Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 73, pp. 1–13.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.73

SHADOWING PROPERTIES OF EVOLUTION EQUATIONS WITH

EXPONENTIAL TRICHOTOMY ON BANACH SPACES

KUN TU, HUI-SHENG DING

Abstract. In this article we investigate the shadowing properties of the semilinear non-autonomous

evolution equation
u′(t) = A(t)u(t) + f(t, u(t)), t ≥ 0

on a Banach space X. Here the linear operator A(t) : D(A(t)) ⊂ X → X may not be bounded,

and the homogeneous equation u′(t) = A(t)u(t) admits a general exponential trichotomy. We

obtain two shadowing properties under BSp type and L2 type Lipschitz conditions on f , re-
spectively. Moreover, a concrete example of parabolic partial differential equation is provided

to illustrate the applicability of our abstract results. Compared with known results, the main

feature of this paper lies in relaxing the Lipschitz conditions on f , considering the shadowing
properties under the framework of general exponential trichotomies, and most importantly, al-

lowing A(t) to be unbounded, which enables the abstract results to be directly applied to partial

differential equations.

1. Introduction and preliminaries

A key characteristic of chaotic dynamical systems, first noted by Poincaré [26], is their sensi-
tivity to initial conditions: even a minor alteration in the initial state can result in a significant
divergence in the output. However, many dynamical systems, such as uniformly hyperbolic dynam-
ical systems, display a remarkable and interesting property: although a small error in the initial
condition can ultimately result in a significant effect, there still exists a true orbit with a slightly
altered initial condition that remains close to the approximate trajectory. This phenomenon is
referred to as the shadowing property or shadowing lemma.

The pioneer works on shadowing property for diffeomorphism can be traced back to [1, 10].
Since then, more and more scholars have begun to focus on shadowing lemma for diffeomorphism
(see, e.g., [2, 8, 9, 13, 20]), shadowing lemma for difference equations (see, e.g., [12, 14, 22, 23, 24]),
shadowing lemma for differential equations (see e.g., [6, 12, 14, 15, 18, 29]), and as well as shadowing
lemma for random dynamical systems (see e.g., [19, 17]).

However, all the aforementioned literature establishes shadowing properties under dichotomous
condition. Despite its importance, the notion of exponential dichotomy is somewhat restrictive.
Does a shadowing property exist without exponential dichotomy condition? Although this is a
tricky question, there are still several interesting results. Palmer [21] obtained shadowing lemma
for the autonomous system of ordinary differential equations x′ = f(x) under a special exponential
trichotomy condition with the constant of center space µ = 0. Thereafter Backes and Dragičević
investigated the shadowing lemma for nonautonomous and nonlinear differential equations

u′(t) = A(t)u(t) + f(t, u(t)), t ≥ 0 (1.1)

on Banach spaces. In [5], shadowing properties for (1.1) was established in exponential trichotomy
condition with the constant of center space µ < 0. Moreover, Backes and Dragičević [4] proved a
weaker version of shadowing lemma for (1.1) in a general exponential trichotomy condition.

2020 Mathematics Subject Classification. 34G20, 37C50, 47J35.

Key words and phrases. Abstract evolution equation; exponential trichotomy; shadowing property.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted May 1, 2025. Published July 15, 2025.

1



2 K. TU , H.-S. DING EJDE-2025/73

We note that {A(t)}t≥0 are bounded linear operators both in [4] and [5]. In fact, to the best
of our knowledge, even in the case of exponential dichotomy, the only known result on shadow-
ing properties for (1.1) with unbounded operators is [12], where a shadowing lemma is obtained
for (1.1) with A(t) being independent of t. This is our main motivation to study the shadow-
ing properties for (1.1) with {A(t)}t≥0 being not necessarily bounded under general exponential
trichotomy.

Throughout this paper, let (X, ∥ · ∥) be an arbitrary Banach space, and B(X) be the space of
all bounded linear operators on X. It is well-known (cf. [25]) that a family T (t, s), t ≥ s ≥ 0,
of operators in B(X) is said to be an evolution system or evolution family on X if the following
properties holds:

(i) T (t, t) = I for all t ≥ 0,
(ii) T (t, s)T (s, τ) = T (t, τ) for all t ≥ s ≥ τ ≥ 0,
(iii) the mapping {(τ, σ) ∈ R+ × R+ : τ ≥ σ} ∋ (t, s) → T (t, s) is strongly continuous, i.e., for

each v ∈ X, (t, s) 7→ T (t, s)v is continuous.

