Integral manifolds for semilinear evolution equations and admissibility of function spaces
We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation in the case where the evolution family (U(t, s)) t≥s has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Li...
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Vu Thi Ngoc Ha Nguyen Thieu Huy Ha Phi 2020-02-09T14:52:06Z 2020-02-09T14:52:06Z 2012 Integral manifolds for semilinear evolution equations and admissibility of function spaces / Vu Thi Ngoc Ha, Nguyen Thieu Huy, Ha Phi // Український математичний журнал. — 2012. — Т. 64, № 6. — С. 772-796. — Бібліогр.: 37 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164414 517.9 We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation in the case where the evolution family (U(t, s)) t≥s has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions, i.e., where φ(t) belongs to some classes of admissible function spaces. Our main method is based on the Lyapunov–Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces. Доведено iснування iнтегральних (стiйких, нестiйких, центральних) многовидiв для розв’язкiв напiвлiнiйного iнтегрального рiвняння у випадку, коли сiм’я еволюцiй (U(t,s))tleqs має експоненцiальну трихотомiю на пiвосi або на всiй осi, а нелiнiйний збурюючий член f задовольняє φ-лiпшицевi умови, тобто належить до деяких класiв допустимих просторiв функцiй. Наш основний метод базується на методах Ляпунова – Перрона, процедурах перемасштабування та технiцi застосування допустимостi просторiв функцiй. On leave from Hanoi University of Science and Technology as a research fellow of the Alexander von Humboldt Foundation at Technical University of Darmstadt. The support by the Alexander von Humboldt Foundation is gratefully acknowledged. The author thanks Prof. Matthias Hieber for his strong support and inspiration. This work is financially supported by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under Project 101.01-2011.25. en Інститут математики НАН України Український математичний журнал Статті Integral manifolds for semilinear evolution equations and admissibility of function spaces Інтегральнi многовиди для напiвлiнiйних еволюцiйних рiвнянь та допустимiсть просторiв функцiй Article published earlier |
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Integral manifolds for semilinear evolution equations and admissibility of function spaces |
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Integral manifolds for semilinear evolution equations and admissibility of function spaces Vu Thi Ngoc Ha Nguyen Thieu Huy Ha Phi Статті |
| title_short |
Integral manifolds for semilinear evolution equations and admissibility of function spaces |
| title_full |
Integral manifolds for semilinear evolution equations and admissibility of function spaces |
| title_fullStr |
Integral manifolds for semilinear evolution equations and admissibility of function spaces |
| title_full_unstemmed |
Integral manifolds for semilinear evolution equations and admissibility of function spaces |
| title_sort |
integral manifolds for semilinear evolution equations and admissibility of function spaces |
| author |
Vu Thi Ngoc Ha Nguyen Thieu Huy Ha Phi |
| author_facet |
Vu Thi Ngoc Ha Nguyen Thieu Huy Ha Phi |
| topic |
Статті |
| topic_facet |
Статті |
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2012 |
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English |
| container_title |
Український математичний журнал |
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Інститут математики НАН України |
| format |
Article |
| title_alt |
Інтегральнi многовиди для напiвлiнiйних еволюцiйних рiвнянь та допустимiсть просторiв функцiй |
| description |
We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation
in the case where the evolution family (U(t, s)) t≥s has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions, i.e.,
where φ(t) belongs to some classes of admissible function spaces. Our main method is based on the Lyapunov–Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.
Доведено iснування iнтегральних (стiйких, нестiйких, центральних) многовидiв для розв’язкiв напiвлiнiйного iнтегрального рiвняння у випадку, коли сiм’я еволюцiй (U(t,s))tleqs має експоненцiальну трихотомiю на пiвосi або на всiй осi, а нелiнiйний збурюючий член f задовольняє φ-лiпшицевi умови, тобто належить до деяких класiв допустимих просторiв функцiй. Наш основний метод базується на методах Ляпунова – Перрона, процедурах перемасштабування та технiцi застосування допустимостi просторiв функцiй.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164414 |
| citation_txt |
Integral manifolds for semilinear evolution equations and admissibility of function spaces / Vu Thi Ngoc Ha, Nguyen Thieu Huy, Ha Phi // Український математичний журнал. — 2012. — Т. 64, № 6. — С. 772-796. — Бібліогр.: 37 назв. — англ. |
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| first_indexed |
2025-11-27T04:38:59Z |
| last_indexed |
2025-11-27T04:38:59Z |
| _version_ |
1850800516776329216 |
| fulltext |
UDC 517.9
Nguyen Thieu Huy∗ (Hanoi Univ. Sci. and Technology, Vietnam;
Techn. Univ. Darmstadt, Germany),
Vu Thi Ngoc Ha (Hanoi Univ. Sci. and Technology, Vietnam),
Ha Phi (Hanoi Univ. Education, Vietnam)
INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS
AND ADMISSIBILITY OF FUNCTION SPACES∗∗
IНТЕГРАЛЬНI МНОГОВИДИ ДЛЯ НАПIВЛIНIЙНИХ ЕВОЛЮЦIЙНИХ
РIВНЯНЬ ТА ДОПУСТИМIСТЬ ПРОСТОРIВ ФУНКЦIЙ
We prove the existence of integral (stable, unstable, center) manifolds for the solutions to the semilinear integral equation
u(t) = U(t, s)u(s) +
∫ t
s
U(t, ξ)f(ξ, u(ξ))dξ in the case where the evolution family (U(t, s))t≥s has an exponential
trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the ϕ-Lipschitz conditions, i.e.,
‖f(t, x)− f(t, y)‖ 6 ϕ(t)‖x− y‖, where ϕ(t) belongs to some classes of admissible function spaces. Our main method
invokes the Lyapunov – Perron methods, rescaling procedures, and the techniques of using the admissibility of function
spaces.
Доведено iснування iнтегральних (стiйких, нестiйких, центральних) многовидiв для розв’язкiв напiвлiнiйного iн-
тегрального рiвняння u(t) = U(t, s)u(s) +
∫ t
s
U(t, ξ)f(ξ, u(ξ))dξ у випадку, коли сiм’я еволюцiй (U(t, s))t≥s має
експоненцiальну трихотомiю на пiвосi або на всiй осi, а нелiнiйний збурюючий член f задовольняє ϕ-лiпшицевi
умови, тобто ‖f(t, x) − f(t, y)‖ 6 ϕ(t)‖x − y‖, де ϕ(t) належить до деяких класiв допустимих просторiв функ-
цiй. Наш основний метод базується на методах Ляпунова – Перрона, процедурах перемасштабування та технiцi
застосування допустимостi просторiв функцiй.
1. Introduction and preliminaries. Consider the semilinear evolution equation of the form
dx
dt
= A(t)x(t) + f(t, x(t)), t ∈ J, (1.1)
where J is a subinterval of the real line R; each A(t) is a (possibly unbounded) linear operator acting
in a Banach space X , x(t) ∈ X, and f(·, ·) : J × X → X is a nonlinear operator. When the linear
part (i.e., the equation dx/dt = A(t)x(t)) of the above equation has an exponential dichotomy (or
trichotomy), one shall try to find conditions imposed on the nonlinear forcing term f such that the
equation (1.1) has an integral manifold (e.g., a stable, unstable, or center manifold).
Such early results can be traced back to Hadamard [10], Perron [29, 30], Bogoliubov and
Mitropolsky [4, 5] for the case of matrix coefficients A(t), to Daleckii and Krein [8] for the case
of bounded coefficients acting on Banach spaces, and to Henry [12] for the case of unbounded
coefficients. At this point, we would like to quote the sentence by Anosov [1]:
,,Every five years or so, if not more often, some one ,,discovers” the theorem
of Hadamard and Perron, proving it by Hadamard’s method of proof or by Perron’s”.
∗ On leave from Hanoi University of Science and Technology as a research fellow of the Alexander von Humboldt
Foundation at Technical University of Darmstadt. The support by the Alexander von Humboldt Foundation is gratefully
acknowledged. The author thanks Prof. Matthias Hieber for his strong support and inspiration.
∗∗ This work is financially supported by the Vietnamese National Foundation for Science and Technology Development
(NAFOSTED) under Project 101.01-2011.25.
c© NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI, 2012
772 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 773
The Hadamard’s method is generalized to the so-called graph transform method which has been
used, e.g., in the works [2, 13, 22] to prove the existence of invariant manifolds. This method is
more far-reaching and related to complicated choices of the transforms between graphs represent-
ing the involved manifolds. Meanwhile, the Perron’s method is now extended to the well-known
Lyapunov – Perron method aimed at the construction of the so-called Lyapunov – Perron equations
(or operators) involving the differential equations under consideration to show the existence of the
integral manifolds. It seems to be more natural to use the Lyapunov – Perron method to handle with
the flows or semiflows which are generated by semilinear evolution equations since in this case it is
relatively simple to construct such Lyapunov – Perron equations or operators. We refer the reader to
[3, 7, 8, 11, 12, 15, 16, 35] and reference therein for more information on the matter.
