Integral manifold of the parabolic

259

<eCH2 f е-*('-')[(|Ф|+7)|у(^)~у(^)| + 11^7-?1Н^1< < eCH2 f e-»(«-) ds\\g - g|| + eCH^m + 1)1 e-'('-)|y(5) - y{s)\ ds =

J —OO ^ —OO

еСНг f e-'<'-') ds\\g - g\\ + eC^H^m +1) f e^-'+'+'^'^'X'-'Hlyo - Уо1+

, Ç^^ ,11) .3 < eCH.Щ^ + Ç^^CIVo - Wl + ^11. - т. a о о — о. — us Hl о.

Also ,

( 13 ) |t/</(f,yo)| < еСЩ f e-'<'-) de = ^^

J—OO ^

From (12) and (13) it is clear that for sufficiently small e U{L{p, 7)) С L{p, 7) and Ug — g has a unique solution ■

Remark 2. Dealing with the problem

- 7 - -h Au = Lut + /(wt) at

u ( 0 ) = ж

where L^X^Y be as in §2 , and / : F -♦ X be a Lipschitz continuous bounded function, we can formulate a solution in the form

( 14 )

z { t ) = (u(t), Щ) = T(t - a)z{a) + j' T{t - 5)[/(ti,), 0] ds,

where T is defined in §2. Then the semigroup T{t) satisfies all assumptions Hi — Я4 from [Pu]. If we define F(x,z) = (/(2),0) a function Z --* Z, aH assumptions for the existence of the center unstable manifold and foliation (see [Pu]) are satisfied and this problem is a special case of the problem studied in [Pu]. Of course, the mam idea to get this manifold was based on the formula (14) from [Mi,2].

References

[ Fo ] Fodéuk V , Iniegral'nyje mnogoobraztja dVa nehnepiych dtfferenctjal'nych uravnentj в za- paxdyvapiiëtm argumeniom, Ukrajinskij matem iurnal 21 (1969), 627-639

[ Ha , l ] Hale J , Integral manifolds о/реНигЬаШ differential systems, Ann Math 73 (1961), 49в>531

[ Ha , 2 ] Hale J , Averaging Methods for Differential Equations with Retarded Arguments and a Small Parameter, Journ Diff Eiquat 2 (1966)

[ He ] Henry D , **Geometric Theory of Semdinear Parabolic Equations,** Lecture Notes In Math 840, Spnnger Verlag, 1981