Definition 1.1 ([11]). An evolution family T (t, s) is said to have exponential trichotomy if there
are projections P i(t), i ∈ {1, 2, 3} and constants M ≥ 1, λ > 0 and µ ∈ (−∞, λ) such that
P i(·) ∈ BC(R+,B(X)), i ∈ {1, 2, 3}, and

(i) P i(t)P j(t) = 0 for all t ≥ 0 and i, j ∈ {1, 2, 3} with i ̸= j,
(ii) P 1(t) + P 2(t) + P 3(t) = I for all t ≥ 0,
(iii) T (t, s)P i(s) = P i(t)T (t, s) for all t ≥ s and i ∈ {1, 2, 3},
(iv) T (t, s)|ker(P 1(s)) : ker(P

1(s)) → ker(P 1(t)) invertible for t ≥ s ≥ 0 and hereafter T (t, s) de-

notes the inverse of the operator T (s, t)|ker(P 1(t)) for t, s ∈ R+ with t ≤ s, where ker(P 1(t))

denotes the null space of P 1(t),
(v) ∥T (t, s)P 1(s)∥ ≤Me−λ(t−s) for all t ≥ s ≥ 0,
(vi) ∥T (t, s)P 2(s)∥ ≤Me−λ(s−t) for all s ≥ t ≥ 0,
(vii) ∥T (t, s)P 3(s)∥ ≤Meµ|t−s| for all t, s ∈ R+.

Remark 1.2. The notion of exponential trichotomy has several variants, and our definition is
the same to that of [11]. For other variants of exponential trichotomy, we refer the reader to
[7, 16, 27, 28] and references therein.

Let A(t) : D(A(t)) ⊂ X → X, t ≥ 0, be a family of linear operators (not necessarily bounded).
We say that T (t, s) is an evolution family associated with x′(t) = A(t)x(t), t ≥ 0, if for each s ≥ 0
and v ∈ D(A(s)), T (·, s)v is a solution of x′(t) = A(t)x(t), t ≥ s with x(s) = v.

Unless otherwise specified, in the rest of this paper, we always assume that T (t, s) is an evolution
family associated with x′(t) = A(t)x(t), t ≥ 0 and T (t, s) has exponential trichotomy with the
constants µ, λ and M as in Definition 1.1.

In this article, we consider the semilinear evolution equation (1.1), i.e.,

x′(t) = A(t)x(t) + f(t, x(t)), t ∈ R+,

on X, where f : R+ × X → X is Lipschitz in the second variable, i.e., there exists a function
L : R+ → R+ such that

∥f(t, x)− f(t, y)∥ ≤ L(t)∥x− y∥, for all t ≥ 0 and x, y ∈ X. (1.2)

Definition 1.3 ([25]). A function x ∈ C(R+, X) such that

x(t) = T (t, 0)x(0) +

∫ t

0

T (t, r)f(r, x(r))dr, ∀t ∈ R+

is called the mild solution of (1.1).

Remark 1.4. It is easy to see that x is a mild solution of (1.1) if and only if x satisfies

x(t) = T (t, s)x(s) +

∫ t

s

T (t, r)f(r, x(r))dr, ∀t ≥ s ≥ 0.



EJDE-2025/73 SHADOWING PROPERTIES OF EVOLUTION EQUATIONS 3

Let p ∈ [1,+∞) and BSp(R+) be the linear space of all Lebesgue measurable functions f :
R+ → R+ with the property that

sup
t∈R+

∫ t+1

t

|f(r)|pdr < +∞.

It is well known that BSp(R+) is a Banach space with the norm

∥f∥BSp = sup
t∈R+

(∫ t+1

t

|f(r)|pdr
)1/p

.

Let L2(R+, X) be the Banach space of all Bochner measurable functions ξ : R+ → X with the
norm

∥ξ∥L2(R+,X) =
(∫

R+

∥ξ(t)∥2dt
)1/2

< +∞.

Let ν ∈ (µ, λ) and

Cν = {ξ ∈ C(R+, X) : ∥ξ∥ν := sup
t∈R+

eνt∥ξ(t)∥ < +∞}.

It is straightforward to verify that (Cν , ∥ · ∥ν) is a Banach space.

2. Main results

In this section, we introduce two definitions of pseudo orbits and their shadowing properties.

Definition 2.1. Let δ > 0 and ν ∈ (µ, λ). We say that a differential function y : R+ → X is a δ
pseudo orbit of (1.1) if y(t) ∈ D(A(t)) for t ≥ 0 and

eνt∥A(t)y(t) + f(t, y(t))− y′(t)∥ ≤ δ, t ∈ R+. (2.1)

Definition 2.2. Let δ ∈ L2(R+,R). We say that a differential function y : R+ → X is a δ − L2

pseudo orbit of (1.1) if y(t) ∈ D(A(t)) for t ≥ 0 and

∥A(t)y(t) + f(t, y(t))− y′(t)∥ ≤ δ(t), a.e. onR+. (2.2)

Theorem 2.3. Suppose p ∈ [1,+∞) and q is the conjugate index of p. If the function L in (1.2)
satisfying L ∈ BSp(R+) and ∥L∥BSp is small enough, then there exists a positive constant C with
the property that for each δ > 0 and δ pseudo orbit y, we have a unique mild solution x of (1.1)
such that

(i) P 1(0)x(0) = P 1(0)y(0),
(ii) ∥x− y∥ν ≤ Cδ.

Proof. Let δ > 0 and y be a δ pseudo orbit. Obviously

y′(t) = A(t)y(t) + y′(t)−A(t)y(t), t ≥ 0,

i.e., y is a classical solution of the equation

u′(t) = A(t)u(t) + y′(t)−A(t)y(t), t ≥ 0. (2.3)

It is easy to see that y is a mild solution of equation (2.3), i.e.,

y(t) = T (t, 0)y(0) +

∫ t

0

T (t, r)(y′(r)−A(r)y(r))dr, t ∈ R+.

If x is a mild solution of (1.1), then

x(t) = T (t, 0)x(0) +

∫ t

0

T (t, r)f(r, x(r))dr, t ∈ R+,

and

x(t)− y(t) = T (t, 0)(x(0)− y(0)) +

∫ t

0

T (t, r)(A(r)y(r) + f(r, x(r))− y′(r))dr, t ∈ R+.



4 K. TU , H.-S. DING EJDE-2025/73

To find a mild solution of (1.1) satisfying (i) is equivalent to finding a function z ∈ Cν such that
P 1(0)z(0) = 0 and

z(t) = T (t, 0)z(0) +

∫ t

0

T (t, r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, t ∈ R+. (2.4)

By (2.4) and P 1(0)z(0) = 0, we have

P 1(t)z(t) =

∫ t

0

T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, (2.5)

for all t ∈ R+. It follows from (2.4) that

z(t) = T (t, s)z(s) +

∫ t

s

T (t, r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, t ≥ s ≥ 0,

which yields that

P i(t)z(t) = T (t, s)P i(s)z(s) +

∫ t

s

T (t, r)P i(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

for all t ≥ s ≥ 0 and i = 2, 3. Since T (t, s)|ker(P 1(s)) is invertible, we have

T (s, t)P i(t)z(t)

= T (s, t)T (t, s)P i(s)z(s) + T (s, t)

∫ t

s

T (t, r)P i(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

= P i(s)z(s) +

∫ t

s

T (s, t)T (t, r)P i(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

= P i(s)z(s) +

∫ t

s

T (s, r)P i(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

for all t ≥ s ≥ 0 and i = 2, 3. Then we have

P i(s)z(s) = T (s, t)P i(t)z(s)−
∫ t

s

T (s, r)P i(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

for all t ≥ s ≥ 0 and i = 2, 3.
Next, we will show that

P i(s)z(s) = −
∫ +∞

s

T (s, r)P i(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

for all s ∈ R+ and i = 2, 3. By exponential trichotomy of T (t, s) and z ∈ Cν , we have

∥T (s, t)P 2(t)z(t)∥ ≤Me−λ(t−s)∥z(t)∥ ≤Me−λ(t−s)e−νt∥z∥ν ,

then T (s, t)P 2(t)z(t) → 0 as t→ ∞. To prove that

lim
t→∞

∫ t

s

T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

=

∫ +∞

s

T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

we need to show that∫ +∞

s

∥T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))∥dr < +∞.

By (1.2), (2.1), and exponential trichotomy of T (t, s), we have∫ +∞

s

∥T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))∥dr

=

∫ +∞

s

∥T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r) + f(r, y(r))− f(r, y(r)))∥dr



EJDE-2025/73 SHADOWING PROPERTIES OF EVOLUTION EQUATIONS 5

≤
∫ +∞

s

∥T (s, r)P 2(r)∥ (∥A(r)y(r) + f(r, y(r))− y′(r)∥+ ∥f(r, y(r) + z(r))− f(r, y(r))∥) dr

≤
∫ +∞

s

Me−λ(r−s)(e−νrδ + L(r)∥z(r)∥)dr

≤
∫ +∞

s

Me−λ(r−s)(e−νrδ + L(r)e−νr∥z∥ν)dr

=
e−νsMδ

λ+ ν
+Meλs∥z∥ν

+∞∑
j=0

∫ s+j+1

s+j

e−(λ+ν)rL(r)dr

≤ e−νsMδ

λ+ ν
+Meλs∥z∥ν

+∞∑
j=0

(∫ s+j+1

s+j

|L(r)|pdr
)1/p(∫ s+j+1

s+j

|e−(λ+ν)r|qdr
)1/q

≤ e−νsMδ

λ+ ν
+Meλs∥z∥ν

+∞∑
j=0

∥L∥BSp

(1− e−(λ+ν)q

(λ+ ν)q

)1/q

e−(λ+ν)(s+j)

=
Me−νsδ

λ+ ν
+
(1− e−(λ+ν)q

(λ+ ν)q

)1/qMe−νs∥L∥BSp∥z∥ν
1− e−(λ+ν)

< +∞.

Then for all s ∈ R+,

P 2(s)z(s) = −
∫ ∞

s

T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr. (2.6)

Similarly, for all s ≥ 0, we have

∫ +∞

s

∥T (s, r)P 3(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))∥dr

≤
∫ +∞

s

Meµ(r−s)(e−νrδ + L(r)e−νr∥z∥ν)dr

=
e−νsMδ

ν − µ
+Me−µs∥z∥ν

+∞∑
j=0

∫ s+j+1

s+j

e−(ν−µ)rL(r)dr

≤ e−νsMδ

ν − µ
+Me−µs∥z∥ν

+∞∑
j=0

(∫ s+j+1

s+j

|L(r)|pdr
)1/p(∫ s+j+1

s+j

|e−(ν−µ)r|qdr
)1/q

≤ e−νsMδ

ν − µ
+Me−νs∥z∥ν

+∞∑
j=0

∥L∥BSp

(1− e−(ν−µ)q

(ν − µ)q

)1/q

e−(ν−µ)(s+j)

=
e−νsMδ

ν − µ
+
(1− e−(ν−µ)q

(ν − µ)q

)1/qMe−νs∥L∥BSp∥z∥ν
1− e−(ν−µ)

< +∞.

Moreover, by exponential trichotomy of T (t, s) and z ∈ Cν , we have

∥T (s, t)P 3(t)z(t)∥ ≤Meµ(t−s)∥z(t)∥ ≤Meµ(t−s)e−νt∥z∥ν , t ≥ s ≥ 0,

then T (s, t)P 3(t)z(t) → 0 as t→ +∞. Obviously, for all s ∈ R+, we have

P 3(s)z(s) = −
∫ +∞

s

T (s, r)P 3(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr. (2.7)



6 K. TU , H.-S. DING EJDE-2025/73

By (2.5), (2.6) and (2.7), we have

z(t) =

∫ t

0

T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

−
∫ +∞

t

T (t, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

−
∫ +∞

t

T (t, r)P 3(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr.

(2.8)

We have showed that (2.4) and (i) imply (2.8). Then, it is straightforward to verify that (2.8) is
equivalent to (2.4) and (i).

Now, we define

(Γw)(t) =

∫ t

0

T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + w(r))− y′(r))dr

−
∫ +∞

t

T (t, r)P 2(r)(A(r)y(r) + f(r, y(r) + w(r))− y′(r))dr

−
∫ +∞

t

T (t, r)P 3(r)(A(r)y(r) + f(r, y(r) + w(r))− y′(r))dr

:= (Γ1w)(t)− (Γ2w)(t)− (Γ3w)(t),

(2.9)

for w ∈ Cν and t ∈ R+. Firstly we need to prove Γ(Cν) ⊂ Cν . For each w ∈ Cν , by (1.2), (2.1)
and exponential trichotomy of T (t, s), we have

sup
t∈R+

eνt∥(Γ1w)(t)∥

≤ sup
t∈R+

eνt
∫ t

0

∥T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + w(r))− y′(r))∥dr

≤ sup
t∈R+

eνt
∫ t

0

∥T (t, r)P 1(r)∥ ∥(A(r)y(r) + f(r, y(r) + w(r))− y′(r))∥dr

≤ sup
t∈R+

eνt
∫ t

0

∥T (t, r)P 1(r)∥
(
∥(A(r)y(r) + f(r, y(r))− y′(r))∥

+ ∥f(r, y(r) + w(r))− f(r, y(r))∥
)
dr

≤ sup
t∈R+

eνt
∫ t

0

Me−λ(t−r)(e−νrδ + L(r)∥w(r)∥)dr

≤ sup
t∈R+

eνt
∫ t

0

Me−λ(t−r)(e−νrδ + L(r)e−νr∥w∥ν)dr

≤ sup
t∈R+

Mδ(1− e−(λ−ν)t)

λ− ν
+ sup

t∈R+

Me−(λ−ν)t∥w∥ν
+∞∑
j=0

∫ t−j

t−j−1

e(λ−ν)rL(r)dr

≤ Mδ

λ− ν
+ sup

t∈R+

Me−(λ−ν)t∥w∥ν
+∞∑
j=0

∫ t−j

t−j−1

e(λ−ν)rL(r)dr

≤ Mδ

λ− ν
+ sup

t∈R+

Me−(λ−ν)t∥w∥ν
+∞∑
j=0

(∫ t−j

t−j−1

|L(r)|pdr
)1/p(∫ t−j

t−j−1

|e(λ−ν)r|qdr
)1/q

≤ Mδ

λ− ν
+ sup

t∈R+

Me−(λ−ν)t∥w∥ν
+∞∑
j=0

∥L∥BSp

(1− e−(λ−ν)q

(λ− ν)q

)1/q

e(λ−ν)(t−j)

≤ Mδ

λ− ν
+
(1− e−(λ−ν)q

(λ− ν)q

)1/qM∥L∥BSp∥w∥ν
1− e−(λ−ν)

< +∞.



EJDE-2025/73 SHADOWING PROPERTIES OF EVOLUTION EQUATIONS 7

Similarly,

sup
t∈R+

eνt ∥(Γ2w)(t)∥ ≤ Mδ

λ+ ν
+

(1− e−(λ+ν)q

(λ+ ν)q

)1/qM∥L∥BSp∥w∥ν
1− e−(λ+ν)

< +∞,

sup
t∈R+

eνt ∥(Γ3w)(t)∥ ≤ Mδ

ν − µ
+
(1− e−(ν−µ)q

(ν − µ)q

)1/qM∥L∥BSp∥w∥ν
1− e−(ν−µ)

< +∞.

Hence

sup
t∈R+

eνt∥(Γw)(t)∥ ≤ sup
t∈R+

eνt∥(Γ1w)(t)∥+ sup
t∈R+

eνt∥(Γ2w)(t)∥+ sup
t∈R+

eνt∥(Γ3w)(t)∥ < +∞.

It is straightforward to verify that Γw is a continuous function, and thus we know that Γ(Cν) ⊂ Cν .
Next we show that Γ has a fixed point on Cν . For each w1, w2 ∈ Cν , by (1.2) and exponential

trichotomy of T (t, s), we have

∥Γ1w1 − Γ1w2∥ν ≤ sup
t∈R+

eνt
∫ t

0

∥T (t, r)P 1(r)(f(r, y(r) + w1(r))− f(r, y(r) + w2(r)))∥dr

≤ sup
t∈R+

eνt
∫ t

0

Me−λ(t−r)L(r)∥w1(r)− w2(r)∥dr

≤ sup
t∈R+

eνt
∫ t

0

Me−λ(t−r)e−νrL(r)∥w1 − w2∥νdr

≤
(1− e−(λ−ν)q

(λ− ν)q

)1/q M∥L∥BSp

1− e−(λ−ν)
∥w1 − w2∥ν ,

(2.10)

and

∥Γ2w1 − Γ2w2∥ν

≤ sup
t∈R+

eνt
∫ +∞

t

∥T (t, r)P 2(r)(f(r, y(r) + w1(r))− f(r, y(r) + w2(r)))∥dr

≤ sup
t∈R+

eνt
∫ +∞

t

Me−λ(r−t)L(r)∥w1(r)− w2(r)∥dr

≤ sup
t∈R+

eνt
∫ +∞

t

Me−λ(r−t)e−νrL(r)∥w1 − w2∥νdr

≤
(1− e−(λ+ν)q

(λ+ ν)q

)1/q M∥L∥BSp

1− e−(λ+ν)
∥w1 − w2∥ν ,

(2.11)

and

∥Γ3w1 − Γ3w2∥ν

≤ sup
t∈R+

eνt
∫ +∞

t

∥T (t, r)P 3(r)(f(r, y(r) + w1(r))− f(r, y(r) + w2(r)))∥dr

≤ sup
t∈R+

eνt
∫ +∞

t

Me−µ(r−t)L(r)∥w1(r)− w2(r)∥dr

≤ sup
t∈R+

eνt
∫ +∞

t

Me−µ(r−t)e−νrL(r)∥w1 − w2∥νdr

≤
(1− e−(ν−µ)q

(ν − µ)q

)1/q M∥L∥BSp

1− e−(ν−µ)
∥w1 − w2∥ν .

(2.12)



8 K. TU , H.-S. DING EJDE-2025/73

Therefore, by (2.10), (2.11) and (2.12) we have

∥Γw1 − Γw2∥ν ≤ ∥Γ1w1 − Γ1w2∥ν + ∥Γ2w1 − Γ2w2∥ν + ∥Γ3w1 − Γ3w2∥ν

≤
(1− e−(λ−ν)q

(λ− ν)q

)1/q M∥L∥BSp

1− e−(λ−ν)
∥w1 − w2∥ν

+
(1− e−(λ+ν)q

(λ+ ν)q

)1/q M∥L∥BSp

1− e−(λ+ν)
∥w1 − w2∥ν

+
(1− e−(ν−µ)q

(ν − µ)q

)1/q M∥L∥BSp

1− e−(ν−µ)
∥w1 − w2∥ν

=: k∥w1 − w2∥ν .

(2.13)

It is easy to see k < 1 provided that ∥L∥BSp is sufficient small. Setting w = 0 in (2.9) implies that

∥Γ0∥ν ≤ Mδ

λ− ν
+

Mδ

λ+ ν
+

Mδ

ν − µ
. (2.14)

Let

C =
M

(1− k)(λ− ν)
+

M

(1− k)(λ+ ν)
+

M

(1− k)(ν − µ)
.

For each w ∈ Cν satisfying ∥w∥ν ≤ Cδ, by (2.14) and (2.13) we have

∥Γw∥ν ≤ ∥Γw − Γ0∥ν + ∥Γ0∥ν ≤ kCδ + (1− k)Cδ = Cδ.

Hence, Γ has a unique fixed point z ∈ Cν satisfying ∥z∥ν ≤ Cδ. It is straightforward to verify that
z is the unique solution of (2.4). Then x = y + z is a unique mild solution of (1.1) which satisfies
(i) and (ii). □

Before presenting the other shadowing property of equation (1.1), let us recall a variant of the
Young’s convolution inequality from [3, Proposition 1.3.2].

Lemma 2.4. Let p ∈ [1,+∞), ϕ ∈ L1(R,R) and ψ ∈ Lp(R+,R). Define

ϕ ∗ ψ(x) =
∫
R+

ϕ(x− y)ψ(y)dy, x ∈ R+.

Then ϕ ∗ ψ ∈ Lp(R+,R) and ∥ϕ ∗ ψ∥Lp(R+,R) ≤ ∥ϕ∥L1(R,R) · ∥ψ∥Lp(R+,R).

Theorem 2.5. Assume that the constant µ in Definition 1.1 is negative and the function L in
(1.2) satisfies L ∈ L2(R+,R). If ∥L∥L2(R+,R) is small enough, then there exists a positive constant

Ĉ with the property that for each δ ∈ L2(R+,R) and δ−L2 pseudo orbit y, we have a unique mild
solution x of (1.1) such that

(i) P i(0)x(0) = P i(0)y(0), i = 1, 3,

(ii) ∥x− y∥L2(R+,X) ≤ Ĉ∥δ∥L2(R+,R).

Proof. Let δ ∈ L2(R+,R) and y be a δ−L2 pseudo orbit. To find a mild solution of (1.1) satisfying
(i) is equivalent to finding a function z ∈ L2(R+, X) such that P i(0)z(0) = 0, i = 1, 3, and

z(t) = T (t, 0)z(0) +

∫ t

0

T (t, r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, a.e. on R+. (2.15)

By (2.15) and P i(0)z(0) = 0, i = 1, 3, we have

P 1(t)z(t) =

∫ t

0

T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, a.e. on R+, (2.16)

and

P 3(t)z(t) =

∫ t

0

T (t, r)P 3(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, a.e. on R+. (2.17)

It follows from (2.15) that

z(t) = T (t, s)z(s) +

∫ t

s

T (t, r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,



EJDE-2025/73 SHADOWING PROPERTIES OF EVOLUTION EQUATIONS 9

for a.e. s ∈ R+ and a.e. t ∈ [s,+∞), which yields that

P 2(t)z(t) = T (t, s)P 2(s)z(s) +

∫ t

s

T (t, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

for a.e. s ∈ R+ and a.e. t ∈ [s,+∞). Since T (t, s)|ker(P 1(s)) is invertible, we have

T (s, t)P 2(t)z(t)

= T (s, t)T (t, s)P 2(s)z(s) + T (s, t)

∫ t

s

T (t, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

= P 2(s)z(s) +

∫ t

s

T (s, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

i.e.

P 2(s)z(s) = T (s, t)P 2(t)z(t)−
∫ t

s

T (s, r)P 2(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

for a.e. s ∈ R+ and a.e. t ∈ [s,+∞). Since z ∈ L2(R+, X), for a.e. s ≥ 0, we can choose a sequence
{tn}n∈N such that tn > max{n, s} and ∥z(tn)∥ ≤ 1. Then

P 2(s)z(s) = T (s, tn)P
2(tn)z(tn)−

∫ tn

s

T (s, r)P 2(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr.

As n→ +∞, by exponential trichotomy of T (t, s), we have

P 2(s)z(s) =

∫ +∞

s

T (s, r)P 2(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr, (2.18)

for a.e. s ≥ 0. By (2.16), (2.17) and (2.18), we have

z(t) =

∫ t

0

T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

−
∫ +∞

t

T (t, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

+

∫ t

0

T (t, r)P 3(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr,

(2.19)

for a.e. t ∈ R+. We have showed that (2.15) and (i) imply that (2.19) holds. It is straightforward
to verify that (2.19) is equivalent to (2.15) and (i). Now, we define

(Γ̂z)(t) =

∫ t

0

T (t, r)P 1(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

−
∫ +∞

t

T (t, r)P 2(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

+

∫ t

0

T (t, r)P 3(r)(A(r)y(r) + f(r, y(r) + z(r))− y′(r))dr

:= (Γ̂1z)(t)− (Γ̂2z)(t) + (Γ̂3z)(t),

for z ∈ L2(R+, X) and t ≥ 0. Similar to the proof of Theorem 2.3, we show that Γ̂ : L2(R+, X) →
L2(R+, X) and Γ̂ has a fixed point on L2(R+, X). For each z1, z2 ∈ L2(R+, X), by (1.2) and
exponential trichotomy of T (t, s), we have(∫ +∞

0

∥(Γ̂z1)(r)− (Γ̂z2)(r)∥2dr
)1/2

≤
(∫ +∞

0

∥(Γ̂1z1)(r)− (Γ̂1z2)(r)− (Γ̂2z1)(r) + (Γ̂2z2)(r)∥2dr
)1/2

+
(∫ +∞

0

∥(Γ̂3z1)(r)− (Γ̂3z2)(r)∥2dr
)1/2



10 K. TU , H.-S. DING EJDE-2025/73

≤
(∫ +∞

0

big∥
∫ t

0

T (t, r)P 1(r)(f(r, y(r) + z1(r))− f(r, y(r) + z2(r)))dr

−
∫ +∞

t

T (t, r)P 2(r)(f(r, y(r) + z1(r))− f(r, y(r) + z2(r)))dr
∥∥∥2dt)1/2

+
(∫ +∞

0

∥∥∥∥∫ t

0

T (t, r)P 3(r)(f(r, y(r) + z1(r))− f(r, y(r) + z2(r)))dr

∥∥∥∥2 dt)1/2

≤
(∫ +∞

0

∣∣ ∫ t

0

Me−λ(t−r)L(r)∥z1(r)− z2(r)∥dr +
∫ +∞

t

Me−λ(r−t)L(r)∥z1(r)− z2(r)∥dr
∣∣2dt)1/2

+
(∫ +∞

0

∣∣∣∣∫ t

0

Meµ|t−r|L(r)∥z1(r)− z2(r)∥dr
∣∣∣∣2 dt)1/2

=
(∫ +∞

0

big|
∫ +∞

0

Me−λ|t−r|L(r)∥z1(r)− z2(r)∥dr
∣∣2dt)1/2

+
(∫ +∞

0

∣∣∣∣∫ t

0

Meµ|t−r|L(r)∥z1(r)− z2(r)∥dr
∣∣∣∣2 dt)1/2

.

By Lemma 2.4 we have

(∫ +∞

0

∣∣∣∣∫ +∞

0

Me−λ|t−r|L(r)∥z1(r)− z2(r)∥dr
∣∣∣∣2 dt)1/2

≤M
(∫ +∞

−∞
e−2λ|r|dr

)1/2
∫ +∞

0

L(r)∥z1(r)− z2(r)∥dr

≤Mλ−1/2∥L∥L2(R+,R)∥z1 − z2∥L2(R+,X),

and

(∫ +∞

0

∣∣∣∣∫ t

0

Me−µ|t−r|L(r)∥z1(r)− z2(r)∥dr
∣∣∣∣2 dt)1/2

≤
(∫ +∞

0

∣∣∣∣∫ +∞

0

Me−µ|t−r|L(r)∥z1(r)− z2(r)∥dr
∣∣∣∣2 dt)1/2

≤M
(∫ +∞

−∞
e2µ|r|dr

)1/2
∫ +∞

0

L(r)∥z1(r)− z2(r)∥dr

≤M(−µ)−1/2∥L∥L2(R+,R)∥z1 − z2∥L2(R+,X).

Then, we have

(∫ +∞

0

∥(Γ̂z1)(r)− (Γ̂z2)(r)∥2dr
)1/2

≤Mλ−1/2∥L∥L2(R+,R)∥z1 − z2∥L2(R+,X) +M(−µ)−1/2∥L∥L2(R+,R)∥z1 − z2∥L2(R+,X) < +∞,

i.e., Γ̂z1 − Γ̂z2 ∈ L2(R+, X) and

∥Γ̂z1 − Γ̂z2∥L2(R+,X) ≤ (λ−1/2 + (−µ)−1/2)M∥L∥L2(R+,R)∥z1 − z2∥L2(R+,X)

=: k̂∥z1 − z2∥L2(R+,X).
(2.20)



EJDE-2025/73 SHADOWING PROPERTIES OF EVOLUTION EQUATIONS 11

It is easy to see that k̂ < 1 provided that ∥L∥L2(R+,R) is sufficient small. Moreover, by (2.2),
Lemma 2.4 and exponential trichotomy of T (t, s), we have(∫ +∞

0

∥(Γ̂0)(r)∥2dr
)1/2

≤
(∫ +∞

0

∥(Γ̂10)(r)− (Γ̂20)(r)∥2dr
)1/2

+
(∫ +∞

0

∥(Γ̂30)(r)∥2dr
)1/2

≤
(∫ +∞

0

∣∣∣∣∫ +∞

0

Me−λ|t−r|δ(r)dr

∣∣∣∣2 dt)1/2

+
(∫ +∞

0

∣∣ ∫ t

0

Meµ|t−r|δ(r)dr
∣∣2dt)1/2

≤
(∫ +∞

0

∣∣∣∣∫ +∞

0

Me−λ|t−r|δ(r)dr

∣∣∣∣2 dt)1/2

+
(∫ +∞

0

∣∣∣∣∫ +∞

0

Meµ|t−r|δ(r)dr

∣∣∣∣2 dt)1/2

≤
(∫ +∞

0

e−λ|t|dr
)(∫ +∞

0

|δ(r)|2dr
)1/2

+
(∫ +∞

0

eµ|t|dr
)(∫ +∞

0

|δ(r)|2dr
)1/2

≤Mλ−1∥δ∥L2(R+,R) −Mµ−1∥δ∥L2(R+,R).