To our best knowledge, the most popular conditions for the existence of invariant manifolds are
the exponential dichotomy (or trichotomy) of the linear part
dx
dt
= A(t)x and the uniform Lipschitz
continuity of the nonlinear part f(t, x) with sufficiently small Lipschitz constants (i.e., ‖f(t, x) −
− f(t, y)‖ 6 q‖x− y‖ for q small enough). The purpose of this paper is establishing the existence of
stable, unstable, and center-stable manifolds when the linear part of equation (1.1) has an exponential
trichotomy on the half-line or on the whole line under more general conditions on the nonlinear term
f(t, x), that is the non-uniform Lipschitz continuity of f, i.e., ‖f(t, x) − f(t, y)‖ 6 ϕ(t)‖x − y‖
for ϕ being a real and positive function which belongs to admissible function spaces defined in
Definition 2.3 below. Under some conditions on ϕ, we will prove the existence of center manifolds
for the equation (1.1) provided that the linear part
dx
dt
= A(t)x has an exponential trichotomy. Our
method is to transform to the case of exponential dichotomy by some rescaling procedures, and
then applying our techniques and results in [15] where we have used the Lyapunov – Perron method
and the characterization (obtained in [14]) of the exponential dichotomy of evolution equations in
admissible spaces of functions defined on the half-line R+ to construct the structures of solutions
of the equation (1.1) in a mild form, which belong to some certain classes of admissible spaces on
which we could implement some well-known procedures in functional analysis such as: constructing
of contraction mapping; using of Implicite Function Theorem, etc. The use of admissible spaces has
helped us to construct the invariant manifolds for equation (1.1) in the case of dichotomic linear
parts without using the smallness of Lipschitz constants of nonlinear forcing terms in classical sense.
Instead, the “smallness” is understood as the sufficient smallness of supt≥0
∫ t+1
t
ϕ(τ)dτ (see the
conditions in Theorem 4.7 in [15]). Consequently, we have obtained the existence of invariant-stable
manifolds for the case of dichotomic linear parts under very general conditions on the nonlinear
term f(t, x) (see [15]). Using these results and rescaling procedures we shall prove, in the present
paper, the existence of center manifolds for the mild solutions of the equation (1.1) in the case of
trichotomic linear parts under the same conditions on the nonlinear term f(t, x) as in [15]. Moreover,
using the same method we can also obtain the existence of unstable and center-unstable manifolds in
the case of dichotomic and trichotomic linear parts (respectively) for the evolution equations defined
on the whole line. Our main results are contained in Theorems 3.1, 4.1, 4.2, and Corollaries 4.1, 4.2,
4.3. We also illustrate our results in the Examples 5.2, 5.3.
We now recall some notions.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
774 NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI
Definition 1.1. Let J be one of the following intervals: R+ or R. A family of operators
{U(t, s)}t≥s, t,s∈J acting on a Banach space X is a (strongly continuous, exponential bounded)
evolution family on J if
(i) U(t, t) = Id and U(t, r)U(r, s) = U(t, s) for all t ≥ r ≥ s and t, s, r ∈ J,
(ii) the map (t, s) 7→ U(t, s)x is continuous on J for every x ∈ X,
(iii) ‖U(t, s)x‖ 6 Keω(t−s)‖x‖ for all t ≥ s, t, s, r ∈ J, and x ∈ X, for some constants K, ω.
The notion of an evolution family arises naturally from the theory of evolution equation which
are well-posed. Meanwhile, if the abstract Cauchy problem
du(t)
dt
= A(t)u(t), t ≥ s, t, s ∈ J,
u(s) = xs ∈ X,
(1.2)
is well-posed, there exists an evolution family (U(t, s))t≥s, t,s∈J such that the solution of the prob-
lem (1.2) is given by u(t) = U(t, s)u(s).
For more details on the notion and some problems focus on properties and applications of evo-
lution family we refer the reader to Pazy [28], Henry [12], and Nagel and Nickel [9]. For a given
evolution family, we have the following concept of an exponential trichotomy of evolution families
on J as follows.
Definition 1.2. Let J be one of the following intervals: R+ or R. A given evolution family
(U(t, s))t≥s, t,s∈J on J is said to have an exponential trichotomy on J if there are three families of
projections (Pj(t))t∈J, j = 1, 2, 3, positive constants N, α, β with α < β such that the following
conditions are satisfied:
(i) supt∈J ‖Pj(t)‖ <∞, j = 1, 2, 3,
(ii) P1(t) + P2(t) + P3(t) = Id for all t ∈ J, and Pj(t)Pi(t) = 0 for all j 6= i,
(iii) Pj(t)U(t, s) = U(t, s)Pj(s), for all t ≥ s ≥ 0, j = 1, 2, 3,
(iv) U(t, s)|ImPj(s) are homeomorphisms from ImPj(s) onto ImPj(t) for all t ≥ s, t, s ∈ J,
and j = 2, 3, respectively; also we denote the inverse of U(t, s)|ImP2(s) by U(s, t)| (here s 6 t),
(v) the following estimates hold:
‖U(t, s)P1(s)x‖ 6 Ne−β(t−s)‖P1(s)x‖,
‖U(s, t)|P2(t)x‖ 6 Ne−β(t−s)‖P2(t)x‖,
‖U(t, s)P3(s)x‖ 6 Neα(t−s)‖P3(s)x‖,
for all t ≥ s, t, s ∈ J, x ∈ X.
The evolution family is said to have an exponential dichotomy on J if it has an exponential
trichotomy for which the family of projections P3(t) is trivial, i.e., P3(t) = 0 for all t ∈ J. In this
case, we remark that the property (i) is a consequence of other properties (see [21], Lemma 4.2), and
we also denote P (t) := P1(t) called dichotomy projections.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 775
2. Function spaces and admissibility. We recall some notions of function spaces and admissibility.
We refer the readers to Massera and Schäffer [20] (Chapter 2) for wide classes of function spaces
that play a fundamental role throughout the study of differential equations in the case of bounded
coefficients A(t) (see also Räbiger and Schnaubelt [31] (§ 1) for some classes of admissible Banach
function spaces of functions defined on the whole line R).
Denote by B the Borel algebra and by λ the Lebesgue measure on R+. As already known, the
set of real-valued Borel-measurable functions on R+ (modulo λ-nullfunctions) that are integrable
on every compact subinterval J ⊂ R+ becomes, with the topology of convergence in the mean on
every such J, a locally convex topological vector space, which we denote by L1,loc(R+). A set of
seminorms defining the topology of L1,loc(R+) is given by pn(f) :=
∫
Jn
|f(t)|dt, n ∈ N, where
{Jn}n∈N = {[n, n+1]}n∈N is a countable set of abutting compact intervals whose union is R+. With
this set of seminorms one can see (see [20], Chapter 2, § 20) that L1,loc(R+) is a Fréchet space.
Let V be a normed space (with norm ‖ · ‖V ) and W be a locally convex Hausdorff topological
vector space. Then, we say that V is stronger than W if V ⊆W and the indentity map from V into
W is continuous. The latter condition is equivalent to the fact that for each continuous seminorm π
of W there exists a number βπ > 0 such that π(x) 6 βπ‖x‖V for all x ∈ V. We write V ↪→ W to
indicate that V is stronger than W. If, in particular, W is also a normed space (with norm ‖ · ‖W )
then the relation V ↪→ W is equivalent to the fact that V ⊆ W and there is a number α > 0 such
that ‖x‖W 6 α‖x‖V for all x ∈ V (see [20], Chapter 2 for detailed discussions on this matter).
We can now define Banach function spaces as follows.
Definition 2.1. A vector space E of real-valued Borel-measurable functions on R+ (modulo
λ-nullfunctions) is called a Banach function space (over (R+,B, λ) if
(1) E is Banach lattice with respect to a norm ‖ · ‖E , i.e., (E, ‖ · ‖E) is a Banach space, and
if ϕ ∈ E and ψ is a real-valued Borel-measurable function such that |ψ(·)| 6 |ϕ(·)| λ-a.e., then
ψ ∈ E and ‖ψ‖E 6 ‖ϕ‖E ,
(2) the characteristic functions χA belong to E for all A ∈ B of finite measure, and
supt≥0 ‖χ[t,t+1]‖E <∞ and inft≥0 ‖χ[t,t+1]‖E > 0,
(3) E ↪→ L1,loc(R+).
For a Banach function space E we remark that the condition (3) in the above definition means
that for each compact interval J ⊂ R+ there exists a number βJ ≥ 0 such that
∫
J
|f(t)|dt 6 βJ‖f‖E
for all f ∈ E.
We state the following trivial lemma which will be frequently used in our strategy.
Lemma 2.1. Let E be a Banach function space. Let ϕ and ψ be real-valued, measurable
functions on R+ such that they coincide with each other outside a compact interval and they are
essentially bounded (in particular, continuous) on this compact interval. Then ϕ ∈ E if and only if
ψ ∈ E.
We then define Banach spaces of vector-valued functions corresponding to Banach function
spaces as follows.
Definition 2.2. Let E be a Banach function space and X be a Banach space endowed with the
norm ‖ · ‖. We set
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
776 NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI
E := E(R+, X) :=
{
f : R+ → X : f is strongly measurable and ‖f(·)‖ ∈ E
}
(modullo λ-nullfunctions) endowed with the norm
‖f‖E := ‖‖f(·)‖‖E .
One can easily see that E is a Banach space. We call it the Banach space corresponding to the
Banach function space E.
We now introduce the notion of admissibility in the following definition.
