On the theory of integral manifolds for some delayed partial differential equations with nondense domain
UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation $$\frac{du}{dt}=(A+B(t))u(t)+f(t,u_t),\quad t\in\mathbb{R},\tag{1}$$ where $(A,D(A))$ satisfies the Hille–Yosida conditi...
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| author | Jendoubi, C. Jendoubi, C. |
| author_facet | Jendoubi, C. Jendoubi, C. |
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Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation
$$\frac{du}{dt}=(A+B(t))u(t)+f(t,u_t),\quad t\in\mathbb{R},\tag{1}$$
where $(A,D(A))$ satisfies the Hille–Yosida condition, $(B(t))_{t\in\mathbb{R}}$ is a family of operators in $\mathcal{L}(\overline{D(A)},X)$ satisfying some measurability and boundedness conditions, and the nonlinear forcing term $f$ satisfies $\|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$;  here, $\varphi$ belongs to some admissible spaces and $\phi,$ $\psi\in\mathcal{C}:=C([-r,0],X)$. 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 such solutions.
Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the admissible functions properties.
 
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| doi_str_mv | 10.37863/umzh.v72i6.6020 |
| first_indexed | 2026-03-24T03:25:20Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i6.6020
UDC 517.9
C. Jendoubi (Univ. Sfax, Tunisia)
ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED
PARTIAL DIFFERENTIAL EQUATIONS WITH NONDENSE DOMAIN
ДО ТЕОРIЇ IНТЕГРАЛЬНИХ МНОГОВИДIВ ДЛЯ ДЕЯКИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ IЗ ЗАПIЗНЕННЯМ
У НЕЩIЛЬНIЙ ОБЛАСТI
Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the
nondensely-defined partial differential equation
du
dt
= (A+B(t))u(t) + f(t, ut), t \in \BbbR , (1)
where (A,D(A)) satisfies the Hille – Yosida condition, (B(t))t\in \BbbR is a family of operators in \scrL (D(A), X) satisfying some
measurability and boundedness conditions, and the nonlinear forcing term f satisfies \| f(t, \phi ) - f(t, \psi )\| \leq \varphi (t)\| \phi - \psi \| \scrC ;
here, \varphi belongs to some admissible spaces and \phi , \psi \in \scrC := 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 such solutions.
Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible
functions properties.
Iнтегральнi многовиди мають велике значення при вивченнi динамiки нелiнiйних еволюцiйних рiвнянь. Ми розгля-
даємо нещiльно визначене диференцiальне рiвняння з частинними похiдними
du
dt
= (A+B(t))u(t) + f(t, ut), t \in \BbbR , (1)
де (A,D(A)) задовольняє умову Хiлла – Йосiди, (B(t))t\in \BbbR є сiм’єю операторiв у \scrL (D(A), X), яка задовольняє деякi
умови вимiрюваностi та обмеженостi, а нелiнiйний доданок f задовольняє умову \| f(t, \phi ) - f(t, \psi )\| \leq \varphi (t)\| \phi - \psi \| \scrC ,
де \varphi належить до деяких допустимих просторiв i \phi , \psi \in \scrC := C([ - r, 0], X). Ми насамперед пропонуємо деякий
результат, що стосується експоненцiальної збiжностi мiж стiйким многовидом та будь-яким слабким розв’язком
рiвняння (1). Далi ми доводимо iснування центральних нестiйких многовидiв для таких розв’язкiв.
Нашi методи доведення посилаються в основному на теорiю екстраполяцiї та метод Ляпунова – Перрона, що
базується на властивостях допустимих функцiй.
1. Introduction. In this paper, we study some integral manifolds properties of the abstract delayed
Cauchy problem
du
dt
= (A+B(t))u(t) + f(t, ut), t \geq s, (1.1)
us = \Phi \in \scrC ,
where
\bigl(
A,D(A)
\bigr)
is a nondensely defined linear operator on a Banach space X, B(t), t \in \BbbR is a
family of linear operators in \scrL (D(A), X), f : \BbbR \times \scrC \rightarrow X is a nonlinear operator, \scrC := C([ - r, 0], X)
and the history function ut is defined for \theta \in [ - r, 0] by ut(\theta ) = u(t+ \theta ). Throughout all this work,
we suppose that A is a Hille – Yosida operator, that is
c\bigcirc C. JENDOUBI, 2020
776 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 777
(H1) There exists w \in \BbbR and M \geq 1 such that (w,+\infty ) \subset \rho (A) and
| R(\lambda ,A)n| \leq M
(\lambda - \omega )n
for all n \in \BbbN and \lambda > w, (1.2)
where \rho (A) denotes the resolvent set of A and R(\lambda ,A) = (\lambda I - A) - 1 for \lambda > w. Without loss
of generality, one assumes that M = 1. Otherwise, we can renorm the space X with an equivalent
norm for which we obtain the estimation (1.2) with M = 1.
Integral manifolds theory plays an important role in the understanding of evolution equations
dynamics. Many works on various types of equations were done in the literature (see, for example,
[1, 4, 10]). Regarding the case of partial differential equations without delay, we refer the reader, for
instance, to [3], where authors investigate invariant manifolds for flows in Banach spaces, by virtue
of the Lyapunov – Perron method. This subject was also of great interest in the case of delayed partial
differential equations. We quote, for example, [2], where the authors investigate inertial manifolds
for retarded semilinear parabolic equations by the Lyapunov – Perron method.
Nevertheless, it is sometimes more convenient in applications, in many contexts, to consider
equations with nondense domain such as in diffusion phenomena and population dynamics. For
instance, we refer the reader to [5, 16, 22, 23]. Concerning the nonautonomous case, several results
about the existence and behaviour of solutions have been studied (see [9, 17, 21] and references
therein). Particularly, many results on the existence of integral manifolds were developed in the
context of the following differential equation:
du
dt
= A(t)u(t) + f(t, ut), t \in [s,+\infty ),
us = \Phi ,
where A(t), t \in \BbbR , is a family of possibly unbounded linear operators on a Banach space X and
f : \BbbR \times \scrC \rightarrow X is a continuous function. The fixed point theory based on the uniform Lipschitzness
of the nonlinear term f was the most powerful tool to investigate such problems. Unfortunately, in
real situations such as some complicated reaction-diffusion phenomena, the function f which can
represent the population size or the source of a material is frequently depending on time (see, for
instance, [19, 20]).
