Almost periodic solutions to nonlinear elliptic and parabolic equations

Nonlinear Andysir, Theory, Metho& Printed in Great Britain.

ALMOST

& Applicntionr,

PERIODIC

Vol. 7, No. 4, pp. 357-363,

SOLUTIONS PARABOLIC

.

1983.

TO NONLINEAR EQUATIONS

036%546&‘83/040357-07 $03.00/O 0 1983 Pergamon Press Ltd.

ELLIPTIC

AND

C. CORDUNEANU* University

of Texas at Arlington,

Texas 76019, U.S.A

(Received 9 September 1982) (This paper

is dedicated

Key words and phrases: Elliptic

paper is devoted nonlinear equations THIS

to Professor

equation,

parabolic

Luigi Amerio, equation,

on his 70th birthday)

almost

periodic

to the problem of almost periodicity u~=Au+~(~,x,u),(~,x)ER

solutions

of bounded

xQ.

CL,; + Au = f(t, X, u), (t, x) E R x R,

solutions for the (El) @I

where Q C R” is a bounded domain with smooth boundary C&J= S. In either case, the boundary value condition tER,

u Is = 0,

PVC)

is associated to the equation. It will be noticed in the sequel that alternate boundary value conditions can be associated with (El) or (Ez), in view of obtaining the almost periodicity of the solution. The kind of almost periodicity we are going to deal with is the L’(Q)-almost periodicity. In other words, if u = u(t, x), (t, x) E R X Q, is a solution of (El) or (E2), then we look at the map t-+ u(t, .) as a map from R into L2(Q). Its almost periodicity, as an abstract function with values in L2(Q), will be our concern in this paper. For the basic properties of the L’(Q)-almost periodic functions, see [3], where the case Q = (a, b) C R is dealt with in detail (chapter 2), while the general case can be easily derived from the theory exposed in chapter 6. Of course, the L2(W)-almost periodicity makes sense only for those solutions of (El) or (Ez) which are L2(Q)-bounded, i.e. there exists M > 0 such that u2(t, X) dx c M, WER. (1) In We shall not approach in this paper the problem of existence of the L’(Q)-bounded solutions of the above mentioned equations. Instead, we will concentrate on finding conditions on the nonlinearity f, among which an almost periodicity condition must be included such that the L’(Q)-boundedness of a solution imply its L2(Q)-almost periodicity. It will follow then that uniqueness of the almost periodic solution is also the consequence of the conditions imposed * Partially

supported

by U.S. Army

Research

Grant

No. DAAG29-80-C-0060 357

358

C. CORDUNEANLJ

on the nonlinearity. Moreover, these conditions will emphasize a certain connection in between the nonlinearity and the spectrum of the Laplace operator occurring in both (El) and (Ez). Our approach to the problem of L2(Q)-almost periodicity is based on two qualitative inequalittes (lemmas) given in [4] and [5]. In those papers, the lemmas were applied to derive almost periodicity in Bohr’s sense. Therefore, heavier conditions had to be imposed on the solution (besides its uniform boundedness). It turns out that the lemmas in [4] and [5] find even more natural applications to the case of L’(Q)-almost periodicity. It is interesting to point out that the estimates from which we shall derive the L’(Q)-almost periodicity, based on the lemmas in [4] and [5], lead also to the periodicity of the L’(R)bounded solutions, provided the nonlinearity f is assumed to be periodic. In such a case, the period of the solution is the same as for f. The problem of almost periodicity in cases of parabolic and elliptic equations has been discussed by many authors, including results on L’(Q)-almost periodicity, as well as variational inequalities instead of equations. For instance, C. Foias and S. Zaidman [6] dealt with the parabolic case, while S. Zaidman [9] considered the problem of L’(Q)-almost periodicity in the linear elliptic case (the special case of (E2), when f(t, x, u) does not contain the unknown function u). The nonlinear parabolic case has been investigated by C. Vaghi in [8], providing conditions that also assure the existence of a bounded solution in the classical sense. Recent papers by R. E. Bruck [l] and A. A. Pankov [7] deal with elliptic and parabolic cases respectively, in an abstract setting. They also contain rather extensive lists of references. See also S. Zaidman [lo] for further developments. In [2], A. Castro and A. C. Lazer consider the periodicity problem in the parabolic case, thoroughly investigating the existence of solutions. More significant contributions to this problem are included in their reference list. Let us point out the fact that the existence problem in the almost periodic case has been discussed in [8] and [7] for the nonlinear parabolic case, under rather restrictive assumptions (including some compactness conditions or bilateral estimates, instead of the unilateral ones required in this paper). Before formulating our results, let us state the basic assumptions to be made on the nonlinear term f(t, x, u), occurring in both equations (E,) and (Ez). C. The mapf: R x Q x respect to u E R.

