Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 97, pp. 1–31.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS OF

A SMOOTH DOMAIN

PRICILA S. BARBOSA, ANTÔNIO L. PEREIRA

Abstract. We consider a family of semilinear parabolic problems with non-

linear boundary conditions

ut(x, t) = ∆u(x, t)− au(x, t) + f(u(x, t)), x ∈ Ωε, t > 0 ,

∂u

∂N
(x, t) = g(u(x, t)), x ∈ ∂Ωε, t > 0 ,

where Ω0 ⊂ Rn is a smooth (at least C2) domain, Ωε = hε(Ω0) and hε is a

family of diffeomorphisms converging to the identity in the C1-norm. Assuming

suitable regularity and dissipative conditions for the nonlinearites, we show
that the problem is well posed for ε > 0 sufficiently small in a suitable scale of

fractional spaces, the associated semigroup has a global attractor Aε and the

family {Aε} is continuous at ε = 0.

1. Introduction

Let Ω = Ω0 ⊂ Rn be a C2 domain, a a positive number, f, g : R → R real
functions, and consider the family of semilinear parabolic problems with nonlinear
Neumann boundary conditions,

ut(x, t) = ∆u(x, t)− au(x, t) + f(u(x, t)), x ∈ Ωε, t > 0 ,

∂u

∂N
(x, t) = g(u(x, t)), x ∈ ∂Ωε, t > 0 ,

(1.1)

where Ωε = Ωhε = hε(Ω0) and hε : Ω0 → Rn is a family of Cm(m ≥ 2) maps
satisfying suitable conditions to be specified later.

One of the central questions concerning this problem is the existence and prop-
erties of global attractors since, as it is well known, they determine the dynamics
of the entire system (see, for example [8] or [19]). The continuity with respect to
parameters present in the equation is also of interest, since it can be seen as a de-
sirable property of “robustness” in the model. In many cases, however, the form of
the equation is fixed, so the ‘parameter‘of interest is the domain where the problem
is posed.

The existence of a global compact attractor for the problem (1.1) has been proved
in [6, 13], under stronger smoothness hypotheses on the domains and growth and
dissipative conditions on the nonlinearities f and g.

2010 Mathematics Subject Classification. 35B41, 35K20, 58D25.
Key words and phrases. Parabolic problem; perturbation of the domain; global attractor;

continuity of attractors.
c©2020 Texas State University.

Submitted December 31, 2019. Published September 21, 2020.

1



2 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

The problem of existence and continuity of global attractors for semilinear par-
abolic problems, with respect to change of domains has also been considered in [3],
for the problem with homogeneous boundary conditions

ut = ∆u+ f(x, u) in Ωε

∂u

∂N
= 0 on ∂Ωε ,

where Ωε, 0 ≤ ε ≤ ε0 are bounded domains with Lipschitz boundary in RN , N ≥ 2.
The authors proved that, if the perturbations are such that the convergence of the
eigenvalues and eigenfunctions of the linear part of the problem can be shown, than
the upper semicontinuity of attractors follow. With the additional assumption that
the equilibria are all hyperbolic, the lower semicontinuity is also obtained.

The behavior of the equilibria of (1.1) was studied in [1, 2]. In these papers,
the authors consider a family of smooth domains Ωε ⊂ RN , N ≥ 2 and 0 ≤ ε ≤ ε0
whose boundary oscillates rapidly when the parameter ε → 0 and prove that the
equilibria, as well as the spectra of the linearized problem around them, converge
to the solution of a “limit problem”.

In [16] the authors prove the continuity of the attractors of (1.1) with respect to
C2-perturbations of a smooth domain of Rn. These results do not extend imme-
diately to the case considered here, due to the lack of smoothness of the domains
considered and the fact that the perturbations do not converge to the inclusion in
the C2-norm.