In recent years, authors have established interesting results in the case of densely defined diffe-
rential equations without delays (see [6, 11, 12]), by investigating the existence of integral manifolds
in view of the Lyapunov – Perron method and the contribution of admissible spaces, without needing
the uniform Lipschitzness of f. More recently, the existence of integral manifolds for densely defined
and delayed differential equations were studied by [7, 8]. Note that the investigation of integral
manifolds for delayed differential equations with nondense domain and where the nonlinear operator
f is not uniformly Lipshitzian was not studied until the author, in [14], investigates the existence
of unstable manifolds for (1.1) and states an attraction result for such unstable manifolds. Then, he
investigates in [15] the existence of stable and center-stable manifolds for (1.1) on the positive half
line.
Motivated by all these works, we aim to prove an attractiveness result between the mild solution
and the stable manifold of (1.1) on the whole line \BbbR . Furthermore, we prove the existence of a
center-unstable manifold for (1.1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
778 C. JENDOUBI
2. Admissible spaces, mild solutions and integral manifolds. We first recall the following
notions and properties of admissible spaces.
Definition 2.1 [8, 13]. Let \scrB denote the Borel algebra and \lambda the Lebesgue measure on \BbbR . A
vector space E of real-valued Borel-measurable functions on \BbbR (modulo \lambda -nullfunctions) is called
a Banach function space (over (\BbbR ,\scrB , \lambda )) if
1) E is Banach lattice with respect to a norm \| .\| E , i.e., (E, \| .\| E) is a Banach space, and if
\varphi \in E and \psi is a real-valued Borel-measurable function such that | \psi (.)| \leq | \varphi (.)| , \lambda -a.e., then
\psi \in E and \| \psi \| E \leq \| \varphi \| E ,
2) the characteristic functions \chi A belong to E for all A \in \scrB of finite measure and
\mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR \| \chi [t,t+1]\| E <\infty , and \mathrm{i}\mathrm{n}\mathrm{f}t\in \BbbR \| \chi [t,t+1]\| E > 0,
3) E \lhook \rightarrow L1,loc(\BbbR ), i.e., for each seminorm pn of L1,loc(\BbbR ) there exists a number \beta pn > 0 such
that pn(f) \leq \beta pn\| f\| E for all f \in E.
Definition 2.2 [8, 13]. The Banach function space E is called admissible if
(i) there is a constant M \geq 1 such that for every compact interval [a, b] \in \BbbR we have
b\int
a
| \varphi (t)| dt \leq M(b - a)
\| \chi [a,b]\| E
\| \varphi \| E ,
(ii) for \varphi \in E, the function \Theta 1\varphi defined by \Theta 1\varphi (t) =
\int t+1
t
\varphi (\tau )d\tau belongs to E,
(iii) E is T+
\tau - and T -
\tau -invariant, where T+
\tau and T -
\tau are defined for \tau \in \BbbR by
T+
\tau \varphi (t) = \varphi (t - \tau ) for t \in \BbbR ,
T -
\tau \varphi (t) = \varphi (t+ \tau ) for t \in \BbbR .
Moreover, there are constants Q, R such that \| T+
\tau \| \leq Q, \| T -
\tau \| \leq R for all \tau \in \BbbR .
Remark 2.1. If \bfS (\BbbR ) :=
\biggl\{
\xi \in L1,loc(\BbbR ) : \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR
\int t+1
t
| \xi (\tau )| d\tau <\infty
\biggr\}
endowed with the norm
\| \xi \| S := \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR
\int t+1
t
| \xi (\tau )| d\tau and E is an admissible Banach function space, it is easy to show
that E \lhook \rightarrow \bfS (\BbbR ).
Proposition 2.1 [8, 13]. Let E be an admissible Banach function space. Then the following
assertions hold:
(a) Let \varphi \in L1,loc(\BbbR ) such that \varphi \geq 0 and \Theta 1\varphi \in E, where \Theta 1 is defined as in Definition
2.2 (ii). For \tau > 0 we define \Theta \prime
\tau \varphi and \Theta \prime \prime
\tau \varphi by
\Theta \prime
\tau \varphi (t) =
t\int
- \infty
e - \tau (t - s)\varphi (s)ds,
\Theta \prime \prime
\tau \varphi (t) =
+\infty \int
t
e - \tau (s - t)\varphi (s)ds.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 779
Then \Theta \prime
\tau \varphi and \Theta \prime \prime
\tau \varphi belong to E. Particularly, if \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR
\int t+1
t
| \varphi (\sigma )| d\sigma <\infty (this will be satisfied
if \varphi \in E (see Remark 2.1)), then \Theta \prime
\tau \varphi and \Theta \prime \prime
\tau \varphi are bounded. Moreover, we have
\| \Theta \prime
\tau \varphi \| \infty \leq Q
1 - e - \tau
\| \Theta 1\varphi \| \infty and \| \Theta \prime \prime
\tau \varphi \| \infty \leq R
1 - e - \tau
\| \Theta 1\varphi \| \infty .
(b) E contains exponentially decaying functions \psi (t) = e - \alpha | t| for t \in \BbbR and any fixed constant
\alpha > 0.
(c) E does not contain exponentially growing functions f(t) = eb| t| for t \in \BbbR and any constant
b > 0.
Definition 2.3 [8, 13]. Let E be an admissible Banach function space and \varphi be a positive
function belonging to E. A function f : \BbbR \times \scrC \rightarrow X is said to be \varphi -Lipschitz if f satisfies
(i) \| f(t, 0)\| \leq \varphi (t) for all t \in \BbbR ,
(ii) \| f(t, \phi 1) - f(t, \phi 2)\| \leq \varphi (t)\| \phi 1 - \phi 2\| \scrC for all t \in \BbbR and all \phi 1, \phi 2 \in \scrC .
Remark 2.2. One can remark that if f(t, \phi ) is \varphi -Lipschitz then \| f(t, \phi )\| \leq \varphi (t)(1 + \| \phi \| \scrC ) for
all \phi \in \scrC and t \in \BbbR .
In the following, we will assume that
(H2) f : \BbbR \times \scrC \rightarrow X is \varphi -Lipschitz, where \varphi is a positive function belonging to an admissible
space E.
We now introduce the following concept.
Definition 2.4. A family of bounded linear operators
\bigl\{
U(t, s)
\bigr\}
t\geq s
on a Banach space X is a
strongly continuous, exponential bounded evolution family if
(i) U(t, t) = Id and U(t, r)U(r, s) = U(t, s) for all t \geq r \geq s,
(ii) the map (t, s) \mapsto \rightarrow U(t, s)x is continuous for every x \in X,
(iii) there are constants K, c \geq 0 such that \| U(t, s)x\| \leq Kec(t - s)\| x\| for all t \geq s and x \in X.
Let
(H3) t \mapsto \rightarrow B(t)x is strongly measurable for every x \in X0 := D(A) and there exists a function
l \in L1
loc(\BbbR ) such that \| B(.)\| \leq l(.).
By [9], if we consider the homogeneous equation
d
dt
u(t) = (A+B(t))u(t), t \geq s, u(s) = x \in X0, (2.1)
then t \mapsto \rightarrow UB(t, s)x is the unique mild solution on [s,+\infty ) of the initial value problem (2.1). Note
that
\bigl\{
UB(t, s)
\bigr\}
t\geq s
is an evolution family on X0. Now, let
\scrC A :=
\bigl\{
\Phi \in \scrC : \Phi (0) \in D(A)
\bigr\}
.
The following result gives a representation of mild solutions of (1.1) in terms of the evolution family
\{ UB(t, s)
\bigr\}
t\geq s
.
Theorem 2.1 [14]. Assume that (H1) - (H3) hold. Let s \in \BbbR and \Phi \in \scrC A. Then equation (1.1)
has a unique mild solution u \in C([s,+\infty [, X0), given by
u(t) = UB(t, s)\Phi (0) + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
s
UB(t, \tau )\lambda R(\lambda ,A)f(\tau , u\tau )d\tau for t \geq s, (2.2)
us = \Phi .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
780 C. JENDOUBI
Furthermore, for every t \geq s, \mathrm{l}\mathrm{i}\mathrm{m}\lambda \rightarrow \infty
\int t
s
UB(t, \tau )\lambda R(\lambda ,A)f(\tau , u\tau )d\tau \in X0 exists uniformly on
compact sets in \BbbR .
Note that we have the following notion of exponential trichotomy for evolution families.
Definition 2.5 [18]. An evolution family
\bigl\{
U(t, s)
\bigr\}
t\geq s
on the Banach space X is said to have
an exponential trichotomy on \BbbR if there are three families of projections \{ Pj(t)\} , t \in \BbbR , j = 1, 2, 3,
and positive constants L, \gamma , \zeta with \gamma < \zeta satisfying the following conditions:
(i) K := \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR \| Pj(t)\| <\infty , j = 1, 2, 3;
(ii) P1(t) + P2(t) + P3(t) = Id for t \in \BbbR and Pj(t)Pi(t) = 0 for all j \not = i;
(iii) Pj(t)U(t, s) = U(t, s)Pj(s) for t \geq s and j = 1, 2, 3;
(iv) U| (t, s) are homeomorphisms from \mathrm{I}\mathrm{m}Pj(s) onto \mathrm{I}\mathrm{m}Pj(t), for all t \geq s and j = 2, 3,
respectively;
(v) for all t, s \in \BbbR and x \in X,
\| U(t, s)P3(s)x\| \leq Le\gamma | t - s| \| P3(s)x\| ,
and, if t \geq s and x \in X, we have
\| U(t, s)P1(s)x\| \leq Le - \zeta (t - s)\| P1(s)x\| ,
\| [U| (t, s)]
- 1P2(t)x\| \leq Le - \zeta (t - s)\| P2(t)x\| .
The projections \{ Pj(t)\} , t \in \BbbR , j = 1, 2, 3, are called the trichotomy projections, and the
constants L, \gamma , \zeta are the trichotomy constants.
Note that the evolution family
\bigl\{
U(t, s)
\bigr\}
t\geq s
is said to have an exponential dichotomy if the
family of projections P3(t) is trivial. That is, P3(t) = 0 for every t \in \BbbR .
We now give the following concept of integral manifolds for (1.1) on the whole line \BbbR .
Definition 2.6 [14, 15]. A set \scrM \subset \BbbR \times \scrC A is said to be an unstable (respectively, a stable)
manifold for solutions of (1.1) if , for every t \in \BbbR , the phase space \scrC A is decomposed into a direct
sum \scrC A = \mathrm{I}\mathrm{m}\scrP (t)\oplus \mathrm{K}\mathrm{e}\mathrm{r}\scrP (t) such that
\mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\| \scrP (t)\| <\infty ,
and there exists a family of Lipschitz continuous mappings
\scrG t : \mathrm{I}\mathrm{m}\scrP (t) \rightarrow \mathrm{K}\mathrm{e}\mathrm{r}\scrP (t), t \in \BbbR ,
with Lipschitz constants \mathrm{L}\mathrm{i}\mathrm{p}(\scrG t) independent of t such that
a) \scrM =
\bigl\{
(t, \xi + \scrG t(\xi )) \in \BbbR \times (\mathrm{I}\mathrm{m}\scrP (t)\oplus \mathrm{K}\mathrm{e}\mathrm{r}\scrP (t)) | t \in \BbbR , \xi \in \mathrm{I}\mathrm{m}\scrP (t)
\bigr\}
, and we denote
\scrM t :=
\bigl\{
\xi + \scrG t(\xi ) : (t, \xi + \scrG t(\xi )) \in \scrM
\bigr\}
,
b) \scrM t is homeomorphic to \mathrm{I}\mathrm{m}\scrP (t) for all t \in \BbbR ,
c) for each s \in \BbbR and \xi \in \scrM s, there corresponds one and only one solution u(t) of (1.1) on
( - \infty , s](respectively, on [s - r,+\infty )) satisfying the conditions that us = \xi and \mathrm{s}\mathrm{u}\mathrm{p}t\leq s \| ut\| \scrC <
< \infty (respectively, \mathrm{s}\mathrm{u}\mathrm{p}t\geq s \| ut\| \scrC < \infty ); furthermore, any two solutions u1(t) and u2(t) of (1.1)
corresponding to different initial functions \xi 1, \xi 2 \in \scrM s attract each other exponentially in the sense
that, there exist positive constants \nu and C\nu independent of s such that for t \leq s (respectively,
t \geq s) \bigm\| \bigm\| u1t - u2t
\bigm\| \bigm\|
\scrC \leq C\nu e
- \nu | s - t| \bigm\| \bigm\| (\scrP (s)\xi 1)(0) - (\scrP (s)\xi 2)(0)
\bigm\| \bigm\| ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 781
d) \scrM is invariant under equation (1.1); that is, if u(t), t \in \BbbR , is a solution of (1.1) satisfying the
conditions that us \in \scrM s and \mathrm{s}\mathrm{u}\mathrm{p}t\leq s \| ut\| \scrC < \infty (respectively, \mathrm{s}\mathrm{u}\mathrm{p}t\geq s \| ut\| \scrC < \infty ), then ut \in \scrM t
for all t \in \BbbR .
Now, suppose that
(H4) The evolution family
\bigl\{
UB(t, s)
\bigr\}
t\geq s
has an exponential dichotomy with projections PB(t),
t \in \BbbR and constants L \mu > 0.
In that case, the corresponding Green function \Gamma B(t, s), (t, s) \in \BbbR 2, given by
\Gamma B(t, s) =
\left\{ PB(t)UB(t, s), t > s,
-
\bigl[
U| B(s, t)
\bigr] - 1
(Id - PB(s)), t < s,
satisfies \bigm\| \bigm\| \Gamma B(t, s)
\bigm\| \bigm\| \leq L(1 +K)e - \mu | t - s| for all t \not = s.
In order to construct unstable and stable manifolds for (1.1), we have considered in [14, 15] the
families of projections
\bigl(
\BbbP B(t)
\bigr)
t\in \BbbR and
\bigl(
\~PB(t)
\bigr)
t\in \BbbR +
defined on \scrC A, respectively, by
(\BbbP B(t)\xi )(\theta ) =
\bigl[
UB| (t, t+ \theta )
\bigr] - 1
(I - PB(t))\xi (0) for all \theta \in [ - r, 0]
and
( \~PB(t)\xi )(\theta ) = UB(t - \theta , t)PB(t)\xi (0) for all \theta \in [ - r, 0].
Let us collect some results about the existence and uniqueness of solutions for (1.1) related to the
choice of the family of projections on \scrC A.
Theorem 2.2 [14]. Assume that (H1) – (H4) hold. Set
H :=
L(1 +K)e\mu r(Q+R)\| \Theta 1\varphi \| \infty
1 - e - \mu
. (2.3)
Then, if H < 1, there exists one and only one solution of equation (1.1) on ( - \infty , s] given by
u(t) = [UB| (s, t)]
- 1\mu 0 + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
s\int
- \infty
\Gamma B(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma
for some \mu 0 \in \mathrm{K}\mathrm{e}\mathrm{r}PB(s) such that \mathrm{s}\mathrm{u}\mathrm{p}t\leq s \| ut\| \scrC < \infty . Besides, for any two solutions u(t), v(t)
corresponding to different initial functions \xi 1 \xi 2 \in \mathrm{I}\mathrm{m}\BbbP B(s), we have the following estimate:
\| ut - vt\| \scrC \leq C\nu e
- \nu (s - t)\| \xi 1(0) - \xi 2(0)\| for all t \leq s,
where \nu is a positive constant satisfying
0 < \nu < \mu + \mathrm{l}\mathrm{n}(1 - L(1 +K)e\mu r(Q+R)\| \Theta 1\varphi \| \infty ),
and
C\nu :=
Le\mu r
1 - L(1 +K)e\mu r(Q+R)\| \Theta 1\varphi \| \infty
1 - e - (\mu - \nu )
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
782 C. JENDOUBI
Remark 2.3. We can prove in a similar way as in Theorem 3.5 of [15] that if we consider
PB(t) defined on the whole line \BbbR then we have under the same assumptions of Theorem 2.2, more
precisely, under conditions (H1) – (H4) and if H < 1, defined by (2.3), the existence and uniqueness
of the solution for equation (1.1) on [s,+\infty ) given by
u(t) = UB(t, s)\mu 0 + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
s
\Gamma B(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma
for some \mu 0 \in \mathrm{I}\mathrm{m}PB(s) such that \mathrm{s}\mathrm{u}\mathrm{p}t\geq s \| ut\| \scrC < \infty , and we have the following estimate for any
two solutions u(t), v(t) corresponding to different initial functions \xi 1 \xi 2 \in \mathrm{I}\mathrm{m} \~PB(s):
\| ut - vt\| \scrC \leq C\nu e
- \nu (t - s)\| \xi 1(0) - \xi 2(0)\| for all t \geq s,
where \nu and C\nu are defined as above.
To get the existence of stable manifolds for (1.1) on the whole line \BbbR , we prove the following
result, which shows that property (d) of Definition 2.6 holds.
Proposition 2.2. Let (\scrP (t))t\in \BbbR be a family of projections on the phase space \scrC A. Let \scrG t, t \in \BbbR ,
be defined by
\scrG t(\xi )(\theta ) = \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
t
\Gamma (t - \theta , \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma for all \theta \in [ - r, 0],
where \scrP (t)ut = \xi , and let E =
\bigl\{
\xi + \scrG t(\xi ) \in (\mathrm{I}\mathrm{m}\scrP (t) \oplus \mathrm{K}\mathrm{e}\mathrm{r}\scrP (t)), t \in \BbbR
\bigr\}
. Then E is invariant
under (1.1). That is, if u(t), t \in \BbbR , is a solution of (1.1) satisfying us \in E and \mathrm{s}\mathrm{u}\mathrm{p}t\geq s \| ut\| \scrC <\infty ,
then ut \in E for all t \in \BbbR .
