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

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348 15. Nonlinear Parabolic Equations

Of course, Tr N<0if >0; meanwhile,

(4.52) det N D d



C d.1  b/



 C a

D p./:

The matrix N will fail to have spectrum in the left half-plane, for some >0,

if p./ is not always >0for >0, hence if p./ has a positive root. From the

quadratic formula, the roots of p./ are

(4.53) r

D

C d.1  b/



C d.1  b/



 4a

Thus p./ will have positive roots if and only if a

C d.1  b/ < 0 and



d.1  b/



>4a

d . Recalling the conditions on a and b to make Tr M<0,we

have the following requirements on the positive numbers b; a; d:

(4.54) b  1<a

<.b 1/d; 2.b C 1/a

d<.b 1/

C a

;

which also requires b>1;d>1. For example, we could choose b D 2;a

D 2,

and d D 20, yielding

(4.55) D D





;MD



2 2



Under these circumstances, M  D will have a positive eigenvalue for

(4.56)  2 .r



/  .0; 1/;

where r

are given by (4.53).

Consequently, if the Laplace operator  on M has an eigenvalue ˛

whose

negative is in the interval (4.56), arbitrarily small initial data of the form yf

will

be magniﬁed exponentially by the solution operator to u

D .L CM/u, provided

y has a nonzero component in the positive eigenspace of M  ˛

D, despite the

fact that the origin is a stable equilibrium for the evolution if the diffusion term is

omitted.

An example of a nonlinear reaction-diffusion equation that exhibits this phe-

nomenon is the “Brusselator,”

(4.57)

D v C v

w  .b C 1/v C a;

D dw  v

w C bv;

governing a certain system of chemical reactions. We assume a; b; d > 0.

The vector ﬁeld X (which incidentally has ﬂow leaving invariant the quadrant

v; w  0) has a critical point at .a; b=a/, and its linearization at this critical point

4. Reaction-diffusion equations 349

is given by the matrix M in (4.50). Thus if u

D .v

/ is a small perturba-

tion of the constant state .a; b=a/, if the estimates (4.54) hold, and if  has an

eigenvalue whose negative is in the range (4.56), then a vector multiple of the

eigenfunction f

will be ampliﬁed by the evolution (4.57). Of course, once this

acquires appreciable size, nonlinear effects take over. In some cases, a spatial pat-

tern emerges, reﬂecting the behavior of the eigenfunction f

.x/. One then has the

phenomenon of “pattern formation.”

In light of the instability just mentioned, we see some limitations on using

invariant rectangular regions to obtain estimates. Consider the following more

general type of Fitzhugh–Nagumo system:

(4.58)

D D

C f.v/C a.v; w/;

D b.v;w/;

where f; a,andb are assumed to be smooth and satisfy

(4.59) ja.v; w/jA



jvjCjwjC1



; jb.v;w/jB



jvjCjwjC1



;

and

(4.60) f.v/  C

v; for v  C

;f.v/ C

v; for v C

;

where A; B; C

,andC

are positive constants. There need be no large invariant

rectangles in such a case, as the example f.v/ D .2A C2B C1/v shows. Never-

theless, one will have global solutions to (4.58) with data in BC

.R; R

/. In fact,

this is a special case of the following result.

To state the result, we use the following family of rectangular solids. Let Q.s/

be the cube in R

centered at 0, with volume .2s/

,andletF

˙j

.s/ be the face of

this cube whose outward normal is ˙e

,wherefe

W 1  j  `g is the standard

basis of R

Proposition 4.6. Let L D D with D D diag.d

;:::;d

/ in (4.1). Assume the

components X

of X satisfy, for some C

2 .0; 1/,

(4.61)

.y/ CC

s for y 2 F

.s/;

.y/ C

s for y 2 F

j

.s/;

for all s  C

.Then(4.1) has a global solution, for any f 2 BC

; R

Proof. We will obtain this as a consequence of (4.33). Use the norm kykD

max

jon R

to construct the norm on function spaces. The hypothesis implies

(4.62) kF

yke

kyk;t 0;

350 15. Nonlinear Parabolic Equations

whenever kykC

. Consequently,

(4.63)





.t=n/L

t=n





 e



kf k

C C



;

so by (4.33) we have the bound on the solution to (4.1):

(4.64) ku.t/k

 e



kf k

C C



Note that this is an application of Proposition 4.4, in a case where fK

g is an

increasing family of rectangular solids. Other proofs of global existence for (4.58)

under the hypotheses (4.59)–(4.60)aregivenin[Rau]and[Rot].