Definition 2.3. The Banach function space E is called admissible if it satisfies
(i) there is a constant M ≥ 1 such that for every compact interval [a, b] ∈ R+ we have
b∫
a
|ϕ(t)|dt 6 M(b− a)
‖χ[a,b]‖E
‖ϕ‖E for all ϕ ∈ E, (2.1)
(ii) for ϕ ∈ E the function Λ1ϕ defined by Λ1ϕ(t) :=
∫ t+1
t
ϕ(τ)dτ belongs to E,
(iii) E is T+
τ -invariant and T−τ -invariant, where T+
τ and T−τ are defined, for τ ∈ R+, by
T+
τ ϕ(t) :=
ϕ(t− τ) for t ≥ τ ≥ 0,
0 for 0 6 t 6 τ,
T−τ ϕ(t) := ϕ(t+ τ) for t ≥ 0.
(2.2)
Moreover, there are constants N1, N2 such that ‖T+
τ ‖ 6 N1, ‖T−τ ‖ 6 N2 for all τ ∈ R+.
Example 2.1. Besides the spaces Lp(R+), 1 6 p 6∞, and the space
M(R+) :=
f ∈ L1,loc(R+) : sup
t≥0
t+1∫
t
|f(τ)|dτ <∞
endowed with the norm ‖f‖M := supt≥0
∫ t+1
t
|f(τ)|dτ, many other function spaces occuring in in-
terpolation theory, e.g. the Lorentz spaces Lp,q, 1 < p <∞, 1 6 q <∞ (see [6, p. 284], Theorem 3,
[36]) and, more general, the class of rearrangement invariant function spaces over (R+,B, λ) (see
[17]) are admissible.
Remark 2.1. If E is an admissible Banach function space then E ↪→ M(R+). Indeed, put
β := inft≥0 ‖χ[t,t+1]‖E > 0 (by Definition 2.1 (2)). Then, from Definition 2.3 (i) we derive
t+1∫
t
|ϕ(τ)|dτ 6
M
β
‖ϕ‖E for all t ≥ 0 and ϕ ∈ E. (2.3)
Therefore, if ϕ ∈ E then ϕ ∈M(R+) and ‖ϕ‖M 6
M
β
‖ϕ‖E . We thus obtain E ↪→M(R+).
We now collect some properties of admissible Banach function spaces in the following proposi-
tion (see [14], Proposition 2.6 and originally in [20]).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 777
Proposition 2.1. Let E be an admissible Banach function space. Then the following assertions
hold.
(a) Let ϕ ∈ L1,loc(R+) such that ϕ ≥ 0 and Λ1ϕ ∈ E, where, Λ1 is defined as in Defini-
tion 2.3(ii). For σ > 0 we define functions Λ′σϕ and Λ′′σϕ by
Λ′σϕ(t) :=
t∫
0
e−σ(t−s)ϕ(s)ds,
Λ′′σϕ(t) :=
∞∫
t
e−σ(s−t)ϕ(s)ds.
Then Λ′σϕ and Λ′′σϕ belong to E. In particular, if supt≥0
∫ t+1
t
ϕ(τ)dτ < ∞ (this will be satisfied
if ϕ ∈ E (see Remark 2.1)) then Λ′σϕ and Λ′′σϕ are bounded. Moreover, denoted by ‖ · ‖∞ for
ess sup-norm, we have
‖Λ′σϕ‖∞ 6
N1
1− e−σ
‖Λ1T
+
1 ϕ‖∞ and ‖Λ′′σϕ‖∞ 6
N2
1− e−σ
‖Λ1ϕ‖∞ (2.4)
for operator T+
1 and constants N1, N2 defined as in Definition 2.3.
(b) E contains exponentially decaying functions ψ(t) = e−αt for t ≥ 0 and any fixed constant
α > 0.
(c) E does not contain exponentially growing functions f(t) := ebt for t ≥ 0 and any fixed
constant b > 0.
Remark 2.2. If we replace the half-line R+ by any infinite (or half-infinite) interval I (e.g.,
I = R−,R, or any (−∞, t0] for fixed t0 ∈ R, etc.), then we have the similar notions of admissible
spaces on the interval I with slight changes as follow:
(1) In Definition 2.3, the translations semigroups T+
τ and T−τ for τ ∈ R+ should be replaced by
T+
τ and T−τ defined for τ ∈ I as
T+
τ ϕ(t) :=
ϕ(t− τ) for t and t− τ belonging to I,
0 for t ∈ I but t− τ /∈ I,
T−τ ϕ(t) := ϕ(t+ τ) for t ∈ I.
(2.5)
(2) In Proposition 2.1 (a), the functions Λ′σ and Λ′′σ should be replaced by
Λ′σϕ(t) :=
t0∫
t
e−σ|t−s|ϕ(s)ds, here t0 =∞ if I = R, and t0 = 0 if I = R−,
Λ′′σϕ(t) :=
t∫
−∞
e−σ|s−t|ϕ(s)ds.
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(3) In Proposition 2.1 (b) and (c) the functions ψ(t) = e−αt (t ≥ 0, and fixed α > 0) should be
replaced by ψ(t) = e−α|t|, t ∈ I and fixed α > 0; and the functions f(t) := ebt for t ≥ 0 and any
fixed constant b > 0 should be replaced by f(t) := eb|t|, t ∈ I and fixed b > 0.
These notions will be used in Section 5. We denote the admissible function space of the functions
defined on I by EI. If I = R+, we denote simply E := ER+ . For a function ϕ defined on the whole
line we denote the restriction of ϕ on I by ϕ|I. It is obvious that, if the function ϕ ∈ ER, then
ϕ|I ∈ EI.
In the case of infinite-dimensional phase spaces, instead of equation (1.1), for an evolution family
(U(t, s))t≥s, t,s∈J where J = R+or R, we consider the integral equation
u(t) = U(t, s)u(s) +
t∫
s
U(t, ξ)f(ξ, u(ξ))dξ for a.e. t ≥ s, t, s ∈ J. (2.6)
We note that, if the evolution family (U(t, s))t≥s, t,s∈J arises from the well-posed Cauchy prob-
lem (1.2) then the function u, which satisfies (2.6) for some given function f, is called a mild
solution of the inhomogeneous problem
du(t)
dt
= A(t)u(t) + f(t, u(t)), t ≥ s, t, s ∈ J,
u(s) = xs ∈ X.
We refer the reader to Pazy [28] for more detailed treatment on the relations between classical and
mild solutions of evolution equations (see also [9, 18, 35]).
To obtain the existence of an integral manifold for equation (2.6), beside the exponential di-
chotomy (or trichotomy) of the evolution family, we also need the properties of (local) ϕ-Lipschitz
of the nonlinear term f in the following definitions in which we suppose as above that J is one of
the infinite intervals R+ or R. Also, we let EJ be an admissible Banach function space on J. When
J = R+, we simply write E instead of ER+ .
Definition 2.4 (Local ϕ-Lipschitz functions). Let ϕ be a positive function belonging to EJ, and
Bρ be the ball with radius ρ in X, i.e., Bρ := {x ∈ X : ‖x‖ 6 ρ}. A function f : J×Bρ → X is said
to be local ϕ-Lipschitz of the class (M,ϕ, ρ) for some positive constants M, ρ if f satisfies
(i) ‖f(t, x)‖ 6Mϕ(t) for a.e. t ∈ J and all x ∈ Bρ,
(ii) ‖f(t, x1)− f(t, x2)‖ 6 ϕ(t)‖x1 − x2‖ for a.e. t ∈ J and all x1, x2 ∈ Bρ.
Remark 2.3. If f(t, 0) = 0 then, the condition (ii) in the above definition already implies that
f belongs to class (ρ, ϕ, ρ).
We next recall the definition of ϕ-Lipschitz functions.
Definition 2.5 (ϕ-Lipschitz functions). Let ϕ be a positive function belongs to EJ. A function
f : J× X→ X is said to be ϕ-Lipschitz if f satisfies
(i) f(t, 0) = 0 for a.e. t ∈ J,
(ii) ‖f(t, x1)− f(t, x2)‖ 6 ϕ(t)‖x1 − x2‖ for a.e. t ∈ J and all x1, x2 ∈ X.
3. Exponential trichotomy and center-stable manifolds on R+. In this section, we will gen-
eralize Theorem 4.7 in [15] to the case that the evolution family (U(t, s))t≥s≥0 has an exponential
trichotomy on R+ and the nonlinear forcing term f is ϕ-Lipschitz.
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INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 779
In this case, the interval J = R+. For an evolution family (U(t, s))t≥s≥0 we rewrite the integral
equation (2.6) for J = R+ as
u(t) = U(t, s)u(s) +
t∫
s
U(t, ξ)f(ξ, u(ξ))dξ for a.e. t ≥ s, t, s ∈ R+. (3.1)
Precisely, we will prove that there exists a center-stable manifold for the solutions of equation (3.1).