R + R is continuous

and L2(Q)-almost

Cl.The map f is of class C(l) in u, and the derivative

periodic

in t, uniformly

with

fu satisfies (2)

where

,U< A,, with A1 the smallest

eigenvalue

of the problem

Av+hv=O

inQ,

vIs=

C2. The map f is of class C (‘1 in u, and the derivative

fu2-p

inRxQxR,

where n < Al, with Ai the same as in condition The main results of this paper can be stated

Cl. as follows.

0.

(3)

fu satisfies (4)

Almost periodic solutions to nonlinear elliptic and parabolic equations

359

THEOREM 1. Let u = u(t, x), (t, x) E R X 9, be a solution of class C”(a) for the parabolic equation (El), under condition (BVC). Assume further that u is L*(Q)-bounded, i.e. verifies

a condition of the form (1). If f(t,x, U) satisfies conditions C and C,, then u(t,x) is L2(S2)almost periodic. If f(t, x, u) is periodic in t, then so is u(t, x). THEOREM 2. Let u = u(t, x), (t, x) E R X a, be a solution of class C(*)(a) for the elliptic equation (E2), under condition (BVC). Assume further that u is L*(Q)-bounded. If f(t, x, u) satisfies conditions C and CZ, then u(t, x) is L*(Q)-almost periodic. If f(t, x, u) is periodic in t, then so is u(t, x). Proof of Theorem 1. Let u(t, x) be the solution whose existence is assumed in theorem 1, and let r~ R be an arbitrary fixed number. Denote u(t, x) = ~(t + t, x) - u(t, x), (t, x) E R x 2. From (El) and its analogue, obtained by changing t in t + t, one obtains v,=

Au + f(t+

x,x,u(t

+ v))

-f(t,x,

u(V)),

(3

as well as the boundary value condition vIs=O,VtER.

(6)

Let us multiply now both sides of (5) by U, and integrate over 52. There results for any tER:

:;/ouZdx=j-ouAudx+

i,[f(t+

v,u(t+t,x))

-f(r,x&n))]u

By application of Green’s formula to the first integral in the right-hand elementary calculations, one obtains v*dx = -

dx.

(7)

side of (7), and

)grad v 1’dx

t + t, x,

u(t + z, x)) -f

t + r, x, u(t, x))

-f@,

(t + z, x, u(t, x))] u dx x, u(t, x))] v dx,

(8)

taking into account the condition (6). On behalf of Poincare’s inequality one can write u2dx,

(9)

with ill as defined in condition Ci. The second integral in the right hand side of (8) is obviously dominated by p Consequently,

in

v* dx.

(10)

(8) with condition (BVC) lead to the following inequality, which holds true

C. CORDUNEANU

360

on the whole

real axis: u’dx

6 -(Ai

+

- p)

v2dx

(L i,2dx)1’2

112

jsupL

If(t+

z,x,u)

-f(t,x,u)

j2dx

.

(11)

with the supremum taken for t, u E R. This supremum is finite due to condition C. The inequality (11) has the form required by the lemma in [5], with regard to the function V(t) = i, u2(t, x) dx, Applying the conclusion the definition of u: sup

I

of the lemma

WER.

(12)

in [5] to the inequality

(ll),

one obtains

on behalf

of

R 1u(t + z, x) - u(t, x) I2 dx c (Ai - p) -2 sup

I

*lf( t + t, x, CL)- f(t, x, u) I* dx.