There are some widely studied reaction-diffusion equations to which Propo-

sition 4.6 does not apply, but for which global existence can nevertheless be

established. For example, the following models the progress of an epidemic,

where v is the density of individuals susceptible to a disease and w is the den-

sity of infective individuals:

(4.65)

Drvw;

D Dw C rvw  aw:

Assume r; a; D > 0. In this model, only the sick individuals wander about. Let’s

suppose  is the Laplace operator on a compact two-dimensional manifold (e.g.,

the surface of a planet). One can see that the domain u;v  0 is invariant; initial

data for (4.65) should take values in this domain. We might consider squares of

side s, whose bottom and left sides lie on the axes, but the analogue of (4.61) fails

for X

D rvw  aw, though of course X

 0 is ﬁne. To get a good estimate on

a short-time solution to (4.65), taking values in the ﬁrst quadrant in R

, note that

(4.66)



v C w



D Dw  aw:

Integrating gives

(4.67)

.v C w/ dV Da

wdV  0:

By positivity,

.v Cw/ dV Dkv.t/k

Ckw.t/k

, which is monotonically de-

creasing; hence both kv.t/k

and kw.t/k

are uniformly bounded. Of course,

we have already noted that kv.t/k

kv

. Thus, inserting these bounds

into the second equation in (4.65), we have

(4.68)

D Dw C g.t; x/;

Exercises 351

where

(4.69) kg.t/k

.M /

 rkv.t/k

kw.t/k

C akw.t/k

 C:

Now use of

(4.70) w.t/ D e

tD

.ts/D

g.s/ ds

plus the estimate

(4.71) ke

s

L.L

 Cs

.11=p/

when dim M D 2, yields an L

-estimate on w.t/, and another application of

(4.70) then yields an L

-estimate on w.t/, hence global existence.

Actually, for a complete argument, we should replace vw in the two parts of

(4.65)byˇ



.vw/,whereˇ



.s/ D s for jsj,andˇ



.s/ D  C1 for s   C1,

get global solvability for such PDE, with L

-estimates, and take  !1.We

leave the details to the reader.

The exercises below contain some other examples of global existence results.

In [Rot] there are treatments of global existence for a number of interesting

reaction-diffusion equations, via methods that vary from case to case.

Exercises

1. Establish the following analogue of Proposition 4.6:

Proposition 4.6A. Let L D D with D D diag.d

;:::;d

/ in (4.1). Assume that

the set C

Dfy 2 R

W each y

 0g is invariant under the ﬂow generated by X.

If F

.s/ is as in Proposition 4.6,setF

.s/ D C

\ F

.s/, and assume that each

component X

of X satisﬁes

.y/  C

s; for y 2 F

.s/;

for all s  C

.Then(4.1) has a global solution, for any f 2 BC

; R

/ taking

values in the set C

2. The following is called the Belousov–Zhabotinski system. It models certain chemical

reactions, exhibiting remarkable properties:

(4.72)

D v C v.1  v  rw/ C Lrw;

D w  bvw  Mw:

Assume r; L; b; M > 0. Show that the vector ﬁeld X has ﬂow that leaves invariant the

quadrant fv; w  0g. Show that Proposition 4.6A applies to yield a global solvability

result.

352 15. Nonlinear Parabolic Equations

3. The following system models a predator-prey interaction:

(4.73)

D Dv Cv.1  v  w/;

D w C aw.v  b/;

where v is the density of prey, w the density of predators. Assume D  0; a > 0,and

0<b<1. Show that the vector ﬁeld X has a ﬂow that leaves invariant the quadrant

fv; w  0gDC

. Show that Proposition 4.6A does not apply to this system.

Demonstrate global existence of solutions to this system, for initial data taking values

in the set C

.(Hint: Start with the identity

(4.74)



v C



D Dv C

w C v  v

 bw

and integrate, to obtain L

-bounds. Also use

(4.75)

 Dv  v

to obtain an L

-bound on v. (If D>0, recall Lemma 2.2.) Then pursue stronger

bounds on w:)

4. If the model (4.65) of an epidemic is extended to cases where susceptible and infective

populations both diffuse, we have

(4.76)

D D

v  rvw;

D Dw C rvw  aw;

where D

;D;r;a > 0. Establish global solvability for this system, for initial data

taking values in C

5. Study global solvability for the Brusselator system (4.57), given initial data with values

in C

.(Hint: After getting an L

-bound on v C w,use

 dw  bv;

and an appropriately modiﬁed version of the argument suggested for Exercise 3,toget

a stronger bound on w. Once you have this, use

(4.77)



v C w



 



v C w



D .d  1/w  v C a

to obtain a stronger bound on v C w:)

6. Consider the following system, modeling a chemical reaction A C B ˛ C :

(4.78)

 D

a D c  ab;

 D

b D c  ab;

 D

c D ab  c:

Note that X leaves invariant the octant C

Dfa; b; c  0g. Assume D

>0. Establish

the global solvability of solutions with initial data in C

. Assume  is the Laplace

5. A nonlinear Trotter product formula 353

operator on M , compact, with dim M  3.(Hint:FirstgetL

-bounds on a C c and

b C c.Thenuse

 D

a  c; b

 D

b  c;

to get L

-bounds on a and b,forsomep>2(if dim M  3). Then use

 D

c  ab

to get L

-bounds on c. Continue. Alternatively, apply an argument parallel to (4.77)to

a C c, and relax the requirement on dim M:)

For a treatment that works for dim M  5 (and @M ¤;), see [Rot].

7. Extend results of this section to the case where L D D,whereD is a diagonal matrix,

D D diag.d

;:::;d

/,and is the Laplace operator on a compact manifold M with

boundary. Consider each of the following boundary conditions:

(a) Dirichlet, u

@M

D 0;

(b) Neumann, @



@M

D 0;



 a

.x/u

@M

D 0:

Apply such a boundary condition only if d

>0.

Also, consider nonhomogeneous boundary conditions.

5. A nonlinear Trotter product formula

In this section we discuss an approach to approximating the solutions to nonlinear

parabolic equations of the form

(5.1)

D Lu C X.u/; u.0/ D f;

and some generalizations, to be mentioned below, by a process involving succes-

sively solving the two simpler equations

(5.2)

D Lu;

D X.u/;

over small time intervals, and composing the resulting solution operators. If F

denotes the nonlinear evolution operator solving the equation @u=@t D X.u/,we

seek to show that the solution to (5.1) satisﬁes

(5.3) u.t/ D lim

n!1



.t=n/L

t=n



.f /:

This is a nonlinear analogue of the Trotter product formula, discussed in

Appendix A of Chap.11. It is a popular tool in the numerical study of nonlinear

evolution equations, where it is also called the “splitting method.”

354 15. Nonlinear Parabolic Equations

We will tackle this via a variant of the analysis used in (A.17)–(A.30) in our

treatment of the Trotter product formula in Chap. 11. The author came to under-

stand this approach through conversations with J. T. Beale on the work [BG].

Other approaches are discussed in [CHMM].

We begin by setting

(5.4) v



.1=n/L

1=n



.f /

and then set

(5.5) v.t/ D e

; for t D

C s; 0  s<

Then (under suitable hypotheses on L,etc.)v.t/ ! v

kC1

as t % .k C1/=n, and,

for k=n < t < .k C 1/=n,

(5.6)

D Lv C e

X.F

/ D Lv C X.v/ C R.t/;

where, again for t D k=n C s; 0  s<1=n,

(5.7)

R.t/ D e

X.F

/  X.v/



 I



X.F

/ C



X.F

/  X.e



To compare v.t/ with the solution u.t/ to (5.1), set w D v  u. Subtracting

(5.1) from (5.6)gives

(5.8)

D Lw C X.v/  X.u/ C R.t/;

and if we write

(5.9) X.v/  X.u/ D



sv C .1  s/u



ds D Y.u;v/w;

we have for w the linear PDE

(5.10)

D Lw C A.t; x/w C R.t/; w.0/ D 0;

where

(5.11) A.t; x/ D Y



u.t; x/; v.t; x/



;

which is an `  ` matrix function if (5.1)isan`  ` system.

5. A nonlinear Trotter product formula 355

We will treat this in a fashion similar to (A.24)–(A.28) in Chap. 11. To recall

some points that arose there, we expect to show that R.t/ is small only in a weaker

norm than that used to measure the size of v  u. In our ﬁrst set of results, we

compensate by exploiting smoothing properties of e

. Thus, for now we assume

that L is a negative-semideﬁnite, second-order, elliptic differential operator. For

the sake of deﬁniteness, let us suppose L acts on (`-tuples of) functions on R

with domain

(5.12) D.L/ D H

We will seek to estimate v  u in some V -norm.