In this work, we follow the general approach of [16], which basically consists in
“pull-backing” the perturbed problems to the fixed domain Ω and then considering
the family of abstract semilinear problems thus generated. We present a brief
overview of this approach in the next section for convenience. Our aim here is
then to prove well-posedness, establish the existence of a global attractor Aε, for
sufficiently small ε ≥ 0 and prove that the family of attractors is continuous at
ε = 0.

These results were obtained in our previous paper [5] for the family of perturba-
tions of the unit square in R2 given by

hε(x1, x2) = (x1, x2 + x2ε sin(x1/ε
α)) (1.2)

with 0 < α < 1 and ε > 0 sufficiently small, (see Figure 1).

hε

x

y

1

1

Ω

x

y

1

1

Ωhε

Figure 1. Perturbed region

In this article, we generalize these results in two directions: we consider the
problem in arbitrary spatial dimensions and, also, instead of a specific family of
perturbations, we consider a general family hε : Ω0 → Rn of Cm, (m ≥ 2) maps
satisfying the following abstract hypotheses:



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 3

(H1) ‖hε− iΩ0
‖C1(Ω) → 0 as ε→ 0, where iΩ0

is the inclusion map of Ω0 into Rn.
(H2) The Jacobian determinant Jhε of hε is differentiable, and

‖∇Jhε‖∞ = sup{‖∇Jhε(x)‖, x ∈ Ω} → 0 as ε→ 0.

In section 4 we show that the family hε considered in [5] satisfies the conditions
(H1) and (H2). Since the domain Ω is not of class C1, the results obtained here do
not immediately apply. However, since the perturbations occur only in a smooth
portion of the boundary, they could easily be adapted to this case. We also give
more general examples of families satisfying our properties.

This article is organized as follows: in section 2 we show how the problem can be
reduced to a family of problems in the initial domain and collect some results needed
later. In section 3 we give some rather general examples of families satisfying our
basic assumptions. In section 4 we show that the perturbed linear operators are
sectorial operators in suitable spaces and study properties of the linear semigroup
generated by them. In section 5 we show that the problem (1.1) can be reformulated
as an abstract problem in a scale of Banach spaces which are shown to be locally
well-posed in section 6, under suitable growth assumptions on f and g. In section
7, assuming a dissipative condition for the problem, we use comparison results to
prove that the solutions are globally defined and the family of associated semigroups
are uniformly bounded. In section 8 we prove the existence of global attractors. In
section 9, we show that these attractors behave upper semicontinuously. Finally,
in section 10, with some additional properties on the nonlinearities and on the set
of equilibria, we show that they are also lower semicontinuous at ε = 0.

2. Reduction to a fixed domain

One of the difficulties encountered in problems of perturbation of the domain is
that the function spaces change with the change of the region. One way to overcome
this difficulty is to effect a “change of variables” in order to bring the problem back
to a fixed region. This approach was developed by D. Henry in [9] and is the one
we adopt here. We describe it briefly here, for convenience of the reader. For a
different approach, see [1, 2, 3].

Given an open bounded Cm region Ω ⊂ Rn, m ≥ 1, denote by Diffm(Ω),m ≥ 0,
the set of Cm embeddings (i.e. diffeomorphisms from Ω to its image).

We define a topology in Diffm(Ω), by declaring that Ω is in a ε neighborhood of
Ω0, if Ω = h(Ω0), with ‖h − iΩ0

‖Cm(Ω0) < ε. It has been shown in [12] that this
topology is metrizable and we denote by Mm(Ω) or simply Mm this (separable)
metric space. We say that a function F defined in the space Mm with values in
a Banach space is Cm or analytic if h 7→ F (h(Ω)) is Cm or analytic as a map
of Banach spaces (h near iΩ in Cm(Ω,Rn)). In this sense, we may express prob-
lems of perturbation of the boundary of a boundary value problem as problems of
differential calculus in Banach spaces.

If h : Ω 7→ Rn is a Ck, k ≤ m embedding, we may consider the ‘pull-back’ of h

h∗ : Ck(h(Ω))→ Ck(Ω) (0 ≤ k ≤ m)

defined by h∗(ϕ) = ϕ ◦ h, which is an isomorphism with inverse h−1∗. Other
function spaces can be used instead of Ck, and we will actually be interested mainly
in Sobolev spaces and fractional power spaces.



