25)–(8.26), if

f 2 H

.M / and s>n=2C 3,then(8.18) has a unique solution

(8.42) u 2 C.Œ0;T /;H

.M // \ C

..0; T / M/;

for some T>0, which persists as long as ku.t/k

1Cr

is bounded, for some r>0.

In order to check the persistence result, we run through (8.31)–(8.41) with u

replaced by the solution u, and with J

replaced by I . In such a case, the analogue

of (8.32)isusefulforanyr>0. The analogue of (8.36) is vacuous, so (8.38)

works for any r>0. An analogue of (8.11)–(8.13) can be applied to (8.39);

recalling that this time we have (8.7)foru 2 C

1Cr

, we also obtain a useful

estimate whenever r>0. This gives the persistence result stated above.

Exercises

In Exercises 1–10, we look at the system

(8.43)

D Mu  ar.urv/;

D Dv C

u Ch

 v:

We assume that M; D; ; a; b,andh are positive constants, and  is the Laplace op-

erator on a compact Riemannian manifold. This arises in a model of chemotaxis,the

attraction of cells to a chemical stimulus. Here, u D u.t; x/ represents the concen-

tration of cells, and v D v.t; x/ the concentration of a certain chemical (see [Grin],

p. 194, or [Mur]).

1. Show that (8.43) is a Petrowski-parabolic system.

2. If .u;v/is a sufﬁciently smooth solution for t 2 Œ0; T /, show that

(8.44) u.0/  0; v.0/  0 H) u.t/  0; v.t/  0; 8 t 2 Œ0; T /:

(Hint: If we can deduce u.t/  0, the result follows for

v.t/ D e

.D/t

v.0/ C

.D/.t/



u./



d; '.u/ D

u Ch

Temporarily strengthen the hypothesis on u to u.0; x/ > 0, and modify the ﬁrst equa-

tionin(8.43)to

D Mu ar.urv/ C ";

with small ">0. Show that u.t; x/ > 0 for t in the interval of existence by considering

the ﬁrst t

at which, for some x

2 M; u.t

/ D 0. Derive the contradictory

394 15. Nonlinear Parabolic Equations

estimate @

u.t

/  ". To pass from the modiﬁed problem to the original, you may

ﬁnd it necessary to work Exercises 3–10 for the modiﬁed problem, which will involve

no extra work.)

3. Show that ku.t/k

.M /

is constant, for t 2 Œ0; T /.(Hint: Integrate the ﬁrst equation

in (8.43) over x 2 M , and use the positivity of u:)

Note: The desired conclusion is slightly different for the modiﬁed problem.

4. Given the conclusion of (8.44), show that, for I D Œ"; T /; r 2 .0; 1/,

(8.45) sup

t2I

kv.t/k

1Cr

.M /

< 1:

(Hint: Regard the second equation in (8.43) as a nonhomogeneous linear equation for

v, with nonhomogeneous term F.t;x/ D bu=.u C h/ 2 L

.I  M/:)

5. Show that, for any ı>0,

(8.46) sup

t2I

ku.t/k

1ı;1

.M /

< 1:

(Hint: Regard the ﬁrst equation in (8.43) as a nonhomogeneous linear equation for

u, with nonhomogeneous term G.t; x/ D arH.t;x/,whereH D urv 2

.I; L

.M //:)

6. Given (8.45)–(8.46), deduce that H 2 L

.I; H

r;1

.M //,foranyr 2 .0; 1/. Hence

improve (8.46)to

(8.47) sup

t2I

ku.t/k

2ı;1

.M /

< 1;

for any ı>0. Consequently, for p 2



1; n=.n  1/



(8.48) sup

t2I

ku.t/k

1;p

.M /

< 1:

7. Now deduce that H 2 L

.I; H

r;p

.M //,foranyr 2 .0; 1/; p 2



1; n=.n  1/



Hence improve (8.48)to

(8.49) sup

t2I

ku.t/k

2ı;p

.M /

< 1:

8. Iterate the argument above, to establish (8.49)forallp<1, hence

(8.50) sup

t2I

ku.t/k

1Cr

.M /

< 1;

for any r<1.

9. Using (8.50), improve the estimate (8.45)to

sup

t2I

kv.t/k

3Cr

.M /

< 1;

for any r<1. Then improve (8.50)tosup

ku.t/k

2Cr

.M /

< 1, and then to

sup

t2I

ku.t/k

3Cr

.M /

< 1:

10. Now deduce the solvability of (8.43)forallt>0,given(8.44).

Exercises 395

In Exercises 11–12, we look at a strongly parabolic K  K system of the form

(8.51) u

D A.u/u

C g.u; u

/; x 2 S

11. If u 2 C



.0; T /  M



solves (8.51), show that

(8.52)

.t/k

C

.t/k

C 2





u.t/; u

.t/





.t/k



.t/k





u.t/; u

.t/





;

where 2A.u/  C

>0.