Proof. First, let t \geq s. Then the result follows analogically as in Theorem 3.7 of [15], by taking
s \in \BbbR . Now, let t \leq s, then, for t - r \leq \tau \leq t \leq s, we have according to Remark 2.3
u\tau ( - \theta ) = UB(\tau - \theta , \tau )\mu 0 + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
\tau
\Gamma B(\tau - \theta , \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma for \mu 0 \in \mathrm{I}\mathrm{m}PB(\tau ).
Furthermore, it follows from Theorem 2.1 that
u(t) = UB(t, \tau )u(\tau ) + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
\tau
UB(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma =
= UB(t, \tau )
\left( \mu 0 + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
\tau
\Gamma B(\tau , \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma
\right) +
+ \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
\tau
UB(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma
for \mu 0 \in \mathrm{I}\mathrm{m}PB(\tau ) = UB(t, \tau )\mu 0 -
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ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 783
- UB(t, \tau ) \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
\tau
[UB| (\sigma , \tau )]
- 1(I - PB(\sigma ))\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma +
+ \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
\tau
UB(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma =
= UB(t, \tau )\mu 0 - \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
\tau
UB(t, \sigma )(I - PB(\sigma ))\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma -
- \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
\tau
[UB| (\sigma , t)]
- 1(I - PB(\sigma ))\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma +
+ \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
\tau
UB(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma =
= UB(t, \tau )\mu 0 + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
\tau
\Gamma B(t, \sigma )\lambda R(\lambda ,A)f(\sigma , u\sigma )d\sigma .
Proposition 2.2 is proved.
Consequently, by virtue of Proposition 2.2 and [14, 15], the following result yields.
Theorem 2.3. Assume that (H1) – (H4) hold. Let
H < \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1,
e\mu r
1 + L
\biggr\}
, (2.4)
where H is defined as in (2.3). Then, there exist a stable and an unstable manifold for solutions of
equation (1.1) on the whole line \BbbR .
3. On the exponential attractiveness of stable manifolds. With the established theory of
stable manifolds for the differential equation (1.1) (see Theorem 2.3), we aim to prove that the stable
manifold \scrS = \{ \scrS t\} t\in \BbbR exponentially attracts all mild solutions of (1.1).
Theorem 3.1. Assume that (H1) – (H4) and (2.4) hold. Let
\~a := Le\mu r
\bigl(
\mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK) + a(1 +K)
\bigr)
< 1,
where a is a constant taking according to properties of admissible spaces. Then the stable manifold
\scrS = \{ \scrS t\} t\in \BbbR exponentially attracts all mild solutions of (1.1). In the sense that for every mild
solution u(.) of (1.1) with initial condition us, there exists a solution \=u(.) in \scrS (that is, \=ut \in \scrS t,
for all t \geq s) and constants \alpha \beta > 0, such that
\| ut - \=ut\| \scrC \leq \alpha e - \beta (t - s)\| us - \=us\| \scrC for all t \geq s.
Proof. For s \in \BbbR , we define the following space:
\scrC s,\mu :=
\bigl\{
x \in C
\bigl(
[s - r,\infty ), X
\bigr)
such that \mathrm{s}\mathrm{u}\mathrm{p}
t\geq s - r
e\mu (t - s)\| x(t)\| <\infty
\bigr\}
,
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784 C. JENDOUBI
equipped with the norm | x| \mu := \mathrm{s}\mathrm{u}\mathrm{p}t\geq s - r e
\mu (t - s)\| x(t)\| . Clearly that \scrC s,\mu is a Banach space. To find
our desired result, we will prove that there exists \=u(.) = u(.) + w(.) such that w \in \scrC s,\mu .
One can see that \=u(.) is a solution of (1.1) if and only if w(.) is a solution of the equation
w(t) = UB(t, s)w(s) + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
s
UB(t, \tau )\lambda R(\lambda ,A)(f(\tau , u\tau + w\tau ) - f(\tau , u\tau ))d\tau . (3.1)
Putting \=f(t, wt) = f(t, ut + wt) - f(t, ut), we can show that \=f : \BbbR \times \scrC \rightarrow X is \varphi -Lipschitz.
Moreover, we have \=f(t, 0) = 0. Hence, we rewrite equation (3.1) as
w(t) = UB(t, s)w(s) + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
s
UB(t, \tau )\lambda R(\lambda ,A) \=f(\tau , w\tau )d\tau . (3.2)
It follows from Remark 2.3 that solutions of (3.2) are bounded on [s - r,+\infty ) if and only if
w(t) = UB(t, s)\mu 0 + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
s
\Gamma B(t, \tau )\lambda R(\lambda ,A) \=f(\tau , w\tau )d\tau (3.3)
for some \mu 0 \in \mathrm{I}\mathrm{m}PB(s) and t \geq s.
Now, let us find \mu 0 \in \mathrm{I}\mathrm{m}PB(s), such that \=us = us + ws \in \scrS s. This means\Bigl(
\~PB(s)(us + ws)
\Bigr)
(\theta ) = \scrG s
\bigl(
(I - \~PB(s))(us + ws)
\bigr)
(\theta ), (3.4)
which gives
\mu 0 = ( \~PB(s)ws)(0)\scrG s
\bigl(
(I - \~PB(s))(us + ws)
\bigr)
(0) - PB(s)u(s). (3.5)
In order to look for \=u(.) satisfying (1.1) and \=us \in \scrS s, we will find solutions for the following
system in the Banach space \scrC s,\mu :
(Cw)(t) =
\left\{
UB(t, s)
\Bigl(
\scrG s
\bigl(
(I - \~PB(s))(us + ws)
\bigr)
(0) - PB(s)u(s)
\Bigr)
+
+ \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
\int +\infty
s
\Gamma B(t, \tau )\lambda R(\lambda ,A) \=f(\tau , w\tau )d\tau for t \geq s,
UB(2s - t, s)
\Bigl(
\scrG s
\bigl(
(I - \~PB(s))(us + ws)
\bigr)
(0) - PB(s)u(s)
\Bigr)
+
+ \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
\int +\infty
s
\Gamma B(2s - t, \tau )\lambda R(\lambda ,A) \=f(\tau , w\tau )d\tau for s - r \leq t \leq s.