4 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

Now, if Fh(Ω) : Cm(h(Ω)) → C0(h(Ω)) is a (generally nonlinear) differential

operator in Ωh = h(Ω) we may consider the operator h∗Fh(Ω)h
∗−1, which is a

differential operator in the fixed region Ω.
Let now hε : Ω0 → Rn be a family of maps satisfying the conditions (H1) and

(H2) and Ωε = hε(Ω) the corresponding family of “perturbed domains”.

Lemma 2.1. If ε > 0 is sufficiently small, the map hε belongs to Diffm(Ω) =
diffeomorphisms from Ω to its image.

The proof of the above lemma is straightforward; we omit it.

Lemma 2.2. If 0 < s ≤ m and ε > 0 is small enough, the map h∗ε : Hs(Ωε) →
Hs(Ω) given by u 7→ u ◦ hε is an isomorphism, with inverse h∗ε

−1 = (h−1
ε )∗.

For a proof of the above lemma see [5]. Using Lemma 2.1 we may bring the
problem (1.1) back to the fixed region Ω0. For this purpose, observe that v(·, t) is a
solution of (1.1) in the perturbed region Ωε = hε(Ω), if and only if u(·, t) = h∗εv(·, t)
satisfies

ut(x, t) = h∗ε∆Ωεh
∗−1

ε u(x, t)− au(x, t) + f(u(x, t)), x ∈ Ω, t > 0,

h∗ε
∂

∂NΩε

h∗
−1

ε u(x, t) = g(u(x, t)), x ∈ ∂Ω, > 0 ,
(2.1)

where h∗ε∆Ωεh
∗
ε
−1 and h∗ε

∂
∂NΩε

h∗ε
−1 are defined by

h∗ε∆Ωεh
∗
ε
−1u(x) = ∆Ωε(u ◦ h−1

ε )(hε(x)),

h∗ε
∂

∂NΩε

h∗ε
−1u(x) =

∂

∂NΩε

(u ◦ h−1
ε )(hε(x))

(in appropriate spaces). In particular, if Aε is the global attractor of (1.1) in

Hs(Ωε), then Ãε = {v ◦ hε : v ∈ Aε} is the global attractor of (2.1) in Hs(Ω) and
conversely. In this way we can consider the problem of continuity of the attractors
as ε→ 0 in a fixed phase space.

For later use, we compute an expression for the differential operator h∗ε∆Ωεh
∗−1
ε

in the fixed region Ω, in terms of hε. Writing

hε(x) = hε(x1, x2, . . . , xn) = ((hε)1(x), (hε)2(x), . . . , (hε)n(x)) = (y1, y2, . . . , yn) = y,

for i = 1, 2, . . . , n, we obtain(
h∗ε

∂

∂yi
h∗−1
ε (u)

)
(x) =

∂

∂yi
(u ◦ h−1

ε )(hε(x))

=

n∑
j=1

[(∂hε
∂xj

)−1]
j,i

(x)
∂u

∂xj
(x)

=

n∑
j=1

bεij(x)
∂u

∂xj
(x) ,

(2.2)



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 5

where bεij(x) is the i, j-entry of the inverse transpose of the Jacobian matrix of hε.
From now on, we omit the ε from the notation for simplicity. Therefore,

h∗ε∆Ωεh
∗−1
ε (u)(x)

=

n∑
i=1

(
h∗ε

∂2

∂y2
i

h∗−1
ε (u)

)
(x)

=

n∑
i=1

( n∑
k=1

bi k
∂

∂xk

( n∑
j=1

bij
∂u

∂xj

))
(x)

=

n∑
k=1

∂

∂xk

( n∑
j=1

n∑
i=1

bijbik
∂u

∂xj

)
(x)−

n∑
j=1

( n∑
i,k=1

∂

∂xk
(bik)bij

) ∂u
∂xj

(x)

=
n∑
k=1

∂

∂xk

( n∑
j=1

Ckj
∂u

∂xj

)
(x)−

n∑
j=1

Aj
∂u

∂xj
(x),

(2.3)

where Ckj =
∑n
i=1 bijbik and Aj =

∑n
i,k=1

∂
∂xk

(bik)bij .

We also need to compute the boundary condition h∗ε
∂

∂NΩε
h∗−1
ε u = 0 in the fixed

region Ω in terms of hε. Let Nhε(Ω) denote the outward unit normal to the boundary
of hε(Ω) := Ωε, and bεij(x) the i, j-entry of the inverse transpose of the Jacobian
matrix of hε. From (2.2), we obtain(

h∗ε
∂

∂NΩε

h∗−1
ε u

)
(x) =

n∑
i=1

(
h∗ε

∂

∂yi
h∗−1
ε u

)
(x)
(
NΩε

)
i
(hε(x))

=

n∑
i=1

∂

∂yi
(u ◦ h−1

ε )(hε(x)) (NΩε)i (hε(x))

=

n∑
i,j=1

bij(x)
∂u

∂xj
(x) (NΩε)i (hε(x)).

(2.4)

Since

NΩε(hε(x)) = h∗εNΩε(x) =
[h−1
ε ]TxNΩ(x)

‖[h−1
ε ]TxNΩ(x)‖

(see [9]), we obtain

(NΩε(hε(x)))i =
1

‖[h−1
ε ]TxNΩ(x)‖

n∑
k=1

bik(NΩ)k(x).

Thus, from (2.4),(
h∗ε

∂

∂NΩε

h∗−1
ε u

)
(x)

=
1

‖[h−1
ε ]TxNΩ(x)‖

n∑
k=1

( n∑
i,j=1

bikbij(x)
∂u

∂xj
(x)
)

(NΩ)k(x)

=
1

‖[h−1
ε ]TxNΩ(x)‖

n∑
k=1

( n∑
j=1

Ckj
∂u

∂xj
(x)
)

(NΩ)k(x)

(2.5)



6 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

Thus, the boundary condition
(
h∗ε

∂
∂NΩε

h∗−1
ε u

)
(x) = 0 becomes

n∑
j,k=1

(NΩ(x))k (CkjDju) = 0 on ∂Ω .

Therefore, the boundary condition is exactly the “oblique normal derivative” with
respect to the divergence part of the operator h∗ε∆Ωεh

∗−1
ε .

3. Basic assumptions and examples on domain perturbations

We assume that the unperturbed domain Ω0 is of class C2, and consider rather
general examples of families hε : Ω0 → Rn of C2 maps satisfying the hypotheses
(H1) and (H2) stated in the introduction.

Example 3.1. The family hε of perturbations of the unit square in R2 considered
in [5], given by

hε(x1, x2) = (x1, x2 + x2ε sin(x1/ε
α)) (3.1)

with 0 < α < 1 and ε > 0 sufficiently small, (see figure (1)) satisfies the conditions
(H1) and (H2). We observe that the unperturbed region is not of class C2 and,
therefore, it does not strictly satisfies our hypothesis. However, since the pertur-
bation occurs only at a smooth portion of the boundary and the elliptic problem
in this case is well posed, (see [7]), the problem can actually be included in the
framework considered here, with only minor modifications.

In fact, hypothesis (H1) was shown in [5, Lemma 2.1]. A simple computation
gives ∇Jhε = (ε(1−α) cos(x1/ε

α), 0), from which (H2) follows easily.
From (H1), it follows that the boundary Jacobian µε = J∂Ωhε

∣∣
∂Ω
→ 1 uniformly

as ε→ 0. It can be checked by explicitly computation, as done in [5]:

µε =



√
1+ε 2−2αcos 2(x1/εα)

1+ε sin(x1/εα) for x ∈ I1 := {(x1, 1) : 0 ≤ x1 ≤ 1},
1

1+ε sin(x1/εα) for x ∈ I3 := {(x1, 0) : 0 ≤ x1 ≤ 1},
1 for x ∈ I2 := {(1, x2) : 0 ≤ x2 ≤ 1}

or x ∈ I4 := {(0, x2) : 0 ≤ x2 ≤ 1}.

Much more general families satisfying the conditions (H1) and (H2) are given in
the examples below.

Example 3.2. Let Ω ⊂ Rn be a C2 domain, and X : U ⊂ Rn → Rn a smooth (say
C1) vector field defined in an open set containing Ω and x(t, x0) the solution of

dx

dt
= X(x)

x(0) = x0.

Then, the map

x : (t, ξ) 7→ x(t, ξ) : (−r, r)× ∂Ω→ V ⊂ Rn

is a diffeomorphism for some r > 0 and some open neighborhood V of ∂Ω. Let W be
a (smaller) open neighborhood of ∂Ω, that is, with W ⊂ V and define hε : W → Rn
by hε(x(t, ξ)) = (x(t + η(t) · θε(ξ), ξ)), where θε : ∂Ω → R is a C1 function, with
‖θε‖C1(∂Ω) → 0 as ε → 0, η : [−r, r] → [0, 1] is a C2 function, with η(0) = 1 and
η(t) = 0 if |t| ≥ r/2. Observe that hε is well defined and {hε, 0 ≤ ε ≤ ε0} is a
family of C1 maps for ε0 sufficiently small, with ‖hε − iBr(∂Ω)‖C1(W ) → 0 as ε→ 0.



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 7

We may extend hε to a diffeomorphism of Rn, satisfying (H1), which we still write
simply as hε by defining it as the identity outside W .

If φ : U ⊂ Rn−1 → Rn is a local coordinate system for ∂Ω in a neighborhood of
x0 ∈ ∂Ω, then the map Ψ(t, y) = x(t, φ(y)) : (−r, r)×U → Rn is a C1 coordinate sys-
tem around the point x0 ∈ Rn and Ψ−1hεΨ(t, y) = (t+η(t)θε(φ(y)), y). By an easy
computation, we find that the Jacobian of Ψ−1hεΨ is given by J(Ψ−1hεΨ(t, y)) =
1+η′(t)θε(φ(y)) and, therefore Jhε(x) = [1 + η′(t(x))θε(φ(π(x)))]·JΨ

(
Ψ−1(hε(x))

)
·

JΨ−1(x) for x ∈W . Since ‖hε − IdRn‖C1 → 0, the condition (H2) follows.
We can also compute J∂Ωhε|∂Ω, the Jacobian of hε restricted to ∂Ω. We drop the

subscript ∂Ω to simplify the notation. Note that the coordinate system Ψ above
takes {0}×U into a neighborhood of x0 ∈ ∂Ω, and Ψ−1hε|∂ΩΨ(0, y) = (θε(φ(y)), y).

A straightforward computation gives J(Ψ−1hε|∂ΩΨ(0, y)) =
√

1 + ‖∇θε(φ(y))‖2
and, therefore

Jhε
∣∣
∂Ω

(φ(y)) =
[√

1 + ‖∇θε(φ(y))‖2
]
JΨ
(
Ψ−1(hε(φ(y)))

)
· JΨ−1(Ψ(0, y))

for y ∈ U , where Ψ0 and Ψε denote the restriction of Ψ to {(0, y) |y ∈ U} and
{(θε(φ(y)), y) |y ∈ U}, respectively. Since ‖hε − IdRn‖C1 and ‖θε(ξ)‖C1(∂Ω) → 0, it
follows that Jhε|∂Ω(φ(y))→ 1 as ε→ 0, uniformly in ∂Ω.

Example 3.3. We can choose the vector field X in the previous example as an
extension of N : ∂Ω → Rn the unit outward normal to ∂Ω, t(x) = ± dist(x, ∂Ω),
(“+” outside, “-” inside), φ(x) = the point of ∂Ω nearest x and Br(∂Ω) = {x ∈
Rn : dist(x, ∂Ω) < r}.