12. Suppose you can establish that the solution u possesses the following property: For

each t 2 .0; T /; ku.t; /k

 C

< 1. Suppose

(8.53) jg.v; p/jC

.1 Cjpj/;

for jvjC

. Show that u extends to a solution u 2 C



.0; T

/  M



of (8.51), for

some T

>T.(Hint: Use Proposition 8.3.)

In Exercises 13–15, we look at a strongly parabolic K  K system of the form

(8.54)

D @

A.u/@

u Cf.u/; x 2 S

13. If u 2 C



.0; T /  M



solves (8.54), show that

(8.55)

.t/k

 .ˇku.t/k

 C

/ku

.t/k

C 2





u.t/





.t/k

;

where

2A.u/  C

>0; ˇD sup

6kDA.u/k:

(Hint: Use the estimate

 3kuk

;

which follows from the p D 1; k D 2 case of Proposition 3.1 in Chap. 13.)

14. Improve the estimate (8.55)to

(8.56)

.t/k





ˇN .u/  C



C 2kf

.u/k

;

where

(8.57) N .g/ D inf

2R

kg  k

osc g:

15. Suppose you can establish that the solution u possesses the following property:

For each t 2 .0; T /; u.t; / takes values in a region K

 R

so small that

396 15. Nonlinear Parabolic Equations



u.t/



 C

=ˇ. Assume kvkC

< 1,forv 2 K

. Show that u extends to

a solution u 2 C



.0; T

/  M



of (8.54), for some T

>T.

Compare with the treatment of (7.59). See also the treatment of (9.61).

16. Rework Exercise 12, weakening the hypothesis (8.53)to

(8.58) jg.v; p/jC

.1 Cjpj/ C ˛C

jpj

;˛<

;

for jvjC

9. Quasi-linear parabolic equations III

(Nash–Moser estimates)

We will be able to get global solutions to a certain class of quasi-linear parabolic

equations by applying the results of  8 together with H¨older estimates for solu-

tions to scalar equations of the form

(9.1)

 Lu D 0; Lu D b

1

j;k





;

where a

;b;b

1

2 L

. The operator L is as in (9.1) of Chap. 14, and we make

the same ellipticity hypothesis as used there; thus we assume

(9.2) 





.t; x/



 



 b.x/  b

;

with

(9.3) 0<

 

< 1;0<b

 b

< 1:

We take b independent of t.H¨older estimates for solutions to (9.1) under these

hypotheses were ﬁrst proved by Nash [Na]. Moser [Mos2] established a Harnack

inequality that yielded such H¨older estimates; a simpler proof is given in [Mos3].

Another treatment of Nash’s results has been given in [FS]. All these arguments

are more elaborate than that used for elliptic equations in Chap.14, partly because

they produce a sharper sort of Harnack inequality. Here, we follow [Kru], who

obtained a parabolic analogue of the weaker Harnack inequality discussed in

Chap. 14, by methods parallel to those in Moser’s ﬁrst treatment of the elliptic

case, in [Mos1].

As in  9 of Chap. 14, which we will refer to as “14” for short, we use a

deﬁne an inner product of vectors in R

(9.4) hV;W iD

9. Quasi-linear parabolic equations III (Nash–Moser estimates) 397

we use the square norm jV j

DhV;V i; and we use bdxD dV to deﬁne the

volume element. Parallel to (9.4) of “14,” we have

(9.5) v D f.u/ H) .@

 L/v D f

.u/.@

 L/u f

.u/jruj

We say v is a subsolution of @

 L provided .@

 L/v  0. Thus we see that

u 7! f.u/ takes solutions to .@

 L/u D 0 to subsolutions if f is convex, while

it takes subsolutions to subsolutions if f is both convex and increasing.

Next, parallel to (9.3) of “14,” we have

(9.6)

“

w.@

 L/u dt dV D

“

u; r

wi dt dV C

“

u dt dV;

where Q D I   D ŒT

   and w vanishes near I  @.Ifweset

w D

u,where .t; x/ is C

and vanishes for x near @, we obtain the

following analogue of (9.5) of “14”:

(9.7)

“

dt dV

D2

“

h r

u; ur

i dt dV C

“

gu dt dV

“

dt dV 

u.T

;x/dV C

u.T

;x/dV;

provided .@

 L/u D g. Consequently, parallel to the estimate (9.6) of “14,” we

have

(9.8)

“

dt dV C

u.T

;x/

 2

“





dt dV

“

gu dt dV C

u.T

;x/

dV:

We now proceed to a Moser iteration argument, parallel to (9.7)–(9.20) of “14.”

Given Q D I , consider nested sequences of regions  D 







j C1

  in R

and intervals I D I

   I

 I

j C1

 , with

intersections O and J , respectively, so we have Q

D I

 

Q D J  O

(see Fig. 9.1). Let us assume I D Œ0; T  and I

D Œ

;T, with J D ŒT =2; T .

We suppose that the distance of any point in @

j C1

to @

is  j

2

and that the

length of I

n I

j C1

is  j

2

. We want to estimate the sup norm of a function v

Q in terms of its L

-norm on Q, assuming

(9.9) v>0 and .@

 L/v  0: