On boundedness, periodicity, and almost periodicity of solutions of some nonlinear parabolic equations

JOURNAL

OF DIFFRRENTIAL

19, 371-385 (1975)

EQUATIONS

On Boundedness, Periodicity, and Almost Periodicity Solutions of Some Nonlinear Parabolic Equations

of

MITSUHIRO NAKAO Department

of Mathematics,

Faculty of Science, Kyushu Fukuoka, Japan 812

University,

Received June 25, 1974

Let Q be an open bounded domain in n-dimensional Euclidean space Rn and asZ its boundary. We shall consider partial differential equations of the form:

in D x R1 together with boundary

condition

u 1$2 = 0. Throughout

the paper we’ make the following

(0.2) assumptions: (0.3)

a.e. in J2 with some constant

C, > 0, a.e.

in 52,

(0.4)

and u&e) EL-q-l).

(0.5)

Bounded or almost periodic solutions to the problem (O.l)-(0.2) have been investigated by several authors, Amerio-Prouse [I], Biroli [2], [3], and others. But in their works the nonlinear terms fi(x, U) are usually assumed to be monotonically increasing in u. For example /3(x, U) = 1u 1% (a 3 0) is a typical one. In such cases the so-called monotonicity method can be applied, and it is known (see e.g. [2]) that bounded or almost periodic 371 Copyright All rights

0 1975 by Academic Press, Inc. of reproduction in any form reserved.

312

MITSUHIRO

NAKAO

solutions to this problem exist uniquely for all f(~, t)‘s that belong to appropriate Banach spaces. However, if /3(x, u) is not monotonic in u, we cannot use such a method, and the problem becomes more delicate. In this paper we shall treat just these cases and show that also in these cases some parallel results to the previous ones hold. Roughly speaking our results are: If f belongs to a certain Banach space V and the norm 11 f Ijy is small, there exists then a bounded solution of (O.l)-(0.2), and moreover if f is almost periodic (0.1~(0.2) admits an almost periodic solution. Also a uniqueness theorem holds in a local sense. The method of potential wells (stable sets) is known to be useful for nonlinear evolution equations with nonmonotonic terms (Sattinger [5], Lions [4], Tsutsumi [6]) and our idea seems to be akin to this method.

1. DEFINITIONS The notations follow to our arguments. LEMMA

Lions

AND PRELIMINARIES

[4]. The next well-known

lemma is the key

1.1 (Sobolev).

where 1 < q < 2n/n - 2 if n > 2 and 1 < q < + CO if n = 1,2, C, E C&n, C?) is th e constant which we call Sobolev’s constant.

and

In what follows for simplicity we shall drop the subscript Q in the notations of function spaces. Let ( , ) denote the duality relationship between W-Is2 and Wi*2. We define the linear operator A: W~~2--f W-1.2 by the equation:

(Av, W> = s ( i Q

iA=1

aij $

z

g

3

+ aOvw) dx

for

and put II u & = (Au, u) for u E Wi9”. Note that by the assumptions (0.3)-(0.5) and Lemma equivalent to II u Ilw;.a and (1.1) is replaced as

v, w E Wi*“,

1.1, I/ u ]lH,l is

(1.1’) Now we shall state our hypotheses.

313

NONLINEAR PARABOLIC EQUATION HYPOTHESIS I.

f(t) EL”(R : L2(Q)) (CLQR

: L2(Q)))

and

f’(t) ELrn(R : w-1.2).

Note that under the Hypothesis I (f(t), q~), v E lVi92, is Lipschitz continuous in t and of course differentiable almost everywhere. Later we shall use this fact. HYPOTHESIS II. For almost all x E Rn, /3(x, u) is continuous in u and there exist constants KI , K, > 0 such that

I ,&, 41 < Kl I u Y+l and I B(x, u) - B(x, v)l < K2(I u I + I a l)a I u - v I,

ae.

x,

forol~0,O,(a!~2/n-2ifn>2andO~or<+03ifn=1,2. By Hypothesis II and Lemma 1.1, we note, B(x, 4 E L2

if

u E W,‘*“.