Taking s - r \leq t \leq s, we have
e\mu (t - s)\| (Cw)(t)\| \leq e\mu (t - s)\| UB(2s - t, s)\mu 0\| +
+e\mu (t - s) \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
s
\| \Gamma B(2s - t, \tau )\lambda R(\lambda ,A) \=f(\tau , w\tau )\| d\tau \leq
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ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 785
\leq Le2\mu (t - s)\| \mu 0\| +
+L(1 +K)e\mu r| w| \mu
\left( 2s - t\int
s
e - 2\mu (s - t)\varphi (\tau )d\tau +
\infty \int
2s - t
e - 2\mu (\tau - s)\varphi (\tau )d\tau
\right) .
Using the fact that \varphi belongs to some admissible spaces, it yields that there exists a > 0 such that,
for all s - r \leq t \leq s,
e\mu (t - s)\| (Cw)(t)\| \leq L\| \mu 0\| + aL(1 +K)e\mu r| w| \mu .
Concerning t \geq s, we get
e\mu (t - s)\| (Cw)(t)\| \leq
\leq e\mu (t - s)\| UB(t, s)\mu 0\| + e\mu (t - s) \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
+\infty \int
s
\| \Gamma B(t, \tau )\lambda R(\lambda ,A) \=f(\tau , w\tau )\| d\tau \leq
\leq L\| \mu 0\| + L(1 +K)e\mu r| w| \mu
\left( t\int
s
\varphi (\tau )d\tau +
+\infty \int
t
e - 2\mu (\tau - t)\varphi (\tau )d\tau
\right) \leq
\leq L\| \mu 0\| + aL(1 +K)e\mu r| w| \mu .
Consequently, we obtain
| Cw| \mu \leq L\| \mu 0\| + aL(1 +K)e\mu r| w| \mu .
Furthermore, by virtue of the Lipschitz condition on \scrG s, we have in view of (3.5)
\| \mu 0\| \leq \| \scrG s((I - \~PB(s))us)(0) - PB(s)u(s)\| +
+\| \scrG s((I - \~PB(s))(us + ws))(0) - \scrG s((I - \~PB(s))us)(0)\| \leq
\leq \| \scrG s((I - \~PB(s))us) - \~PB(s)us\| \scrC + \mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK)e\mu r| w| \mu .
Hence,
| Cw| \mu \leq L\| \scrG s
\bigl(
(I - \~PB(s))us
\bigr)
- \~PB(s)us\| \scrC + L\mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK)e\mu r| w| \mu +
+aL(1 +K)e\mu r| w| \mu \leq
\leq L\| \scrG s((I - \~PB(s))us) - \~PB(s)us\| \scrC + \~a| w| \mu . (3.6)
This means that Cw belongs to \scrC s,\mu .
Now, we propose to prove that C is a contraction. Let v, w \in \scrC s,\mu and t \in [s - r, s], then
e\mu (t - s)\| (Cv)(t) - (Cw)(t)\| \leq Le\mu (t - s)e - \mu (s - t)\| \epsilon 0 - \zeta 0\| +
+L(1 +K)e\mu (t - s)
+\infty \int
s
e - \mu | 2s - t - \tau | \varphi (\tau )\| v\tau - w\tau \| \scrC d\tau \leq
\leq L\| \epsilon 0 - \zeta 0\| + aL(1 +K)e\mu r| v - w| \mu .
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786 C. JENDOUBI
For t \geq s,
e\mu (t - s)\| (Cv)(t) - (Cw)(t)\| \leq L\| \epsilon 0 - \zeta 0\| +
+L(1 +K)e\mu (t - s)
+\infty \int
s
e - \mu | t - \tau | \varphi (\tau )\| v\tau - w\tau \| \scrC d\tau \leq
\leq L\| \epsilon 0 - \zeta 0\| + aL(1 +K)e\mu r| v - w| \mu .
Consequently,
| Cv - Cw| \mu \leq L\| \epsilon 0 - \zeta 0\| + aL(1 +K)e\mu r| v - w| \mu .
Furthermore,
\| \epsilon 0 - \zeta 0\| = \| \scrG s
\Bigl( \bigl(
I - \~PB(s)
\bigr)
(us + vs)
\Bigr)
(0) - \scrG s((I - \~PB(s))(us + ws))(0)\| \leq
\leq \mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)\| (I - \~PB(s))(vs - ws)\| \scrC \leq \mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK)e\mu r| v - w| \mu .
Hence, we get
| Cv - Cw| \mu \leq L\mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK)e\mu r| v - w| \mu + aL(1 +K)e\mu r| v - w| \mu \leq
\leq Le\mu r(\mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK) + a(1 +K))| v - w| \mu .
Since \~a < 1, C is a contraction on \scrC s,\mu and so, it has a unique fixed w, which belongs to \scrC s,\mu .
Using (3.6), it follows that
| w| \mu \leq L
1 - \~a
\bigm\| \bigm\| \bigm\| \scrG s((I - \~PB(s))us) - \~PB(s)us
\bigm\| \bigm\| \bigm\|
\scrC
.