Theorem 3.1. Let the evolution family (U(t, s))t≥s≥0 have an exponential trichotomy with
the corresponding constants N, α, β (α < β), and projections (Pj(t))t≥0, j = 1, 2, 3, given in
Definition 1.2. Suppose that f : R+ × X → X be ϕ-Lipschitz, where ϕ is the positive function
belonging to E such that k < min
{
1
N + 1
,
1− eα−β
1− e−β
}
, here k is defined by
k :=
(1 +H)N
1− e−β
(
N1‖Λ1T
+
1 ϕ‖∞ +N2‖Λ1ϕ‖∞
)
. (3.2)
Then there exists a center-stable manifold C =
{
(t,Ct) | t ∈ R+ and Ct ⊂ X
}
for the solutions
of equation (3.1), with the family (Ct)t≥0 being the graphs of the family of Lipschitz continuous
mappings (gt)t≥0 (i.e., Ct := graph(gt) =
{
x + gtx | x ∈ Im(P1(t) + P3(t))
}
for each t ≥ 0)
where gt : Im (P1(t) + P3(t)) → ImP2(t) has the Lipschitz constant
Nk
1− k
independent of t, such
that the following properties hold:
(i) to each x0 ∈ Ct0 there corresponds one and only one solution u(t) of equation (3.1) on
[t0,∞) and it satisfies u(t0) = x0 and ess supt≥t0 ‖e
−γtu(t)‖ <∞, where γ :=
α+ β
2
,
(ii) Ct is homeomorphism to X1(t)⊕X3(t) for all t ≥ 0, where X1(t) = P1(t)X, X3(t) = P3(t)X,
(iii) C is invariant under the equation (3.1) in the sense that, if u(t) is the solution of equa-
tion (3.1) satisfying u(t0) = x0 ∈ Ct0 and ess supt≥t0 ‖e
−γtu(t)‖ < ∞, then u(s) ∈ Cs for all
s ≥ t0,
(iv) every two solutions u1(t), u2(t) on the center-stable manifold C satisfy the condition that
there exist positive constants µ and Cµ independent of t0 ≥ 0 such that
‖x(t)− y(t)‖ 6 Cµe
(γ−µ)(t−t0)
∥∥(P1(t0) + P3(t0))x(t0)− (P1(t0) + P3(t0))y(t0)
∥∥
for all t ≥ t0.
Proof. Set P (t) := P1(t) + P3(t) and Q(t) := P2(t) = I − P (t). We consider the following
rescaling evolution family:
Ũ(t, s)x := e−γ(t−s)U(t, s)x for all t ≥ s ≥ 0, x ∈ X,
where γ :=
α+ β
2
.
It is easy to check that (Ũ(t, s))t≥s≥0 is an evolution family on X.
We now claim that (Ũ(t, s))t≥s≥0 has an exponential dichotomy with the projection P (t) and
Q(t) on the half-line. Infact, it suffices to verify the estimates in Definition 1.2.
By the definition of exponential trichotomy we have
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780 NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI
‖Ũ(s, t)Q(t)x‖ 6 Ne−(β−γ)(t−s)‖Q(t)x‖ = Ne−
(β−α)
2 (t−s)‖Q(t)x‖
for all t ≥ s ≥ 0, x ∈ X.
On the other hand,
‖Ũ(t, s)P (s)x‖ = e−γ(t−s)‖U(t, s)[P1(s) + P3(s)]x‖ 6
6 Ne−(γ+α)(t−s)‖P1(s)x‖+ e−(γ−α)(t−s)‖P3(s)x‖ 6
6 Ne−
(β−α)
2 (t−s)(‖P1(s)x‖+ ‖P3(s)x‖) =
= Ne−
(β−α)
2 (t−s)(‖P1(s)(P1(s) + P3(s))x‖+ ‖P3(s)(P1(s) + P3(s))x‖
)
6
6 NHe−
(β−α)
2 (t−s)(‖(P1(s) + P3(s))x‖+ ‖(P1(s) + P3(s))x‖
)
=
= 2NHe−
(β−α)
2 (t−s)‖P (s)x‖ for all t ≥ s ≥ 0, x ∈ X
(here we use the fact that H := supt≥0{‖P1(t)‖, ‖P2(t)‖, ‖P3(t)‖} <∞).
We finally obtain the following estimate:
‖Ũ(t, s)P (s)x‖ 6 2NHe−
(β−α)
2 (t−s)‖P (s)x‖ for all t ≥ s ≥ 0, x ∈ X.
Therefore, (Ũ(t, s))t≥s≥0 has an exponential dichotomy with the projections (P (t))t≥0 and the di-
chotomy constants N ′ := max{N, 2NH}, β′ = β − α
2
> 0.
Put x̃(t) := e−γtx(t), and define the mapping F as follows:
F : R+ × X→ X,
F (t, x) = e−γtf(t, eγtx) for all t ≥ 0, x ∈ X.
We can easily verify that the operator F is also ϕ-Lipschitz. Thus, we can rewrite the equation (3.1)
in the new form
x̃(t) = Ũ(t, s)x̃(s) +
t∫
s
Ũ(t, ξ)F (ξ, x̃(ξ))dξ for a.e. t ≥ s ≥ 0. (3.3)
Hence, by [15] (Theorem 4.7), we obtain that, if
k =
(1 +H)N(N1‖Λ1T
+
1 ϕ‖∞ +N2‖Λ1ϕ‖∞)
1− e−β
<
1
1 +N
,
then there exists an invariant stable manifold C for the solutions of equation (3.3). Return to equa-
tion (3.1) by using the relation x(t) := eγtx̃(t) we can easily verify the properties of C which are
stated in (i), (ii), (iii), and (iv). Thus, C is an invariant center-stable manifold for the solutions of
equation (3.1).
Theorem 3.1 is proved.
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INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 781
Remark 3.1. In case the evolution family has an exponential trichotomy and the nonlinear term
f satisfies the local ϕ-Lipschitz of the class (M,ϕ, ρ) with f(t, 0) = 0 and the positive function
ϕ ∈ E satisfying k < min
{
ρ
2M
,
1
N + 1
,
1− eα−β
1− e−β
}
(here k is defined as in (3.2)), then by the
similar ways as above and using the results in [15] (Theorem 3.8) we can obtain the existence of a
local center-stable manifold for the solutions of equation (3.1), that is a set C ⊂ R+ ×X such that
there exist positive constants ρ, ρ0, ρ1 and a family of Lipschitz continuous mappings
gt : Bρ0 ∩ Im(P1(t) + P3(t))→ Bρ1 ∩ ImP2(t), t ∈ R+,
with Lipschitz constants independent of t satisfying:
(i) C = {(t, x+gt(x)) ∈ R+×
(
Im(P1(t)+P3(t))⊕ ImP2(t)
)
| t ∈ R+, x ∈ Bρ0 ∩ Im(P1(t)+
+ P3(t))}, and we denote by Ct := {x+ gt(x) | (t, x+ gt(x)) ∈ C},
(ii) Ct is homeomorphic to Bρ0 ∩ Im(P1(t) +P3(t)) = {x ∈ Im(P1(t) +P3(t)) | ‖x‖ 6 ρ0} for
all t ≥ 0,
(iii) to each x0 ∈ Ct0 there corresponds one and only one solution u(t) of equation (3.1) on
[t0,∞) and it satisfies u(t0) = x0 and ess supt≥t0 ‖e
−γtu(t)‖ <∞, where γ :=
α+ β
2
,
(iv) every two solutions u1(t), u2(t) on the local center-stable manifold C satisfy the condition
that there exist positive constants µ and Cµ independent of t0 ≥ 0 such that
‖x(t)− y(t)‖ 6 Cµe
(γ−µ)(t−t0)
∥∥(P1(t0) + P3(t0))x(t0)− (P1(t0) + P3(t0))y(t0)
∥∥ (3.4)
for all t ≥ t0.
4. Unstable manifolds for equations defined on the whole line. We now consider the case that
the evolution family (U(t, s))t≥s and the nonlinear forcing term f are defined on the whole line (i.e.,
the case J = R). That is to say, we will consider the integral equation
x(t) = U(t, s)x(s) +
t∫
s
U(t, ξ)f(ξ, x(ξ))dξ for a.e. t ≥ s, t, s ∈ R. (4.1)
As in Section 1, the solutions of the equation (4.1) is called the mild solutions of the equation
dx
dt
= A(t)x+ f(t, x), t ∈ R, x ∈ X, (4.2)
where A(t), t ∈ R (in general case), are unbounded operators in X, which are coefficients of a
well-posed Cauchy problem
du(t)
dt
= A(t)u(t), t ≥ s,
u(s) = xs ∈ X,
whose solutions are given by x(t) = U(t, s)x(s) as mentioned in Section 1. In this case, the exis-
tences of (local- or invariant-) stable manifolds on R are defined and proved by the same way as in
the case of equations defined on a half-line R+ (see [15], Theorem 4.7). Therefore, we will pay our
attention to the case of the unstable manifolds which are defined below.
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4.1. Local-unstable manifolds on R. We shall prove the existence of the local-unstable mani-
fold under the conditions that the evolution family (U(t, s))t≥s has an exponential dichotomy and the
nonlinear term f is local ϕ-Lipschitz of the class (M,ϕ, ρ) for a relevant positive function ϕ ∈ ER.
We now give the description of a local-unstable manifold for the solutions of the integral equa-
tion (4.1) in the following definition in which we remind that by Br we denote the ball in X with
radius r centered at 0, i.e., Br = {x ∈ X | ‖x‖ 6 r}.
Definition 4.1. A set U ⊂ R × X is said to be a local-unstable manifold for the solutions of
equation (4.1) if for every t ∈ R the phase spaces X splits into a direct sum X = X1(t)⊕ X2(t) such
that
inf
t∈R+
Sn(X1(t),X2(t)) := inf
t∈R+
inf
{
‖x1 + x2‖ : xi ∈ Xi(t), ‖xi‖ = 1, i = 1, 2
}
> 0,
and if there exist positive constants ρ, ρ0, ρ1 and a family of Lipschitz continuous mappings
ht : Bρ0 ∩ X2(t)→ Bρ1 ∩ X1(t), t ∈ R,
with the Lipschitz constants independent of t such that
(i) U = {(t, x + ht(x)) ∈ R × (X2(t) ⊕ X1(t)) | x ∈ Bρ0 ∩ X2(t)}, and we denote by Ut :=
:= {x+ ht(x) | (t, x+ ht(x)) ∈ U},
(ii) Ut is homeomorphic to Bρ0 ∩ X2(t) for all t ∈ R,
(iii) to each x0 ∈ Ut0 there corresponds one and only one solution x(t) of equation (4.1)
satisfying the conditions x(t0) = x0 and ess supt6t0 ‖x(t)‖ 6 ρ.
Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the corresponding
projection P (t), t ∈ R, and the dichotomy constants N, β > 0. Then, we can define the Green’s
function as follows:
G(t, τ) :=
P (t)U(t, τ) for t ≥ τ,
−U(t, τ)|[I − P (τ)] for t < τ.
(4.3)
Thus, we have
‖G(t, τ)‖ 6 (1 +H)Ne−β|t−τ | for all t 6= τ, where H = sup
t∈R
‖P (t)‖ <∞.
We now prove the existence of a local-unstable manifold. To do that, we first construct the form
of the solutions of the equation (4.1) which are bounded on the half-line (−∞, t0]. We denote by
‖ · ‖∞ the sup-norm on the half-line (−∞, t0].
Lemma 4.1. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Suppose that ϕ is the
positive function which belongs to ER. Let f : R×Bρ → X be local ϕ-Lipschitz of the class (M,ϕ, ρ)
for some positive constants M, ρ. Let x(t) be a solution of (4.1) such that ess supt6t0 ‖x(t)‖ 6 ρ
for some fixed t0. Then, for t 6 t0, we have that x(t) can be rewritten in the form
x(t) = U(t, t0)|v +
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ for all t 6 t0, (4.4)
and some v ∈ X2(t0) = (I − P (t0))X, where G(t, τ) is the Green’s function defined above.
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INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 783
Proof. Let
y(t) :=
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ for all t 6 t0. (4.5)
Then the function y(·) is bounded. Indeed, by the estimates of the Green’s function G and the function
f we have
‖y(·)‖∞ 6
t0∫
−∞
(1 +H)Ne−β|t−τ |‖f(τ, x(τ))‖dτ 6
6 (1 +H)NM
t∫
−∞
e−β(t−τ)‖ϕ(τ)‖dτ +
t0∫
t
eβ(t−τ)‖ϕ(τ)‖dτ
6
6 (1 +H)NM
[
N1‖Λ1ϕ‖∞ +N2‖Λ1T
+
1 ϕ‖∞
1− e−β
]
<∞.
Next, by computing directly we verify that y(·) satisfies the integral equation
y(t0) = U(t0, t)y(t) +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ for all t 6 t0. (4.6)
Indeed, subtituting y from (4.5) to the right-hand side of (4.6) we obtain
U(t0, t)y(t) +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ =
= U(t0, t)
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ =
= U(t0, t)
t∫
−∞
U(t, τ)P (τ)f(τ, x(τ))dτ−
−U(t0, t)
t0∫
t
U(t, τ)|(I − P (τ))f(τ, x(τ))dτ +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ =
=
t∫
−∞
U(t0, τ)P (τ)f(τ, x(τ))dτ−
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784 NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI
−
t0∫
t
U(t0, t)U(t, τ)|(I − P (τ))f(τ, x(τ))dτ +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ =
=
t0∫
−∞
U(t0, τ)P (τ)f(τ, x(τ))dτ =
t0∫
−∞
G(t0, τ)f(τ, x(τ)) = y(t0),
here we use the fact U(t0, t)U(t, τ)|(I − P (τ)) = U(t0, τ)(I − P (τ)) that for all t 6 τ 6 t0.
Thus, we have
y(t0) = U(t0, t)y(t) +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ.
On the other hand,
x(t0) = U(t0, t)x(t) +
t0∫
t
U(t0, τ)f(τ, x(τ))dτ.
Then x(t0)− y(t0) = U(t0, t)[x(t)− y(t)]. We need to prove that x(t0)− y(t0) ∈ (I − P (t0))X.
Applying the operator P (t0) to the expression x(t0)− y(t0) = U(t0, t)[x(t)− y(t)], we have
‖P (t0)[x(t0)− y(t0)]‖ = ‖U(t0, t)P (t)[x(t)− y(t)]‖ 6 Ne−β(t0−t)‖P (t)‖.‖x(t)− y(t)‖.
Since supt∈R ‖P (t)‖ < ∞ and ‖x(t) − y(t)‖ 6 ‖x(·)‖∞ + ‖y(·)‖∞ < ∞, letting t → −∞ we
obtain that
‖P (t0)[x(t0)− y(t0)]‖ = 0.
It means that, v := x(t0)− y(t0) ∈ (I − P (t0))X = X2(t0) finishing the proof.
Remark 4.1. By computing directly, we can see that the converse of Lemma 4.1 is also true. It
means, all solutions of equation (4.4) satisfied the equation (4.1) for t 6 t0.
Lemma 4.2. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Suppose that ϕ is
the positive function which belongs to E. Put
k :=
(1 +H)N
1− e−β
[
N1‖Λ1ϕ‖∞ +N2‖Λ1T
+
1 ϕ‖∞
]
. (4.7)
Let f : R× Bρ → X be local ϕ-Lipschitz of the class (M,ϕ, ρ) such that k < min
{
1,
ρ
2M
}
. Then
there corresponds to each v ∈ Bρ/2N ∩ X2(t0) one and only one solution x(t) of the equation (4.1)
on (−∞, t0] satisfying the conditions that (I − P (t0))x(t0) = v and ess supt6t0 ‖x(t)‖ 6 ρ.
Proof. We consider in the space L∞((−∞, t0], X) the ball
Bρ :=
{
x(·) ∈ L∞((−∞, t0], X) : ‖x(·)‖∞ := ess sup
t6t0
‖x(t)‖ 6 ρ
}
.
For v ∈ Bρ/2N ∩X2(t0) we will prove the transformation T defined by
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(Tx)(t) = U(t, t0)|v +
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ for all t 6 t0,
acts from Bρ into Bρ and is a contraction. In fact, for x(·) ∈ Bρ we have that ‖f(t, x(t))‖ 6Mϕ(t).
Therefore, putting
y(t) = U(t, t0)|v +
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ for all t 6 t0,
we obtain that ‖y(t)‖ 6 Ne−β(t0−t)‖v‖ + (1 + H)NM
∫ t0
−∞
e−β|t−τ |ϕ(τ)dτ. It follows from the
admissibility of L∞ that, y(·) ∈ L∞ and
‖y(·)‖∞ 6 N‖v‖+
(1 +H)NM
1− e−β
(
N1‖Λ1T
+
1 ϕ‖∞ +N2‖Λ1ϕ‖∞
)
.
Using now the fact that ‖v‖ 6 ρ
2N
and
(1 +H)N
1− e−β
(N1‖Λ1T
+
1 ϕ‖∞ +N2‖Λ1ϕ‖∞) <
ρ
2M
,
we have that ‖y(·)‖∞ 6 ρ. Therefore, the transformation T acts from Bρ to Bρ.
It follows from the estimates of G and U(t, s) that
‖T (x)− T (y)‖∞ 6
6
(1 +H)N‖x(·)− y(·)‖∞
1− e−β
[
N1‖Λ1ϕ‖∞ +N2‖Λ1T
+
1 ϕ‖∞
]
= k‖x(·)− y(·)‖∞.
Since k < 1, we obtain that T is a contraction. By the Banach contraction mapping theorem, the
lemma follows.
From Lemmas 4.1, 4.2 and using the same arguments as in [15] (Theorem 3.8) we obtain the
existence of an unstable manifold in the following theorem.
Theorem 4.1. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Then, for any ρ > 0
and M > 0, we have that, if f is local ϕ-Lipschitz of the class (M,ϕ, ρ) with the positive function
ϕ ∈ ER such that k < min
{
ρ
2M
,
1
N + 1
}
, here k is defined as in 4.7, there exists a local unstable
manifold for the solutions of equation (4.1). Moreover, for any two solution x1(·) and x2(·) belonging
to this manifold we have∥∥x1(t)− x2(t)∥∥ 6 Ceµ(t−t0)
∥∥(I − P (t0))x1(t0)− (I − P (t0))x2(t0)
∥∥ for all t 6 t0, (4.8)
where C, µ be the positive constants independent of t0, x1(·) and x2(·).
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Proof. The proof of this theorem can be done by the same way as in [15] (Theorem 3.8) replacing
R+ by R and using the structures of bounded solutions as in Lemmas 4.1, 4.2. We just note that the
family of Lipschitz mappings (ht)t∈R determining the local-unstable manifold is define by
ht : Bρ/2N ∩X2(t)→ Bρ/2 ∩X1(),
ht(y) =
t∫
−∞
G(t, s)f(s, x(s))ds
for y ∈ Bρ/2N ∩ X2(t), where x(·) is the unique solution in L∞((−∞, t], X) of equation (4.1) on
(−∞, t] satisfying (I − P (t))x(t) = y (note that the existence and uniqueness of x(·) is obtained in
Lemma 4.2). Furthermore, the Lipschitz constant of ht is
kN
1− k
< 1 which is the same as that of gt
determining the local-stable manifold (see [15], Theorem 3.8).
Theorem 4.1 is proved.
From the existence of the local-stable and local-unstable manifolds of equation (4.1) defined on
the whole line we have the following important corollary which describes the geometric picture of
solutions to equation (4.1).