Using the operator A, (O.l)-(0.2) can be written formally in the form: u’(t) + Au(t) + B(*, 4 = f(*, t)

(1.2)

u(t) E w,‘*“,

(1.3)

where ’ denotes the differentiation with respect to t. u(x, t) is said to be a strong solution of (1.2)-(1.3) if the following conditions are fulfilled: DEFINITION I.

(i)

u(t) E LfO,(R: w1s2)

(ii)

Au(t) ELF,,,,(R: L2(Q))

(iii)

u’(t) ELF&R: L2(Q)),

and R =$)

u/(x, t) f Au(x,

t)

+ 13(x,u(x,

t))

= f(x,

t)

a.e. on Q

x

R, where

DEFINITION II. Let u(t) be a measurable function with values in a Banach space V with norm II * /IV. We say u(t) to be V-almost periodic if for any E > 0 there exists a relatively dense set {T}< CR such that

es;gp II u(t + 7) - @NV < 6

for 7 E {TX .

MITSUHIRO NAKAO

374 DEFINITION III. S2(I’)-bounded if

Let u(t) be as in Definition

II. We say u(t) to be

and to be S2( V)- a1most periodic if for any E > 0, there exists a relatively dense set {T}~ such that

In what follows we shall use Definitions II and III in the cases V = IV1s2, wy, L2. 2. BOUNDED SOLUTION First we prove the existence of bounded solutions. Under the Hypotheses I and II we assume that there are numbers a, b > 0 such that THEOREM I (Existence).

M < a(1 - KrC~+“C~$$5,oI)/C,“C,2.

(2.1)

In addition, we assume if

M < a-lb2( 1 - KIC,“+lC~~&V-l)

,X21,

(2.2)

and 1 - &c;+sc;~$;

> 0,

(2.3)

where M = ess;up

Lo = I tia(M

IIfWL~

+ L2))li2,

,

if LX>1 zy l>a:>O,

and L, = +(l - fx)(a(a + I))ar+l’lea (K1C~+1C~~.2)2’1-4 Then (1.2~(1.3) has a strong solution u which sati$es ess y-v II u(t)ll,l ess fUP st

<

LO

,

t+l 11Au(s)ll;, ds < Lo2 + 2(M + KlC,“+1C;;;&+1)2

(2.4) (2.5)

NONLINEAR

PARABOLIC

375

EQUATION

where N = ess supt Ilf’(t)jlH-~.

For the proof of Theorem I, we prepare some lemmas. Let {wi}F be the basis of Wip2 consisting of the eigen functions of A, and consider the system of ordinary differential equations:

(4,&), 4 + Wn,&)~ wd + (8(*,~m.r(O~ 4 = (f(t), 4, j = 1, 2,..., m

(2.7)

with %J-r>

= 0,

(2.8)

where

Strictly speaking ai,r (i = l,..., m) depend on m, i.e., CQ~= ai,+,m , but we drop the subscript m for simplicity. Since (f(t), wi) is continuous in t (in fact Lipschitz continuous) and @(a, u&t), wj) is Lipschitz continuous in (oli,r), the system (2.7)-(2.8) h as a unique solution ar&t) (i = 1, 2 ,..., m), i.e., umsr(t), on some interval C-r, tm]. As usual we shall estimate u~,,( t ) in various norms. All of the assumptions in Theorem I are made for this purpose. LEMMA

[-r,

+a)

2.1.

Under the assumptions and the inequalities

II%r(t>ll~a< a

and

(2.1) and (2.2), um,Jt)

II%d~>ll*~ < Lll

exists on

P-9)

hold. Proof.

For the proof of the lemma, it suffices to prove (2.9) for

-r,(t
Multiplying the jth equation of (2.7) by CY~,? and summing over j from 1 to m, we have

505/I9/2-12

376

MITSUHIRO

Schwartz’s

inequality,

NAKAO

the assumptions

Next, multiplying thejth equation overj from 1 to 111,we have

and (1.1’) give

of (2.7) by hj (eigen value) and summing

< (M + JGc,“+1c;::2II%AIl;~~ - IIJ.ka,7(~)ll~ze) (2.11)

x II&n&Il~~ *

We shall prove (2.9) by contradiction. Suppose that (2.9) is false. Then, considering the initial condition and the continuity of +Jt) in t with respect to the norms 11* llLp and 11* Ilx,l , there are numbers ir and (or) is < t, such that

II%,?4t)ll~z< a

for

< t < &

-r

and

11u,),(Q~~~

= a,

(2.12)

and (or)

II%&)llql

G Lo

for

-r

< t < iz

and II~,&)llHol > Lo (2.13)

for fa
The Case i2 > il .