Now, by Duhamel’s principle,

(5.13) w.t/ D

.t/L



A./w./ C R./



d:

Pick T>0;2 .0; 1/, and Banach spaces of functions V; W for which we have

an estimate of the form

(5.14) ke

 Ct



kgk

;0<t T:

The next step is to estimate the W -norm of R.t/,givenby(5.7). We have sepa-

rated this into two parts:

(5.15) R

.t/ D .e

 I/X.F

/; R

.t/ D X.F

/  X.e

where t D k=n C s; 0  s<1=n. Parallel to (A.26), we need an estimate of the

form

(5.16) ke

 I k

L.V;W /

 Cs

;ı>0;

to estimate R

.t/. Of course, this requires that W have a weaker topology than V .

Granted this estimate, since s 2 Œ0; 1=n in (5.15), we obtain

(5.17) kR

.t/k

 Cn

ı

kˆ

.t/k

;

where

(5.18) ˆ

.t; x/ D X



‰

.t; x/



;

with

(5.19) ‰

.t; x/ D F

.x/ D F



.x/



;

356 15. Nonlinear Parabolic Equations

where F

is the ﬂow on R

generated by X, a vector ﬁeld on R

. Meanwhile,

(5.20) R

.t/ D ˆ

.t/  ˆ

.t/;

where ˆ

.t/ is as in (5.18)–(5.19)and

(5.21) ˆ

.t; x/ D X



‰

.t; x/



;‰

.t/ D e

‰

.t/:

Thus, with Y.u;v/given by (5.9),

(5.22) R

.t/ D Z.t/.I  e

/‰

.t/; Z.t / D Y



‰

.t/; ‰

.t/



;

so, again using (5.16), we have

(5.23) kR

.t/k

 Cn

ı

kZ.t/k

L.W /

k‰

.t/k

;

where Z.t/ denotes the operation of left multiplication by the matrix-valued func-

tion Z.t; x/.

There remains the task of estimating the right sides of (5.17)and(5.23). This

involves estimating the V -norms of

(5.24)



.1=n/L

1=n



f; ‰

.t/ D F

;

‰

.t/ D e

‰

.t/; ˆ

.t/ D X



‰

.t/



;

and the L.W /-norm of Z.t/ D Y



.t/; ˆ

.t/



. Thus, we want the estimates

(5.25) ke

L.V /

 e

; kF

.f /k

 e

kf k

;0 t  T;

rather than weaker estimates in which e

is replaced by C

. On most of our

favorite Banach spaces V; e

is frequently a contraction semigroup, while the

second estimate in (5.25) may require more work to establish. Actually, we need

this second estimate only for kf k

 some constant C

. We get a good estimate

on all the quantities in (5.24) if the estimates (5.25) hold and also

(5.26) X W V ! V is bounded,

where Xf.x/ D X.f .x//.By(5.26) we mean that X.S/ is bounded in V when-

ever S is a bounded subset of V . In most examples, X will be locally Lipschitz,

which is more than sufﬁcient. Granted these hypotheses, to get a good bound on

Z.t/ it sufﬁces to have that

(5.27) Y W V  V ! L.W / is bounded;

where Y.f; g/.x/ D Y.f .x/; g.x//. Let us summarize this analysis:

5. A nonlinear Trotter product formula 357

Proposition 5.1. Let V and W be Banach spaces of (`-tuples of) functions for

which e

satisﬁes the estimates

(5.28) ke

L.V /

 e

; ke

L.W;V /

 Ct



; ke

 I k

L.V;W /

 Ct

;

for 0<t T , with some ı>0;0<<1.LetX be a vector ﬁeld, generating a

ﬂow F

on R

, satisfying

(5.29) kF

.f /k

 C

; for kf k

 C

;

and, for kf k

 C

(5.30) kF

.f /k

 e

kf k

;

for 0  t  T ,whereF

f.x/ D F

.f .x//. Assume also that

(5.31) X W V ! V and Y W V  V ! L.W / \ L.V / are bounded,

where Xf.x/ D X.f.x// and

Y.f; g/.x/ D Y



f.x/;g.x/





sg.x/ C .1  s/f .x/



ds:

If f 2 V; u 2 C



Œ0; T ; V



is a solution to (5.1), and v 2 C



Œ0; T ; V



is deﬁned

by (5.4)–(5.5), then, for 0  t  T ,

(5.32) kv.t/  u.t /k

 C



kf k



n

ı

Proof. The hypothesis (5.31) also yields an L.V /-bound on A./ in (5.13), so we

have

(5.33) kw.t/k

 A

c.t/

kw./k

d CkF.t/k

;

where A is a constant and

F.t/ D

.t/L

R./ d :

From the hypotheses (5.28)–(5.31) and the consequent estimates in (5.17)and

(5.23), we have

(5.34) kF.t/k

 C



kf k



.t  /



d  n

ı

D B



kf k



n

ı

1