(13)

In (13), the first supremum is taken with respect to t E R, while the second one is the same as in (11). The inequality (13) proves that u(t, x) is L2(Q)-almost periodic. Moreover, when f (t, x, CL) is periodic in t, say with period T > 0, the same holds true for u(t, x). Simultaneously, the inequality (13) provides a very simple connection between the almost periods off (t, x, u), and those of u(t, x). n Proof of Theorem 2. Let u = u(t, x) be the solution of equation (Ez), under boundary condition (BVC), and such that (1) holds true for some M > 0. One considers again the function u(t, x) from the proof of theorem 1, where r is a fixed real number. From (El) and its analogue obtained by changing t in t + z, one easily finds urt + Au = f(t + z, x, u(t + z, x)) - f(t, x, u(t, x)). Let us multiply the identity

now both sides of (14) by 2v, and integrate

and then apply the Green’s d2 -1 u2dx-2 dt2 o

formula,

(14)

over 52. If one takes into account

one obtains

IQ (u,)~ dx - 2 LI grad u I2dx f(t + z,x, u(t + z, x)) - f(t, x, u(t, x))] u du.

Using

again

Poincare’s

inequality,

and neglecting

the second

(negative)

(16)

term in the left hand

Almost

periodic

solutions

side of (16), one derives d* -1 u2dx>23L1 dt* o

to nonlinear

elliptic

and parabolic

361

v2dx

t+z,

x, u(t + r,

x)) - f(t, x, u(t, x))] v dx.

The last integral in the right hand side of (17) can be transformed proof of theorem 1 above. There results d2 -jv2dx>2(Ar-p) dt2 Q

(17)

in the same manner as in

v2 dx

- 2(/,U?dxj”*(sup~o where the supremum

equations

lf(t+ GX,U>-fkx,d*dx

(18)

is taken with respect to t, u E R. Of course, (18) holds true for any

t E R.

Since (18) is an inequality of the type considered in the lemma of [4], one can apply the conclusion of that lemma. One obtains sup p 1u(t + z, x) - u(t, x) I* dx I

c

@l

-

Pr2 sup Izl lf( t + z, x, u) - f(t, x, u) I*dx,

(19)

which proves the assertion of theorem 2 (both almost periodic and periodic cases). Let us discuss now a few problems related to the results obtained in theorems 1 and 2. Uniqueness. As mentioned above, the hypotheses of theorems 1 and 2 assure the uniqueness of the L*(Q)-bounded solution (if such a solution exists). The proof of the uniqueness can be easily obtained using the same kind of arguments as in the proofs of theorems 1 and 2. For instance, in the case of equation (Ez) with condition (BVC), if one admits the existence of two L*(Q)-bounded solutions u and ti, then v = u zi satisfies an inequality similar to (17), namely

d* v* dx, Vt E R. -jv2dx32& (20) dt* o I This inequality leads immediately to the concltsion that v = 0 in Q. Indeed, it shows the convexity of Jo v2 dx on the whole real axis, and since the integral is bounded there, one obtains the assertion. An alternate approach could be based on the lemma in [4]. One can strengthen the inequality (20) by subtracting an arbitrary small positive number from its right-hand side. Then, letting this number tend to zero in the estimate for Jo u* dx obtained from the lemma [4]. Under our assumptions of regularity, one obtains v = 0 in R. Other boundary value conditions. If instead of condition (BVC) one attaches to (Er) or (E2) another boundary value condition, for instance

au $1, =huls,

(BVC)’

C. CORDUNEANU

362

where n represents the inward normal to S, and h is a nonnegative function on S, then inequality of the form (17) is still in force. Of course, ;i, will be chosen accordingly.

More general f. In our assumptions on f, it was stipulated that into R. A more general approach would start from the assumption from a function space into another. For instance, in the equation f of the form f

(t; u> (x>=

jQ k(t, x -

f

is a map from R x R x R that f is an operator acting (El) one could choose an

E) u(y. 5) d&

(21)

which acts from the product R X L'(Q) into the space L*(Q), provided adequate are made on the kernel k(t, x). A simple examination of the proof of theorem a condition of the form

I If (c u>(x>- f (t; u>@>I[u(t,x>- u(t> R

x)1

[u(t, x) - u(t, x)]*dx, with p < )3i, will suffice for the validity

an

assumptions 1 shows that

dx V’t E R,

(22)

of the result.