In this case, for -r

< t < ii, we have by (2.10)

NONLINEAR

PARABOLIC

377

EQUATION

noteing 1 - K,LouC,“+2C,“$ > 0,

< (M - (1 -

- IIum.r(t)ll,J

&-W~+2C:%‘02C22

x II%n,Ml,z > which implies

II%.&>llLz<

- woT+*c~;)

J4T32C22/u

< a,

for

< t < $ .

-Y

This is a contradiction to (2.12). (ii)

The Case il > i2 . In this case, for

< t < i1 , we have by (2.11)

--I

(2.14) If 0 < OL< 1, this and Young’s inequality give

where L, is the number given in the statement of Theorem I. This implies for which is a contradiction to (2.13). Next if 1 < 01,we have by (2.14), for

-Y

<

t

-Y

< zz

<

t

< fl,

378

MITSUHIRO

NAKAO

from which we can conclude /I u,,,(t)lg~

< M/(a-1 - K,C,“+lC,*,+:,b”-l) < b2,

for

-Y < t < f2 .

This is a contradiction since 11~,,,(t~)j1~~r = L, = 6. Next we shall estimate Au,,,(t) LEMMA 2.2. &Jr)

Q.E.D.

in S2(L2) norm.

is S2(L2)-bounded and

f-t+1 J, II Aum,r(s)ll;z ds ,< LOX+ (M + KlC;+1C;J-$L$+1)2. Proof.

(2.15)

By Lemma 2.1 we get SUPII%,&)ll~,’
Then similarly as in the proof of the previous lemma,

;-g II%n.r(m;~ < (M +

w,"+'CE2G"

-

II

&n.,Wll,*) II&&)ll,*

*

Integration from t to t + 1 and Schwartz’s inequality give t+1 st

II-kA~)II~~ ds < St + (M + W,“+lC;L$G+l) (s,“’ II&n,&)ll;~ df”,

from which (2.15) follows.

Q.E.D.

Our final lemma is LEMMA

2.3.

Under the assumptionsof Theorem I, II u&,,(t)llLs is bounded and

II4z,&)llp < Proof.

Gc2w~

-

~2coa+2czG).

(2.16)

By differentiation of both sides of (2.7) we get

(4Jt), Wj)+ (Ada&)> 4 + (Bu(*,%&)) 4&r(t), 4 = (f’ (th WA j = 1, 2,..., m.

NONLINEAR

PARABOLIC

EQUATION

379

Multiplication by &Jr), summation overj from 1 to m, Schwartz’s inequality, and Holder’s inequality give

+ K, 5, I um,r(x, W I 4rz,r(x,912dx -II 4&)l12Ho’

Here we used also Lemma 2.1 and (2.3). This differential inequality implies Q.E.D. (2.16). We are now ready to prove Theorem I. Proof of Theorem I. Using the above Lemmas 2.1, 2.2, and 2.3, by standard compactness arguments (Lions [4], Biroli [2]) we can extract an appropriate subsequence, which we denote also by {urn,?>,such that weakly star in Lm(R: H,,l), strongly in L&(R: L2), weakly star in Lm(R: L2), weakly in L&,(R: L’),

pz %*,(X, 4 = %(X,4

a.e. in

Sz x R

and F-i /3(., urn,(t)) = p(., urn(t)) weakly in L&(R: L2).