Consequently, we obtain from (3.4)
\| \=ut - ut\| \scrC = \| wt\| \scrC \leq e\mu re - \mu (t - s)| w| \mu \leq
\leq e\mu re - \mu (t - s) L
1 - \~a
\bigl(
\| \scrG s((I - \~PB(s))\=us) - \scrG s((I - \~PB(s))us)\| \scrC + \| \~PB(s)(\=us - us)\| \scrC
\bigr)
\leq
\leq e\mu re - \mu (t - s) L
1 - \~a
(\mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK)\| us - \=us\| \scrC + LK\| us - \=us\| ) \leq
\leq e\mu re - \mu (t - s) L
1 - \~a
(\mathrm{L}\mathrm{i}\mathrm{p}(\scrG s)(1 + LK) + LK) \| us - \=us\| \scrC
for all t \geq s.
Theorem 3.1 is proved.
4. On the existence of center-unstable manifolds. This section is devoted to investigating the
existence of center-unstable manifolds for solutions of (1.1), in the case that the evolution family\bigl\{
UB(t, s)
\bigr\}
t\geq s
has an exponential trichotomy on \BbbR and the nonlinear term f is \varphi -Lipschitz. In the
following, we assume that
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ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 787
(H5) The evolution family
\bigl\{
UB(t, s)
\bigr\}
t\geq s
has an exponential trichotomy with the trichotomy
projections PB,j(t), t \in \BbbR , j = 1, 2, 3, and constants L, \gamma , \zeta > 0.
Using the trichotomy projections PB,j(t), t \in \BbbR , j = 1, 2, 3, we define the families of projections
\{ \BbbP B,j(t)\} , t \in \BbbR , j = 1, 2, 3, on \scrC A as follows:
(\BbbP B,j(t)\xi )(\theta ) =
\bigl[
UB | (t, t+ \theta )
\bigr] - 1
(I - PB,j(t))\xi (0) for all \theta \in [ - r, 0] and \xi \in \scrC A. (4.1)
Now, we give our main result of this section, which proves the existence of center-unstable manifold
for solutions of (1.1).
Theorem 4.1. Assume that (H1) – (H3) and (H5) hold. Set
\=K := \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\| PB,j(t)\| : t \geq 0 j = 2, 3
\bigr\}
, \=L := \mathrm{m}\mathrm{a}\mathrm{x}\{ L, 2L \=K\} (4.2)
and
H :=
\=L(1 + \=K)e\mu r(Q+R)\| \Theta 1\varphi \| \infty
1 - e - \~\mu
for some \~\mu fixed below. Then if H < \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1,
e\~\mu r
1 + \=L
\biggr\}
for each fixed \rho > \gamma , there exists a center-
unstable manifold \scrC \scrU =
\bigl\{
(t, \scrC \scrU t)
\bigr\}
t\in \BbbR \subset \BbbR \times \scrC A for solutions of (1.1), which is represented by a
family of Lipschitz continuous mappings
\scrG t : \mathrm{I}\mathrm{m}(\BbbP B,2(t)\oplus \BbbP B,3(t)) \rightarrow \mathrm{I}\mathrm{m}\BbbP B,1(t)
with Lipschitz constants independent of t such that \scrC \scrU t satisfies the following conditions:
a) \scrC \scrU t is homeomorphic to \mathrm{I}\mathrm{m}(\BbbP B,2(t) + \BbbP B,3(t)) for all t \in \BbbR .
b) To each \xi \in \scrC \scrU s, there corresponds one and only one solution of (1.1) on ( - \infty , s] such
that e - \tau (s+\theta )us(\theta ) = \xi (\theta ) for \theta \in [ - r, 0] and \mathrm{s}\mathrm{u}\mathrm{p}t\leq s \| e - \tau (t+.)ut(.)\| \scrC < \infty , where \tau =
\rho + \gamma
2
.
Furthermore, for any two solutions u(t) and v(t) of equation (1.1) corresponding to different initial
functions \xi 1, \xi 2 \in \scrC \scrU s, there exist positive constants \nu and C\nu independent of s \in \BbbR such that
\| ut - vt\| \scrC \leq C\nu e
(\tau - \nu )(t - s)
\bigm\| \bigm\| (\BbbP B(s)\xi 1)(0) - (\BbbP B(s)\xi 2)(0)
\bigm\| \bigm\| for t \geq s,
where \BbbP B(t) = \BbbP B,2(t) + \BbbP B,3(t).
c) \scrC \scrU is invariant under equation (1.1). That is, if u(t), t \in \BbbR , is a solution of (1.1) such that
e - \tau (s+.)us \in \scrC \scrU s and \mathrm{s}\mathrm{u}\mathrm{p}t\leq s \| e - \tau (t+.)ut(.)\| \scrC <\infty , then e - \tau (t+.)ut(.) \in \scrC \scrU t for all t \in \BbbR .
Proof. Put PB(t) := PB,2(t) + PB,3(t) and QB(t) := PB,1(t) = Id - PB(t) for t \in \BbbR .
Then PB(t) and QB(t) are projections complemented to each other on X0. We then define the
corresponding projections \{ \BbbP B,j(t)\} , t \in \BbbR , j = 1, 2, 3, on \scrC A as in equality (4.1). Let \BbbP B(t) :=
:= \BbbP B,2(t) + \BbbP B,3(t) and \BbbQ B,t := \BbbP B,1(t) = Id - \BbbP B(t) for t \in \BbbR . Hence, \BbbP B(t) and \BbbQ B(t) are
complemented on \scrC A for each t \in \BbbR . Now, consider the following evolution family:
\=UB(t, s) = e - \tau (t - s)UB(t, s) for all t \geq s.
Let us prove that the evolution family \=UB(t, s) has an exponential dichotomy with the dichotomy
projections QB(t), t \in \BbbR . In fact, we have, for t \geq s,
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788 C. JENDOUBI
QB(t) \=UB(t, s) = e - \tau (t - s)PB,1(t)UB(t, s) =
= e - \tau (t - s)UB(t, s)PB,1(s) = \=UB(t, s)QB(s).