(2.21)

By (2.20) and (2.21), for each z ∈ L2(R+, X), we have(∫
R+

∥(Γ̂z)(r)∥2dr
)1/2

≤ ∥Γ̂0∥L2(R+,X) + k̂∥z∥L2(R+,X) < +∞,

i.e., Γ̂z ∈ L2(R+, X). Let

Ĉ =
M(λ−1 − µ−1)

λ(1− k̂)
.

For each w ∈ L2(R+, X) satisfying ∥w∥L2(R+,X) ≤ C∥δ∥L2(R+,R), by (2.20) and (2.21), we have

∥Γ̂w∥L2(R+,X) ≤ ∥Γ̂w − Γ̂0∥L2(R+,X) + ∥Γ̂0∥L2(R+,X)

≤ k̂Ĉ∥δ∥L2(R+,R) + (1− k̂)Ĉ∥δ∥L2(R+,R)

= Ĉ∥δ∥L2(R+,R).

Hence, Γ̂ has a unique fixed point z ∈ L2(R+, X) satisfying ∥Γ̂z∥L2(R+,X) ≤ Ĉ∥δ∥L2(R+,R). Then
x = y + z is a unique mild solution of (1.1) which satisfies (i) and (ii). □

3. Example

As an application of the abstract results in this article, we consider the partial differential
equation

∂tw(t, x) = a(t)∂2xw(t, x) + a(t)ηw(t, x) + h(t) sinw(t, x), (t, x) ∈ R+ × (0, 1),

w(t, 0) = w(t, 1) = 0,
(3.1)

where a ∈ L1
loc(R,R+), η ∈ R+ and h ∈ BSp(R+) with p ∈ [1,+∞).

Let X = L2(0, 1) and A : D(A) → X;φ 7→ ∂2xφ, where D(A) = H1
0 (0, 1) ∩ H2(0, 1). By [28,

Section 3.8], A has eigenvalues

βn = −n2π2, for n ∈ N,
and the corresponding eigenvectors en(x) =

√
2sin(nπx), n ∈ N, which form an orthonormal basis

for the space X. Moreover, A generates an analytic semigroup eAt with the form

(eAtϕ)(x) =

+∞∑
n=1

e−n2π2t < ϕ, en > en(x), for all ϕ ∈ X and t ∈ R+.

Now, equation (3.1) can be written in the abstract form

u′(t) = A(t)u(t) + f(t, u(t)), for t ∈ R+. (3.2)



12 K. TU , H.-S. DING EJDE-2025/73

on X, where u(t) = w(t, ·) is regarded as an abstract function of t with values in X, and linear
operator A(t) = a(t)(A+η) for t ≥ 0, as well as nonlinear term f : R×X → X; (t, ϕ) 7→ h(t) sinϕ(·).
Furthermore, the associated evolution family T (t, s) of

u′(t) = A(t)u(t), t ≥ 0

is the form of

T (t, s)ϕ =

+∞∑
n=1

e(βn+η)
∫ t
s
a(r)dr < ϕ, en > en, ϕ ∈ X, t ≥ s ≥ 0.

If βn0
+η = 0 where n0 ∈ N with n0 > 1, then evolution family T (t, s) has exponential trichotomy

with constants M = 1, λ = (2n0 − 1)π2, µ = 0, and projections

P 1(t)ϕ =

+∞∑
n=n0+1

< ϕ, en > en, P 2(t)ϕ =

n0−1∑
n=1

< ϕ, en > en, P 3(t)ϕ =< ϕ, en0 > en0 ,

for all t ≥ 0. In addition, the nonlinear form f is Lipschitz in the second variable since

∥f(t, ϕ)− f(t, ψ)∥ =
(∫ 1

0

|h(t) sinϕ(x)− h(t) sinψ(x)|2dx
)1/2

≤ h(t)∥ϕ− ψ∥.

for ϕ, ψ ∈ X and t ∈ R+.
Late ν ∈ (µ, λ). If ∥h∥BSp is small enough, we can apply Theorem 2.3 to equation (3.2).

Acknowledgments. This work was supported by the NSFC (12361023) and by the Key Project
of Jiangxi Provincial NSF (20242BAB26001).

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Acta Math. Sci. Ser. A (Chinese Ed.), 2025, to appear.

Kun Tu

College of Mathematics and Statistics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
Email address: tukun@jxnu.edu.cn

Hui-Sheng Ding (corresponding author)

College of Mathematics and Statistics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
Email address: dinghs@mail.ustc.edu.cn


	1. Introduction and preliminaries
	2. Main results
	3. Example
	Acknowledgments

	References