Then, the map ρ : (t, ξ) 7→ ξ+tN(ξ) : (−r, r)×∂Ω→ Br(∂Ω) is a diffeomorphism,
for some r > 0, with inverse x 7→ (t(x), π(x)) (see [9]).

Define hε : Br(∂Ω)→ Rn by hε(ρ(t, ξ)) = ξ + tN(ξ) + η(t)θε(ξ)N(ξ) = ρ(t, ξ) +
η(t)θε(ξ)N(ξ), where θε : ∂Ω → R is a C1 function, with ‖θε‖C1(∂Ω) → 0 as ε →
0, η : [−r, r] → [0, 1] is a C2 function, with η(0) = 1 and η(t) = 0 if |t| ≥ r

2 .

Then, {hε, 0 ≤ ε ≤ ε0} is a family of C1 maps for ε0 sufficiently small, with
‖hε − iBr(∂Ω)‖C1 → 0 as ε → 0. We may extend hε to a diffeomorphism of Rn,
satisfying (H1), which we still write simply as hε by defining it as the identity
outside Br(∂Ω).

If φ : U ⊂ Rn−1 → Rn is a local coordinate system for ∂Ω in a neighborhood of
x0 ∈ ∂Ω, then the map Ψ(t, y) = φ(y) + tN(φ(y)) = ρ(t, φ(y)) : (−r, r) × U → Rn
is a C1 coordinate system around the point x0 ∈ Rn and Ψ−1hεΨ(t, y) = (t +
η(t)θε(φ(y)), y). The condition (H2) can now be checked as in the previous example.

Remark 3.4. We may choose the function θε with “oscillatory behavior”, so the
example above essentially includes the case considered in [5], since the perturbation
there is nonzero only in a smooth portion of the boundary.

4. Linear semigroup

In this section we consider the linear semigroups generated by the family of
differential operators −h∗ε∆Ωεh

∗−1
ε + aI, appearing in (2.1).

4.1. Strong form in Lp spaces. Consider the operator in Lp(Ω), p ≥ 2, given by

Aε :=
(
− h∗ε∆Ωεh

∗−1
ε + aI

)
(4.1)



8 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

with domain

D(Aε) =
{
u ∈W 2,p(Ω) : h∗ε

∂

∂NΩε

h∗
−1

ε u = 0, on ∂Ω
}
. (4.2)

We will denote simply by A the unperturbed operator
(
−∆Ω + aI

)
.

Theorem 4.1. If ε > 0 is sufficiently small and hε ∈ Diff1(Ω), then the operator
Aε =

(
− h∗ε∆Ωεh

∗−1
ε + aI

)
defined by (4.1) and (4.2) is sectorial.

Proof. Consider the operator −∆Ωε defined in Lp(hε(Ω)), with domain

D(−∆Ωε) =
{
u ∈W 2,p(Ωε) :

∂

∂NΩε

u = 0 on ∂Ωε
}
,

where Ωε = hε(Ω). It is well known that −∆Ωε is sectorial, with the spectra
contained in the interval (0,∞) ⊂ R.

If λ ∈ C and f ∈ L2(Ω), we have

(h∗ε∆Ωεh
∗−1
ε + λI)u(x) = f(x)

⇔ (∆Ωε + λI)u ◦ h−1
ε (hε(x)) = f ◦ h−1

ε (hε(x))

⇔ (∆Ωε + λI)v(y) = g(y).