Corollary 4.1. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Then, for any ρ > 0
and M > 0, we have that, if f is local ϕ-Lipschitz of the class (M,ϕ, ρ) with the positive function
ϕ ∈ ER such that k < min
{
ρ
2M
,
1
N + 1
,
ρ
2MN
}
, here k is defined as in (4.7), then there exist a
local-stable manifold S and a local-unstable manifold U for the solutions of equation (4.1) having
the following properties:
(a) for each t0 the intersection St0 ∩Ut0 contains the unique element zt0 ,
(b) the solution u0(t) of equation (4.1) with initial condition u0(t0) = zt0 is bounded on the
whole line R,
(c) the solutions u(t) of equation (4.1) satisfying u(t0) ∈ St0 exponentially approach u0(t) as
t→∞,
(d) the solutions u(t) of equation (4.1) satisfying u0(t) ∈ Ut0 exponentially approach u0(t) as
t→ −∞.
Proof. (a) The condition that x ∈ St0 ∩Ut0 is equivalent to the fact that there are w ∈ Bρ0 ∩
∩X1(t0) and y ∈ Bρ0 ∩X2(t0) such that x = w+ gt0w = ht0y + y where gt0 and ht0 are members
of the families of Lipschitz continuous mappings (gt)t∈R determining S and (ht)t∈R determining U,
respectively. Then w − ht0y = y − gt0w ∈ X1(t0) ∩X2(t0) = {0}. This follows that w = ht0y and
y = gt0w. Therefore, w = ht0(gt0w) = (ht0 ◦ gt0)w. We now estimate gt0w for w ∈ Bρ0 ∩X1(t0)
by using the formula (see [15], equation (18))
gt0(w) =
∞∫
t0
G(t0, s)f(s, x(s))ds, (4.9)
where w ∈ Bρ/2N∩X1(t0) and x(·) is the unique solution in Bρ of equation (4.1) on [t0,∞) satisfying
P (t0)x(t0) = w (note that the existence and uniqueness of x(·) is obtained in [15] (Theorem 3.7)).
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By (4.9) we have that
‖gt0(w)‖ 6
∞∫
0
‖G(t0, s)‖‖f(s, x(s))‖ds 6 (1 +H)NM
∞∫
0
e−|t0−s|ϕ(s)ds 6
6
(1 +H)NM
1− e−β
(N1‖Λ1T
+
1 ϕ‖∞ +N2‖Λ1ϕ‖∞) = kM <
ρ
2N
(
since k <
ρ
2MN
)
.
Therefore, we obtain that gt0 : Bρ/2N ∩X1(t0)→ Bρ/2N ∩X2(t0). Similarly, we have ht0 : Bρ/2N ∩
∩X2(t0)→ Bρ/2N ∩X1(t0). This follows that
ht0 ◦ gt0 : Bρ/2N ∩X1(t0)→ Bρ/2N ∩X1(t0).
Since the mappings gt0 and ht0 are both Lipschitz continuous with the same Lipschitz constant
kN
1− k
< 1 (see the proof of [15] (Theorem 3.8)), we obtain that ht0 ◦gt0 is a contraction. Thus, there
exists a unique w0 such that w0 = (ht0 ◦ gt0)w0. Putting zt0 = w0 + gt0w0 we obtain that zt0 is the
unique element of the intersection St0 ∩Ut0 .
The property (b) follows from the definitions of the local-stable and local-unstable manifolds,
respectively.
The properties (c) and (d) are consequences of the inequalities in [15] (Theorem 3.7, ineq. (13))
and (4.8), respectively.
4.2. Invariant unstable manifolds on R. In this subsection we consider the existence of the
invariant unstable manifold under the conditions that the evolution family has an exponential di-
chotomy, and the nonlinear term f is ϕ-Lipschitz continuous.
We now give the definition of an invariant unstable manifold for the solutions of the integral
equation (4.1).
Definition 4.2. A set S ⊂ R× X is said to be an invariant unstable manifold for the solutions
of equation (4.1) if for every t ∈ R the phase spaces X splits into a direct sum X = X1(t) ⊕ X2(t)
such that
inf
t∈R+
Sn(X1(t),X2(t)) := inf
t∈R+
inf
{
‖x1 + x2‖ : xi ∈ Xi(t), ‖xi‖ = 1, i = 1, 2
}
> 0,
and if there exists a family of Lipschitz continuous mappings
gt : X2(t)→ X1(t), t ∈ R,
with the Lipschitz constants independent of t such that
(i) S = {(t, x+ gt(x)) ∈ R× (X2(t)⊕X1(t)) | x ∈ X2(t)}, and we denote by St := {x+ gt(x) |
(t, x+ gt(x)) ∈ S},
(ii) St is homeomorphic to X2(t) for all t ∈ R,
(iii) to each x0 ∈ St0 there corresponds one and only one solution x(t) of equation (4.1) satisfying
the conditions x(t0) = x0 and ess supt6t0 ‖x(t)‖ <∞,
(iv) S is invariant under the equation (4.1) in the sense that, if x(·) is a solution of equation (4.1)
satisfying x(t0) ∈ St0 and ess supt6t0 ‖x(t)‖ <∞, then x(t) ∈ St for all t 6 t0.
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As in the previous subsection, we can construct the form of the solutions of equation (4.1) which
are bounded on the half-line (−∞, t0] in the following lemma whose proof can be done by the same
way as in Lemma 4.1.
Lemma 4.3. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Suppose that ϕ is
the positive function which belongs to ER. Let f : R× X→ X be ϕ-Lipschitz. Let x(t) be a solution
of (4.1) such that ess supt6t0 ‖x(t)‖ <∞ for some fixed t0. Then, for t 6 t0, we have that x(t) can
be rewritten in the form
x(t) = U(t, t0)|v +
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ for all t 6 t0, (4.10)
and some v ∈ X2(t0) = (I − P (t0))X, where G(t, τ) is the Green’s function defined above.
Remark 4.2. By computing directly, we can see that the converse of Lemma 4.3 is also true. It
means, all solutions of equation (4.10) satisfied the equation (4.1) for t 6 t0.
Similarly to Lemma 4.2 we have the following lemma which describes the existence and unique-
ness of certain bounded solutions.
Lemma 4.4. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Suppose that ϕ
is the positive function which belongs to E. Let f : R × X → X be ϕ-Lipschitz satisfying k <
< 1, where k is defined as in (4.7). Then there corresponds to each v ∈ X2(t0) one and only one
solution x(t) of the equation (4.1) on (−∞, t0] satisfying the condition (I − P (t0))x(t0) = v and
ess supt6t0 ‖x(t)‖ <∞.
Proof. For each t0 ∈ R, v ∈ X2(t0) we consider the operator
T : L∞((−∞, t0],X)→ L∞((−∞, t0],X),
x 7→ (Tx)(t) = U(t, t0)|v +
t0∫
−∞
G(t, τ)f(τ, x(τ))dτ for all t 6 t0.
It follows from the estimates of G and U(t, s) that
‖T (x)− T (y)‖∞ 6
6
(1 +H)N‖x(·)− y(·)‖∞
1− e−β
[
N1‖Λ1ϕ‖∞ +N2‖Λ1T
+
1 ϕ‖∞
]
= k‖x(·)− y(·)‖∞.
Since k < 1, we obtain that T is a contraction. By the Banach contraction mapping theorem, the
lemma follows.
From Lemmas 4.3, 4.4 and using the same arguments as in [15] (Theorem 4.7) we obtain the
existence of an invariant unstable manifold in the following theorem.
Theorem 4.2. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Suppose that
f : R × X → X be ϕ-Lipschitz, where ϕ is the positive function which belongs to ER such that
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k < 1, here k defined as in (4.7). Then there exists an invariant unstable manifold for the solutions
of equation (4.1). Moreover, for any two solution x1(·) and x2(·) belonging to this unstable manifold
we have
‖x1(t)− x2(t)‖ 6 Ceµ(t−t0)‖(I − P (t0))x1(t0)− (I − P (t0))x2(t0)‖ for all t 6 t0,
where C, µ be the positive constants independent of t0, x1(·) and x2(·).
Proof. The proof of this theorem can be done by the same way as in [15] (Theorem 4.7) replacing
R+ by R and using the structures of bounded solutions as in Lemmas 4.3, 4.4. We just note that the
family of Lipschitz mappings (gt)t∈R determining the unstable manifold is define by
gt : X2(t)→ X1(t),
gt(y) =
t∫
−∞
G(t, s)f(s, x(s))ds
for y ∈ X2(t), where x(·) is the unique solution in L∞((−∞, t], X) of equation (4.1) on (−∞, t] sat-
isfying (I−P (t))x(t) = y (note that the existence and uniqueness of x(·) is obtained in Lemma 4.4).
Theorem 4.2 is proved.
Using now the similar arguments as in Corollary 4.1, we easily obtain the following corollary
which describes the relations of solutions of equation (4.1) with initial values lying on the invariant
stable or unstable manifolds and the solution lying on the intersection of the two manifolds.