Monotonic@. Both conditions (Ci) and (C,) have been formulated in such a way that the operator Au + f (t, x, u/l,) in case of equation (El), or -Au + f (t, x, u) in case of equation (Ez), verify a monotonicity type condition. Such a condition seems to play an important role in regard to the almost periodicity. In the papers [l] and [7] for instance, this assertion is substantiated. Further references can be found in [l] and [7]. Nevertheless, it has to be pointed out that in [l] the operator A is chosen to be stationary, while in [7] there are some hypotheses not required in obtaining the results of this paper (specifically, compactness and bilateral estimates). S-almost periodicity. In the papers [l] and [7] the authors deal with various concepts of almost periodicity. Better results than those given in theorems 1 and 2 can be obtained, assuming less than L*(Q)-almost periodicity on f, and still assuring the L2(Q)-almost periodicity of the solutions which are L2(G?)-bounded. For instance, in regard to f one could assume only S2-almost periodicity. In order to deal with the case described above, some variants of the lemma in [4] and (51 are necessary. If one looks at the inequality (ll), one easily finds out that it is reducible to a linear one, namely to the form 2s with (Y> 0 and f E S. Assuming of the form

QtER,

-0% +f(t),

that u(t) is bounded

(23)

on R, then (23) must imply an estimate

1+1 sup/u(t)) with both supremums

taken

s K sup

on R. The constant

I

t

I+>1

ds,

K must depend

(24) on Q only

363

AlmOst periodic solutions to nonlinear elliptic and parabolic equations Indeed,

we shall

prove

that

any bounded

(on R) solution

of (23) satisfies

(24), with

K =

eE(l - een)-‘. In order to obtain the estimate (24), let us remark first that (23) implies for any solution the estimate 1u(t) / s 1u(t,J (e-“(‘-tn) + c re-n(r-s) If(s) ( ds, Jtn

(25)

for t > tn. Let us choose now a sequence of real numbers {t,}, such that t,, + - co as n + 03. For every fixed t E R, (25) is verified for all sufficiently large n. Taking into account the boundedness of the sequence 1u(tn) 1, (25) implies at the limit

lu(t)l c sup

f II -m

e-“(‘-S)/f(s)idsl,

WER.

(26)’

the supremum being considered on R (forf E S2 C S, it is always finite). It is well known that the right hand side of (26) is dominated by K sup i~“llf(s)ldS~>

(27)

where K is the one indicated above. Hence, (26) implies (24), which has been our aim. A forthcoming paper will be devoted to the generalization of the results given above, to some classes of abstract differential equations. REFERENCES 1. BRUCK R. E., On the weak asymptotic almost periodicity of bounded solutions of u” E Au + f, for monotone, A, J. diff, Eqns 37, 309-317 (1980). 2. CASTROA. & LAZER A. C., Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. Un. mat. ital. to appear. 3. CORDUNEANUC., Almost Periodic Functions, John Wiley, New York (1968). 4. CORDUNEANUC., Bounded and almost periodic solutions to nonlinear elliptic equations, Tbhoku Math. J. 32, 26.5-278 (1980). 5. CORDLJNEANU C., Bounded and almost periodic solutions to nonlinear parabolic equations, Libertas Mathematics,

Tomus II, pp. 131-139 (1982). 6. FOIAS C. & ZAIDMANS., Almost periodic solutions of parabolic systems, Ann. Scuola Norm. Sup. Pka 15(3), 247-262 (1961). 7. PANKOVA. A., Bounded and almost periodic solutions of evolutionary variational inequalities, Math. Sb. 36, 519-533 (1980). 8. VAGHI C., Soluzioni limitate, o quasi-periodiche, di un’equazione di tipo parabolic0 non lineare, Boll. Un. mat. ital. l(4), 559-580 (1968). 9. ZAIDMANS., Sohuioni limitate o quasi-periodiche dell’equazione di Poisson, Annali mat. pura appl. 64, 365-405 (1965). 10. ZAIDMANS., Solutions presque-periodiques des equations diffkrentielles abstraites, Ens. Math. XXIV, 87-l 10

(1978).