Since u 7→ h∗εu := u ◦ hε is an isomorphism from L2(Ωε) to L2(Ω) with inverse
(h−1
ε )∗, it follows that the first equation is uniquely solvable in L2(Ω) if, and only

if, the last equation is uniquely solvable in L2(Ωε).
Suppose λ belongs to ρ(−∆Ωε), the resolvent set of −∆Ωε . Then

‖u‖pLp(Ω) =

∫
Ω

|u(x)|p dx

=

∫
Ω

|v ◦ hε(x)|p dx

=

∫
Ωε

|v(y)|p|Jh−1
ε (y)|dy

≤ ‖Jh−1
ε ‖∞‖v‖

p
Lp(Ωε)

≤ ‖Jh−1
ε ‖∞‖ (∆Ωε + λI)

−1 ‖L(Lp(Ωε))‖g‖
p
Lp(Ωε)

.

On the other hand,

‖g‖pLp(Ωε)
=

∫
Ωε

|g(x)|p dy

=

∫
Ωε

|f ◦ h−1
ε (y)|p dy

=

∫
Ω

|f(x)|p|Jhε(x)|dx

≤ ‖Jhε‖∞‖f‖pLp(Ω).

It follows that

‖u‖pLp(Ω) ≤ ‖Jhε‖∞‖Jh
−1
ε ‖∞‖(∆Ωε + λI)−1‖L(Lp(Ωε))‖f‖

p
Lp(Ω).

Therefore, λ ∈ ρ(−h∗ε∆Ωεh
∗−1
ε ) and

‖(h∗ε∆Ωεh
∗−1
ε +λI)−1‖L(Lp(Ω)) ≤ ‖Jhε‖∞‖Jh−1

ε ‖∞‖(∆Ωε +λI)−1‖L(Lp(Ωε)). (4.3)



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 9

It can be proved similarly that λ ∈ ρ(−h∗ε∆Ωεh
∗−1
ε )⇒ λ ∈ ρ(−∆Ωε).

Finally, if Bε = −∆Ωε +aI is sectorial with ‖(λ−Bε)−1‖ ≤ M
|λ−a′| for all λ in the

sector Sa′,φ0
= {λ : φ0 ≤ | arg(λ− a′)| ≤ π, λ 6= a′}, for some a′ ∈ R and 0 ≤ φ0 <

π/2, it follows from (4.3) that Aε = a− h∗ε∆Ωεh
∗−1
ε satisfies ‖(λ− A)−1‖ ≤ M ′

|λ−a′|
for all λ in the sectoriality of Aε follows from the sectoriality of Bε. �

Remark 4.2. From Theorem 4.1 and results in [10], it follows that Aε generates
a linear analytic semigroup in Lp(Ω), for each ε ≥ 0.

4.2. Weak form in Lp spaces. One would like to prove that the operators Aε de-
fined by (4.1) and (4.2) become close to the operator A as ε→ 0 in a certain sense.
This is possible when the perturbation diffeomorphisms hε converge to the identity
in the C2-norm (see, for example [14, 16]). To obtain similar results here, we need
to consider the problem in weaker topologies, that is, we need to extend those oper-
ators. To this end, we now want to consider the operator Aε =

(
−h∗ε∆Ωεh

∗−1
ε + aI

)
as an operator Ãε in (W 1,q(Ω))′ with D(Ãε) = W 1,p(Ω), where q is the conjugate
exponent of p, that is 1

p + 1
q = 1.

If u ∈ D(Aε) = {u ∈ W 2,p(Ω) : h∗ε
∂

∂NΩε
h∗−1
ε u = 0}, ψ ∈ W 1,q(Ω), and v =

u ◦ h−1
ε , integrating by parts, we obtain

〈Aεu, ψ〉−1,1

= −
∫

Ω

(h∗ε∆Ωεh
∗−1
ε u)(x)ψ(x) dx+ a

∫
Ω

u(x)ψ(x) dx

= −
∫

Ω

∆Ωε(u ◦ h−1
ε )(hε(x))ψ(x) dx+ a

∫
Ω

u(x)ψ(x) dx

= −
∫

Ωε

∆Ωεv(y)ψ(h−1
ε (y))

1

|Jhε(h−1
ε (y))|

dy

+ a

∫
Ωε

u(h−1
ε (y))ψ(h−1

ε (y))
1

|Jhε(h−1
ε (y))|

dy

= −
∫
∂Ωε

∂v

∂NΩε

(y)ψ(h−1
ε (y))