Since UB | (t, s) is a homeomorphism from \mathrm{I}\mathrm{m}PB,j(s) onto \mathrm{I}\mathrm{m}PB,j(t), j = 2, 3, then it is a home-
omorphism from \mathrm{I}\mathrm{m}PB(s) onto \mathrm{I}\mathrm{m}PB(t). As \mathrm{I}\mathrm{m}PB(s) = \mathrm{K}\mathrm{e}\mathrm{r}QB(s), s \in \BbbR , then \=UB| (t, s)
is a homeomorphism from \mathrm{K}\mathrm{e}\mathrm{r}QB(s) onto \mathrm{K}\mathrm{e}\mathrm{r}QB(t). By using the definition of exponential tri-
chotomy, we obtain
\| \=UB(t, s)QB(s)x\| \leq Le - (\zeta +\tau )(t - s)\| QB(s)x\| for all t \geq s,
besides
\| [ \=UB| (t, s)]
- 1PB(t)x\| = e - \tau (t - s)\| [UB| (t, s)]
- 1(PB,2(t) + PB,3(t))x\| \leq
\leq Le - \tau (t - s)
\bigl(
e - \zeta (t - s)\| PB,2(t)PB(t)x\| + e\gamma (t - s)\| PB,3(t)PB(t)x\|
\bigr)
for all t \geq s and x \in X0. By (4.2), we obtain\bigm\| \bigm\| [ \=UB| (t, s)]
- 1PB(t)x
\bigm\| \bigm\| \leq 2L \=Ke -
\rho - \gamma
2
(t - s)\| PB(t)x\| .
Consequently, \=UB(t, s) has an exponential dichotomy with the dichotomy projections PB(t), t \in \BbbR ,
and constants \=L, \~\mu :=
\rho - \gamma
2
.
Set z(t) := e - \tau tu(t) and let the mapping G : \BbbR \times \scrC \rightarrow X be defined by
G(t, \xi ) = e - \tau tf(t, e\tau (t+.)\xi (.)) for (t, \xi ) \in \BbbR \times \scrC .
It is easy to show that G is also \varphi -Lipschitz. Hence, equation (2.2) can be rewritten as
z(t) = \=UB(t, s)z(s) + \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow \infty
t\int
s
\=UB(t, \sigma )\lambda R(\lambda ,A)G(\sigma , z\sigma )d\sigma for all t \geq s,
(4.3)
zs(.) = e - \tau (s+.)\Phi (.) \in \scrC A.
By virtue of Theorem 2.3, we obtain that, if H < \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1,
e\~\mu r
1 + \=L
\biggr\}
, then there exists an unstable
manifold \scrM for solutions of (4.3). Returning to equation (2.2), by the relation u(t) := e\tau tz(t), in
view of Theorems 2.2 and 2.3, it is easy to check properties of \scrC \scrU . Therefore, \scrC \scrU is a center-unstable
manifold for solutions of (1.1).
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equations, Nonlinear Anal., 34, 907 – 925 (1998).
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ON THE THEORY OF INTEGRAL MANIFOLDS FOR SOME DELAYED PARTIAL . . . 789
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ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
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| id | umjimathkievua-article-6020 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:20Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/11/16580f12af90c9fb9a4c6ffb21258c11.pdf |
| spelling | umjimathkievua-article-60202022-03-26T11:01:51Z On the theory of integral manifolds for some delayed partial differential equations with nondense domain On the theory of integral manifolds for some delayed partial differential equations with nondense domain Jendoubi, C. Jendoubi, C. UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation $$\frac{du}{dt}=(A+B(t))u(t)+f(t,u_t),\quad t\in\mathbb{R},\tag{1}$$ where $(A,D(A))$ satisfies the Hille–Yosida condition, $(B(t))_{t\in\mathbb{R}}$ is a family of operators in $\mathcal{L}(\overline{D(A)},X)$ satisfying some measurability and boundedness conditions, and the nonlinear forcing term $f$ satisfies&nbsp;$\|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$;&nbsp;&nbsp;here, $\varphi$ belongs to some admissible spaces and $\phi,$ $\psi\in\mathcal{C}:=C([-r,0],X)$.&nbsp;We first present an exponential convergence result between the stable manifold and every mild solution of (1).&nbsp;&nbsp;Then we prove the existence of center-unstable manifolds for such solutions. Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the admissible functions properties. &nbsp; &nbsp; УДК 517.9 Інтегральні многовиди мають велике значення при вивченні динаміки нелінійних еволюційних рівнянь.&nbsp;&nbsp;Ми розглядаємо нещільно визначене диференціальне рівняння з частинними похідними $$\frac{du}{dt}=(A+B(t))u(t)+f(t,u_t),\quad t\in\mathbb{R},\tag{1}$$ де $(A,D(A))$ задовольняє умову Хілла–Йосіди, $(B(t))_{t\in\mathbb{R}}$ є сім'єю операторів у $\mathcal{L}(\overline{D(A)},X),$ яка задовольняє деякі умови вимірюваності та обмеженості, а нелінійний доданок $f$ задовольняє умову $\|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$, де $\varphi$ належить до деяких допустимих просторів і $\phi,$ $\psi\in\mathcal{C}:=C([-r,0],X)$.&nbsp;Ми насамперед пропонуємо деякий результат, що стосується експоненціальної збіжності між стійким многовидом та будь-яким слабким розв'язком рівняння (1).&nbsp;&nbsp;Далі ми доводимо існування центральних нестійких многовидів для таких розв'язків. Наші методи доведення посилаються в основному на теорію екстраполяції та метод Ляпунова–Перрона, що базується на властивостях допустимих функцій.&nbsp; Institute of Mathematics, NAS of Ukraine 2020-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6020 10.37863/umzh.v72i6.6020 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 6 (2020); 776-789 Український математичний журнал; Том 72 № 6 (2020); 776-789 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6020/8714 |
| spellingShingle | Jendoubi, C. Jendoubi, C. On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title | On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title_alt | On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title_full | On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title_fullStr | On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title_full_unstemmed | On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title_short | On the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| title_sort | on the theory of integral manifolds for some delayed partial differential equations with nondense domain |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6020 |
| work_keys_str_mv | AT jendoubic onthetheoryofintegralmanifoldsforsomedelayedpartialdifferentialequationswithnondensedomain AT jendoubic onthetheoryofintegralmanifoldsforsomedelayedpartialdifferentialequationswithnondensedomain |