Corollary 4.2. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, β > 0. Suppose that f is
ϕ-Lipschitz with the positive function ϕ ∈ ER such that k <
1
N + 1
, here k defined as in (4.7). Then
there exist an invariant stable manifold S and an invariant unstable manifold U for the solutions of
equation (4.1) having the following properties:
(a) for each t0 the intersection St0 ∩Ut0 contains the unique element zt0 ,
(b) the solution u0(t) of equation (4.1) with initial condition u0(t0) = zt0 is bounded on the
whole line R,
(c) the solutions u(t) of equation (4.1) satisfying u(t0) ∈ St0 exponentially approach u0(t) as
t→∞,
(d) the solutions u(t) of equation (4.1) satisfying u(t0) ∈ Ut0 exponentially approach u0(t) as
t→ −∞.
4.3. Invariant center-unstable manifolds on R. Using Theorem 4.2 and rescaling procedures
similar to Theorem 3.1 to transform the trichotomy case to the dichotomy case, we can easily obtain
the exsitence of an invariant center-unstable manifolds in the following theorem.
Theorem 4.3. Let the evolution family (U(t, s))t≥s have an exponential trichotomy with the
corresponding constants K, α, β (α < β), and projections (Pj(t))t∈R, j = 1, 2, 3, given in Def-
inition 1.2. Suppose that f : R+ × X → X be ϕ-Lipschitz, where ϕ is the positive function which
belongs to ER such that k < min
{
1
N + 1
,
1− eα−β
1− e−β
}
, here k is defined by (4.7). Then there exists
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a center-unstable manifold Cu = {(t,Cu
t ) | t ∈ R+ and Cu
t ⊂ X} for the solutions of equa-
tion (3.1), with the family (Cu
t )t∈R being the graphs of the family of Lipschitz continuous mappings
(ht)t∈R
(
i.e., Cu
t := graph(ht) =
{
x + htx | x ∈ Im(P2(t) + P3(t))
}
for each t ∈ R
)
where
ht : Im(P2(t) + P3(t))→ ImP1(t) has the Lipschitz constant
Nk
1− k
independent of t, such that the
following properties hold:
(i) to each x0 ∈ Cu
t0 there corresponds one and only one solution u(t) of equation (3.1) on
(−∞, t0] satisfying u(t0) = x0 and ess supt6t0 ‖e
γtu(t)‖ <∞, where γ :=
α+ β
2
,
(ii) Cu
t is homeomorphism to X2(t)⊕ X3(t) for all t ∈ R, where X2(t) = ImP2(t) and X3(t) =
= ImP3(t),
(iii) Cu is invariant under the equation (3.1) in the sense that, if u(t) is the solution of equa-
tion (3.1) satisfying u(t0) = x0 ∈ Cu
t0 and ess supt6t0 ‖e
γtu(t)‖ < ∞, then u(s) ∈ Cu
s for all
s 6 t0,
(iv) every two solutions u1(t), u2(t) on the center-unstable manifold Cu satisfy the condition
that there exist positive constants µ and Cµ independent of t0 ≥ 0 such that
‖x(t)− y(t)‖ 6 Cµe
(µ−γ)(t−t0)
∥∥(P1(t0) + P3(t0))x(t0)− (P1(t0) + P3(t0))y(t0)
∥∥ (4.11)
for all t 6 t0.
Note that the existence of an invariant center-stable manifold on R is defined and proved by the
same ways as in the case of half-line R+ (see Theorem 3.1).
From the existence of the invariant center-stable and center-unstable manifolds of equation (4.1)
defined on the whole line we have the following important corollary describing the behavior of
solutions to equation (4.1).
Corollary 4.3. Let the evolution family (U(t, s))t≥s have an exponential dichotomy with the
corresponding projections P (t), t ∈ R, and the dichotomy constants N, α, β > 0. Suppose that f is
ϕ-Lipschitz with the positive function ϕ ∈ ER such that
k < min
{
1
N + 1
,
1− eα−β
1− e−β
,
√
2− 1
N +
√
2− 1
}
,
here k defined as in (4.7). Then there exist an invariant center-stable manifold C and an invariant
center-unstable manifold Cu for the solutions of equation (4.1) having the following properties:
(a) for each t0 ∈ R the intersection Ct0 ∩Cu
t0 is homeomorphism to X3(t0) = P3(t0)X,
(b) the solution u0(t) of equation (4.1) with initial condition u0(t0) ∈ Ct0 ∩ Cu
t0 satisfies that
ess supt∈R ‖e−γ|t|u(t)‖ <∞, where γ :=
α+ β
2
,
(c) for the solution u(t) of equation (4.1) satisfying u(t0) ∈ Ct0 we have that e−γtu(t) exponen-
tially approaches e−γtu0(t) as t→∞,
(d) for the solution u(t) of equation (4.1) satisfying u(t0) ∈ Cu
t0 we have that eγtu(t) exponen-
tially approaches eγtu0(t) as t→ −∞.
Proof. (a) Let us first prove that for each z ∈ X3(t) there exists a unique w ∈ X1(t) ⊕ X3(t)
such that w = ht(z + gt(w)) + z, where gt and ht are the members of the Lipschitz mapping
families (gt)t∈R and (ht)t∈R determining the invariant center-stable and center-unstable manifolds,
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respectively. Indeed, the mapping
L : X1(t)⊕ X3(t)→ X1(t)⊕ X3(t),
y 7→ ht(z + gt(y)) + z
satisfies that
‖Ly1 − Ly2‖ = ‖ht(z + gt(y1))− ht(z + gt(y2))‖ 6
Nk
1− k
‖gt(y1)− gt(y2)‖ 6
6
(
Nk
1− k
)2
‖y1 − y2‖.
Since
Nk
1− k
< 1 we obtain that L is a contraction. Let w be its unique fixed point. Then w is the
unique element in X1(t)⊕ X3(t) such that w = ht(z + gt(w)) + z.
Define now the mapping D : X3(t) → Ct ∩ Cu
t by D(z) = w + gt(w), where w is the unique
element in X1(t)⊕ X3(t) such that w = ht(z + gt(w)) + z. Then we have w+ gt(w) = z + gt(w) +
+ ht(z + gt(w)) ∈ Ct ∩Cu
t . The uniqueness of w yields that D is a well-defined mapping.
We next prove the sujectiveness of D. For x ∈ Ct ∩Cu
t we have that there is u ∈ X1(t)⊕ X3(t)
and v ∈ X2(t) ⊕ X3(t) such that x = u + gt(u) = v + ht(v). Then we have u − ht(v) = v −
− gt(u) ∈
(
X1(t) ⊕ X3(t)
)
∩
(
X2(t) ⊕ X3(t)
)
= X3(t). Therefore, there is a z ∈ X3(t) such that
u− ht(v) = v − gt(u) = z. This follows that u− ht(z + gt(u)) = z. As shown above, this relation
means that Dz = u+ gt(u) = x. Therefore, D is surjevtive.
We now prove that D is a Lipschitz mapping. In fact, by the definition of D we have D(z1) =
= w1 + gt(w1) and D(z2) = w2 + gt(w2) for w1 and w2 being the unique solutions in X1(t)⊕X3(t)
of equations w1 = ht(z1 + gt(w1)) + z1 and w2 = ht(z2 + gt(w2)) + z2, respectively. Then putting
l =
Nk
1− k
(the Lipschitz constant of gt and ht), we have
(1− l)‖w1 − w2‖ 6 ‖D(z1)−D(z2)‖ =
= ‖z1 + ht(z1 + gt(w1)) + gt(w1)−
(
z2 + ht(z2 + gt(w2)) + gt(w2)
)
‖ 6
6 ‖z1 − z2‖+ l‖z1 − z2‖+ l‖gt(w1)− gt(w2)‖+ ‖gt(w2)− gt(w2)‖ 6
6 (1 + l)‖z1 − z2‖+ l(l + 1)‖w2 − w2‖.
Therefore, we obtain that ‖D(z1)−D(z2)‖ 6 (1 + l)‖z1 − z2‖+
l(l + 1)
1− l
‖D(z1)−D(z2)‖. Thus,
‖D(z1)−D(z2)‖ 6
1− l2
2− (1 + l)2
‖z1 − z2‖,
here we note that 2 − (1 + l)2 > 0 since k <
√
2− 1
N +
√
2− 1
. Hence, we obtain that D is a Lipschitz
mapping with Lischitz constant
1− l2
2− (1 + l)2
. This follows that D is continuous and injective. As
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792 NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI
shown above D is already surjective, therefore, D is bijetive. The inverse D−1 of D is defined as
D−1 : Ct ∩ Cu
t → X3(t) with D−1(w + gt(w)) = z if z = w − ht(z + gt(w)) (note that, by the
contraction-mapping arguments we can easily show that for each w ∈ X1(t) ⊕ X3(t) there exists a
unique z ∈ X3(t) such that z = w − ht(z + gt(w))). We then prove that D−1 is also a Lipschitz
mapping. Indeed, for x1 = u+ gt(u) and x2 = v + gt(v) belonging to Ct ∩Cu
t we have that
‖D−1x1 −D−1x2‖ = ‖z1 − z2‖ 6
6 ‖w1 − ht(z1 + gt(w1))−
(
w2 − ht(z2 + gt(w2))
)
‖ 6
6 ‖w1 − w2‖+ l‖z1 − z2‖+ l2|‖w1 − w2‖ =
= (1 + l2)‖w1 − w2)‖+ l‖D−1x1 −D−1x2‖ 6
6
1 + l2
1− l
‖w1 + gt(w1)− w2 − gt(w2)‖+ l‖D−1x1 −D−1x2‖ =
=
1 + l2
1− l
‖x1 − x2‖+ l‖D−1x1 −D−1x2‖.