MITSUHIRONAM0

380

Thus urn(t) becomes the solution of

(%‘@>,Wj)+ w4&), Wi)+ (fx*, %n(t)),PO31 = (f(t), WA j = 1) 2 ,...) 171, (2.17) where

u,(t) = 5 aj(t)wj. j=l

Furthermore (2.4), (2.9 and (2.6) are valid for u = u, . We can again extract a subsequence of (urn}, which is denoted also by {urn>, such that

;z u,(t) = u(t)

weakly star in Lm(R: Hal),

ii

z&(t) = u(t)

strongly in L&(2?: L’),

&

u,‘(t) = u’(t)

weakly star in Lm(R: L2),

jii

Au,(t)

weakly in Lt,,,(R: L2),

= A,(t)

lim u,(x, t) = u(x, t) m-m

a.e. in

~2 x R

/ii

weakly in LT,,(R: L2),

and fi(*, urn(t)) = /I(*, u(t))

and u(t) satisfies (2.17) for j = 1, 2 ,..., or u’(x, t) + (Au)(x, t) + /3(x, u(x, t)) = f(x, t)

a.e. on fJ x R.

(2.4), (2.5), and (2.6) are of course satisfied. Thus u(t) is a required strong Q.E.D. solution. After the existence theorem is proved, the question of uniqueness of the bounded solution arises naturally. We do not know at this time whether the global uniqueness is valid or not, but some local uniqueness theorems hold. THEOREMII (Local Uniqueness).

Assume that

2°LK2L0aC;+2C,a=22 < 1.

(2.18)

Then the strong solution of (1.2)-( 1.3), if it exists, is uniquely determined in B L,, where BLO = (u(t) EL~(R: f&l) / ess ;up II u(t)ilHOl < L,}.

NONLINEAR

Proof.

PARABOLIC

381

EQUATION

Let z+(t) and z+(t) be strong solutions of (1.2H1.3) in BL, . Then

; $ II %W - uaw;*

Hence

II dt)

-

u~(W~

is monotonically

decreasing. Moreover

using

Lemma 1.1 we have

where

Therefore II %(h) - ~2(~1)/1~2< II u1(t,) -

U2(t,)ll2,

for

- 2~5,It1 II u,(t) - uz(t)ll;z dt

tz

t, > t,.

(2.19)

Supposing ur + u2 , there exists a number t, such that II u&J - U&,)ll;, >, 6 > 0, and then, since /I n,(t) - u2(t)jlLeis decreasing, we have II w

- u,(t)ll;*

2 E > 0,

for

t E (-co,

tl].

(2.19) implies 6 < II u&J

- u,(tl)ll;z

< II U&2) - u,(t,)ll;*

- 2L&

-

t2)

or II u,(t,) - u2(t,)ll;z a E + 2L&

- t2) --t 00

as t23

This is a contradiction that ui(t) (i = 1, 2) belong to BL, .

--co. Q.E.D.

382

MITSUHIRO

NAKAO

Remark. If M = ess sup, Ijf(t)llLa is sufficiently small, we can choose a and b to be small (Theorem I) so that a bounded strong solution of (1.2)(1.3) may exist and satisfy (2.18). In the cases 0 < a < 4/n,

if

n>2,

and

0 <(Y < 2,

if

n = 1, 2,

(2.20)

we can give a slightly different uniqueness theorem. THEOREM

II’.

Let (Ybe as (2.20) and assumethat 2”a”K,C,2C,2,,-, < 1.

(2.21)

Then the strong solution is unique in S, , where S, = {u EL~(R: L2) j ess sup 11u(t)llr2 & CZ}. Proof, Let ui(t) (i = 1,2) be strong solutions in S, . Similarly as in the proof of Theorem II, we obtain

; & IIs(t) -

uz(t)ll;e < -II 111- ~2 ll;o~+ 2aK2~aG2C:,2-u

= -(l GO

- 2”aaK,c,2c,,2-,)

IIu1 - ~2ll$

/I ui - Ua11;;

by (2.21).

(2.22)

Here we used the fact 412 - a ,< 2nln - 2

if

(2.20) holds.

From (2.22) it is easily verified that t+(t) = u2(t).

Q.E.D.

3. PERIODIC OR ALMOST PERIODIC SOLUTION In this last section we shall investigate the periodicity or almost periodicity of strong solutions of (1.2)-(1.3). The following theorem is an immediate consequence of the results of the previous section. THEOREM III. In addition to the assumptions of TheoremsI and II (01 II’), we assumef (x, t) is w-periodic in t. Then (1.2)-(1.3) has at least one w-periodic strong solution.

