On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations with Nondense Domain
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ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL DIFFERENTIAL EQUATIONS WITH NONDENSE DOMAIN C. Jendoubi
UDC 517.9
Integral manifolds are very useful in studying the dynamics of nonlinear evolution equations. We consider a nondensely defined partial differential equation du = (A + B(t))u(t) + f (t, ut ), dt
(1)
t 2 R,
where (A, D(A)) satisfies the Hille–Yosida condition, (B(t))t2R is a family of operators in L(D(A), X) satisfying certain measurability and boundedness conditions, and the nonlinear forcing term f satisfies the inequality kf (t, φ) − f (t, )k '(t)kφ − kC , where ' belongs to admissible spaces and φ, 2 C := C([−r, 0], X) . We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for these solutions. Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the properties of admissible functions .
1. Introduction In the present paper, we study some properties of integral manifolds for the abstract delayed Cauchy problem du = (A + B(t))u(t) + f (t, ut ), dt
(1.1)
t ≥ s,
us = Φ 2 C, � � where A, D(A) is a nondensely defined linear operator on a Banach space X, B(t), t 2 R, is a family of linear operators in L(D(A), X), f : R ⇥ C ! X is a nonlinear operator, C := C([−r, 0], X), and the history function ut is defined for ✓ 2 [−r, 0] by ut (✓) = u(t + ✓). Throughout the paper, we suppose that A is a Hille– Yosida operator, i.e., (H1 ) There exist w 2 R and M ≥ 1 such that (w, +1) ⇢ ⇢(A) and |R(λ, A)n |
M (λ − !)n
for all n 2 N
and
λ > w,
(1.2)
where ⇢(A) denotes the resolvent set of A and R(λ, A) = (λI − A)−1 for λ > w. Without loss of generality, we can assume that M = 1. Otherwise, we can renormalize the space X with an equivalent norm for which we obtain estimate (1.2) with M = 1. University of Sfax, Tunisia; e-mail: [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 776–789, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.6020. Original article submitted May 9, 2017. 900
0041-5995/20/7206–0900
c 2020 �
Springer Science+Business Media, LLC
O N THE T HEORY OF I NTEGRAL M ANIFOLDS FOR S OME D ELAYED PARTIAL D IFFERENTIAL E QUATIONS
901
The theory of integral manifolds plays an important role in understanding of the dynamics of evolution equations. Numerous works on various types of equations were done in the literature (see, e.g., [1, 4, 10]). Regarding the case of partial differential equations without delay, we refer the reader, e.g., to [3], where the authors investigated invariant manifolds for flows in Banach spaces by using the Lyapunov–Perron method. This subject was also of great interest in the case of delayed partial differential equations. We quote, e.g., [2], where the authors studied inertial manifolds for retarded semilinear parabolic equations by the Lyapunov–Perron method. Nevertheless, it is sometimes more convenient in applications, in various contexts, to consider
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