Therefore, we obtain that ∥∥D−1x1 −D−1x2
∥∥ 6
1 + l2
(1− l)2
‖x1 − x2‖.
Hence, D−1 is also Lipschitz mapping. This follows that D is a homeomorphism, and we obtain that
Ct ∩Cu
t is homeomorphism to X3(t) for all t ∈ R.
The property (b) follows from the definitions of the invariant center-stable and center-ustable
manifolds, respectively.
The properties (c) and (d) are consequences of the inequalities (3.4) and (4.11), respectively.
Corollary 4.3 is proved.
5. Examples. In this section, we give some concrete examples of reaction-diffusion equations to
illustrate our abstract results.
The reaction-diffusion processes are modeled by the following equation:
dx(t)
dt
= A(t)x(t) + f(t, x),
where x(t) is the density of material, the partial differential operators A(t) represent the diffusion,
and f represents the source of material which, in many contexts, depends on time in diversified
manners (see [23] (Chapter 11), [24, 37]). Therefore, sometimes one may not hope to have the
uniformly Lipschitz continuity of f. Our theoretical results hence give a chance to consider the
above reaction-diffusion equation in general cases. Let us start by the following equation.
Example 5.1. Consider the reaction-diffusion equation of the form
dx(t)
dt
= Ax(t) + f(t, x),
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INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 793
where A is a sectorial operator satisfying that the spectrum σ(A) of A is decomposed in three disjoint
sets that are {λ ∈ σ(A) | Reλ < 0}, {λ ∈ σ(A) | Reλ > 0}, and {λ ∈ σ(A) | Reλ = 0} such
that σ(A)∩ iR is of finitely many points. Then, A is a generator of an analytic semigroup (T (t))t≥0.
We define the evolution family U(t, s) := T (t − s) for all t ≥ s ≥ 0. We now claim that it has an
exponential trichotomy with an appropriate choice of projections. By the spectral mapping theorem
for analytic semigroups we have that, for fixed t0, the spectrum of the operator T (t0) splits into three
disjoint sets σ1, σ2, σ3, where σ1 ⊂ {|z| < 1}, σ2 ⊂ {|z| > 1}, σ3 ⊂ {|z| = 1} with σ3 consisting
of finitely many points.
Next, we choose P1 = P1(t0), P2 = P2(t0), P3 = P3(t0) be the Riesz projections corresponding
to the spectral sets σ1, σ2, σ3, respectively. Clearly, P1, P2 and P3 commute with T (t) for all t ≥ 0.
Obviously, P1 + P2 + P3 = I and PiPj = 0 for i 6= j, and there are positive constants M, δ
such that ‖T (t)P1‖ 6 Me−δt for all t ≥ 0. Furthermore, let Q := P2 + P3 = I − P1 and consider
the strongly continuous semigoup (TQ(t))t≥0 on the space ImQ, where TQ(t) := T (t)Q. Since
σ2 ∪ σ3 = σ(TQ(t0)), (TQ(t))t≥0 can be extended to a group (TQ(t))t∈R in ImQ. As well-known
in the semigroup theory, there are positive constants K, α, γ such that α can be chosen as small as
required (we may let α < γ), and the following estimates hold:
‖TQ(−t)P2‖ 6 Ke−γt for all t ≥ 0,
‖TQ(t)P3‖ 6 Keα|t| for all t ∈ R.
Summing up the above discussions, we conclude that the evolution family (U(t, s))t≥s has an expo-
nential trichotomy with projections Pj , j = 1, 2, 3, and positive constants N, α, β, where
β := min{δ, γ},
N := max{K,M}.
Thus, if f is ϕ-Lipschitz for some positive function ϕ satisfying that supt∈R
∫ t+1
t
ϕ(τ)dτ is small
enough, then the integral equation
x(t) = U(t, s)x(s) +
t∫
s
U(t, ξ)f(ξ, x(ξ))dξ for all t ≥ s,
has a center manifold.
Example 5.2. For fixed n ∈ N∗, consider the equation
wt(x, t) = wxx(x, t) + n2w(x, t) + ϕ(t) sin(w(x, t)), 0 6 x 6 π, t ∈ R,
w(0, t) = w(π, t) = 0, t ∈ R,
(5.1)
where the step function ϕ(t) is defined as in formula (5.2).
We define X := L2[0, π], and let A : X→ X be defined by A(y) = y′′ + n2y, with
D(A) =
{
y ∈ X : y and y′′ are absolutely continuous, y′′ ∈ X, y(0) = y(π) = 0
}
.
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794 NGUYEN THIEU HUY, VU THI NGOC HA, HA PHI
The equation (5.1) can now be rewritten as
du
dt
= Au+ f(t, u) for u(t) = w(·, t),
where f : R× X→ X, with f(t, u) = ϕ(t) sin(u) for ϕ being defined for a constant c > 1 by
ϕ(t) =
|k| if t ∈
[
2k + 1
2
− 1
2|k|+c
,
2k + 1
2
+
1
2|k|+c
]
for k = 0,±1,±2, . . . ,
0 otherwise.
(5.2)
Here, we note that ϕ can take any arbitrarily large value but we still have that
sup
t∈R
t+1∫
t
|ϕ(τ)|dτ 6 2 sup
k∈Z
2k+1
2 +
1
2|k|+c∫
2k+1
2 − 1
2|k|+c
|k|dt = 2 sup
k∈Z
|k|
2|k|+c−2
6
1
2c−1
.
Therefore, ϕ ∈M(R) which is an admissible space.
It can be seen that (see [9]) that A is the generator of an analytic semigroup (T (t))t≥0.
Since σ(A) = {−1+n2,−4+n2, . . . , 0,−(1+n)2+n2, . . .}, applying the spectral mapping theorem
for analytic semigroups we get
σ(T (t)) = etσ(A) =
= {et(n2−1), et(n
2−4), . . . , et((n−1)
2−n2)} ∪ {1} ∪ {e−t((1+n)2−n2), e−t((2+n)
2−n2), . . .}.
One can see easily that the nonlinear forcing term f is ϕ-Lipschitz. Using Example 5.1 we obtain
that, if supt∈R
∫ t+1
t
ϕ(τ)dτ, which is less than
1
2c−1
, is sufficient small (or c is sufficiently large),
then there exists a center manifold for mild solutions of equation (5.1).
Example 5.3. For fixed n ∈ N∗, consider the equation
wt(x, t) = a(t)[wxx(x, t) + n2w(x, t)] + ϕ(t) sin(w(x, t)), 0 6 x 6 π, t ∈ R,
w(0, t) = w(π, t) = 0, t ∈ R,
(5.3)
where ϕ is defined as in (5.2); the function a(·) ∈ L1,loc(R) and satisfies the condition γ1 ≥ a(t) ≥
≥ γ0 > 0 for fixed γ0, γ1 and a.e. t ∈ R.
We put X := L2[0, π], and let A : X→ X be defined by A(y) = y′′ + n2y, with
D(A) =
{
y ∈ X : y and y′′ are absolutely continuous, y′′ ∈ X, y(0) = y(π) = 0
}
.
Putting A(t) := a(t)A, the equation (5.3) can now be rewritten as
du
dt
= A(t)u+ f(t, u) for u(t) = w(·, t),
where f : R× X→ X, with f(t, u) = ϕ(t) sin(u).
Thus, as the above examples, A is a sectorial operator and generates an analytic semigroup
(T (t))t≥0, and σ(A) satisfies the conditions as in Examples 5.1 and 5.2. Therefore, A(t) “generates”
the evolution family (U(t, s))t≥s which is defined by the formula
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INTEGRAL MANIFOLDS FOR SEMILINEAR EVOLUTION EQUATIONS AND ADMISSIBILITY . . . 795
U(t, s) = T
t∫
s
a(τ)dτ
.
Using the above arguments as in Examples 5.1 and 5.2 we have that the analytic semigroup (T (t))t≥0
has an exponential trichotomy with the projections Pk, k = 1, 2, 3, and the trichotomy constants N,
α, β where α is as small as required. Also, the following estimates hold:
(i) ‖T (t)|P1X‖ 6 Ne−βt,
(ii) ‖T2(−t)‖ = ‖(T (t)|P2X)−1‖ 6 Ne−βt,
(iii) ‖T (t)|P3X‖ 6 Neαt for all t ≥ 0.
From this, it is straightforward to check that the evolution family (U(t, s))t≥s has an exponential
trichotomy with the trichotomy projection Pk, k = 1, 2, 3, and the trichotomy constants N, β, α by
the following estimates:
‖U(t, s)|P1X‖ =
∥∥∥∥∥∥T
t∫
s
a(τ)dτ
∣∣∣∣∣∣
P1X
∥∥∥∥∥∥ 6 Ne−β(t−s),
‖U(s, t)|‖ = ‖(U(t, s)|P2X)−1‖ =
∥∥∥∥∥∥T
− t∫
s
a(τ)dτ
∣∣∣∣∣∣
P2X
∥∥∥∥∥∥ 6 Ne−β(t−s),
‖U(t, s)|P3X‖ =
∥∥∥∥∥∥T
t∫
s
a(τ)dτ
∣∣∣∣∣∣
P3X
∥∥∥∥∥∥ 6 Neα(t−s)
for all t ≥ s ≥ 0. Therefore, we obtain that, if supt∈R
∫ t+1
t
ϕ(τ)dτ =
1
2c−1
is sufficient small, then
there exists a center manifold for mild solutions of equation (5.3).
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Received 02.06.11,
after revision — 20.04.12
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