NONLINEAR

PARABOLIC

383

EQUATION

Proof. By Theorem I, (1.2)-(1.3) admits a strong solution u EL~(R: We*“) which satisfies II WI,2

< a

and

II Nll,l

< Lo -

The periodicity of f(t) implies zl(t + W, X) is also a bounded solution with the same bounds as u, and by the local uniqueness Theorem II (or II’) we conclude Q.E.D.

u(t) = u(t + w). The main result in this section is

THEOREM IV. In addition to the assumptions of Theorems I and II we assumef(t) is W- 1*2-almostperiodic. Then the bounded solution u(t) constructed in Theorem I is L2-almost periodic and S2(W,$2)-almostperiodic.

Proof. By the almost periodicity assumption on is a relatively dense set {T}~ such that

IIf@ + 4 -f(t>ll,-I

< E>

for

f,

for any E > 0 there

7 E {T}~-.

(3.1)

Let urn(t) be the solution of (2.17). We have then similarly as in the proof of Theorem II,

; $ II%n(t+ 4 - %n(t>ll;~ < (E- co%% 11%n(X+ 7) - %&>lq> 11Urn@ + T) - %n(t)ll, < (E- L, 11%n(t+ T>- %(t>il,2>11‘& + T>- %(t>&,; 3

(3.2)

where we recall L, = (1 - 2aK2C~+2C:~~,ol)/C,2C,2 > 0

by (2.18).

From this we obtain

d-& . 11%n(t+ 7) - %(t)ilLa < ’ Hence urn(t) is L2-almost periodic. Next, integration of (3.2) gives

< E s,I 11%(t + 7 + s) - u,(t + s)ll H,’ ds - co%%

s’ 11%(t + T + s> - %(t + s)ll$ ds. 0

(3.3)

384

MITSUH~O

NAKAO

By (3.3), Schwartz’s inequality and Young’s inequality we have

or

This implies the S2(W~92)-almost periodicity of urn(t). From (3.3) and (3.4) the limit u(t) of a subsequence of (~&t)> is also La and ~*(~~~‘~-aImost periodic. Q.E.D. Remark. In the assumptions of Theorem IV, if Theorem II’ is valid instead of Theorem II, we obtain only L2-almost periodicity of g(t). THEORIXM IV’. In addition to the assumptions of Theorem IV, we assume f(t) is La-almost periodic. Then the bounded solution u(t) is W,“2-almost periodic.

Proof, We give the proof briefly. Since u,(t) is La-almost periodic, similar arguments in Lemma 2.1 and Theorem II give

II%ntt + 4 - %?z(t)il,~ < wo ,4 *6, are almost periods of f(t) in L2. This yields the

where {& periodicity of u(t).

(3.5) W$2-almost Q.E.D.

Remark. As is easily verified, under the assumptions of Theorems I and II (or II’), the unique bounded solution depends on continuously. Indeed if Theorem II is valid, we can obtain

f(t)

ess;up II44

- f42Ww~.a \<

W.

, a, M) es.2 SUP IL&(t) -f2(t)llL2

,

which is essentially the same inequality as (3.5), where ai(t) {i = 1,2) is the corresponding bounded solution to f =fi(t).

NONLINEAR

PARABOLIC EQUATION

385

REFERENCES 1. L. AMERIO AND G. PROUSE,“Almost Periodic Functions and Functional Equations,” Van Nostrand, New York, 1971. 2. M. BIROLLI, Sulla esistenza ed unicita della soluzione limitata e della soluzione quasi periodica per una equazione parabolica con termine dissipativo non lineare discontinue, Ricer&e Mat. XIX (1970), 93-l 10. 3. M. BIROLLI, Solutions bornees et presque periodiques des equations et inequations devolution, Ann. Mat., Ser. 4, XC111 (1972), l-79. 4. J. L. LIONS, “Quelques MCthodes de Resolution des Problemes aux Limites Nonlintaire,” Dunod, Paris, 1969. 5. D. H. SATTINGER, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148-172. 6. M. TSUTSUMI, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. R. I. M. S., Kyoto Univ. 8, 1973, 21 l-229.