1

|Jhε(h−1
ε (y))|

dσ(y)

+

∫
Ωε

∇Ωεv(y) · ∇Ωε

[
ψ(h−1

ε (y))
1

|Jhε(h−1
ε (y))|

]
dy

+ a

∫
Ωε

u(h−1
ε (y))ψ(h−1

ε (y))
1

|Jhε(h−1
ε (y))|

dy

=

∫
Ωε

∇Ωεv(y) · ∇Ωε

[
ψ(h−1

ε (y))
1

|Jhε(h−1
ε (y))|

]
dy

+ a

∫
Ωε

u(h−1
ε (y))ψ(h−1

ε (y))
1

|Jhε(h−1
ε (y))|

dy

=

∫
Ω

∇Ωεv(hε(x)) · ∇Ωε

[
ψ ◦ h−1

ε

1

|Jhε ◦ h−1
ε |

(hε(x))
]
‖Jhε(x)| dx+ a

∫
Ω

u(x)ψ(x)dx

=

∫
Ω

(h∗ε∇Ωεh
∗−1
ε u)(x) ·

[
h∗ε∇Ωεh

∗−1
ε

ψ

Jhε

]
(x)|Jhε(x)| dx + a

∫
Ω

u(x)ψ(x) dx

=

∫
Ω

(h∗ε∇Ωεh
∗−1
ε u)(x) · h∗ε∇Ωεh

∗−1
ε ψ(x) dx+ a

∫
Ω

u(x)ψ(x) dx



10 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

+

∫
Ω

(h∗ε∇Ωεh
∗−1
ε u)(x)(h∗ε∇Ωεh

∗−1
ε Jhε)(x)

1

Jhε
ψ(x) dx. (4.4)

Since (4.4) is well defined for u ∈W 1,p(Ω), we define an extension Ãε of Aε, with
domain W 1,p(Ω) and values in (W 1,q(Ω))′, by

〈Ãεu , ψ〉−1,1

:=

∫
Ω

(h∗ε∇Ωεh
∗−1
ε u)(x) · h∗ε∇Ωεh

∗−1
ε ψ(x) dx+ a

∫
Ω

u(x)ψ(x) dx

+

∫
Ω

(h∗ε∇Ωεh
∗−1
ε u)(x) · (h∗ε∇Ωεh

∗−1
ε Jhε)(x) · 1

Jhε
ψ(x) dx,

(4.5)

for any Ψ ∈ (W 1,q(Ω)).

Remark 4.3. If u is regular enough, then Ãu = Au implies that u must satisfy

the boundary condition h∗ε
∂

∂NΩε
h∗
−1

ε u = 0, on ∂Ω but, since this is not well defined

in (W 1,q(Ω)), the domain of à does not incorporate this boundary condition.

For simplicity, we still denote this extension by Aε, whenever there is no danger
of confusion. Also, from now on, we drop the absolute value in |Jhε(x)|, since the
Jacobian of hε is positive for sufficiently small ε.

Next we now prove the following basic inequality.

Theorem 4.4. D
(
Aε
)
⊃ D

(
A
)

for any ε ≥ 0 and there exists a positive function
τ(ε) such that ∣∣∣∣(Aε −A)u ∣∣∣∣W 1,q(Ω)′

≤ τ(ε)
∣∣∣∣Au ∣∣∣∣

W 1,q(Ω)′
,

for all u ∈ D
(
A
)
, with limε→0+ τ(ε) = 0.

Proof. The assertion about the domain is immediate. The inequality is equivalent
to

|〈(Aε −A)u, ψ〉−1,1| ≤ τ(ε)‖Au‖(W 1,q(Ω))′‖ψ‖W 1,q(Ω) ,

for all u ∈W 1,p(Ω), ψ ∈W 1,q(Ω), with limε→0+ τ(ε) = 0. For ε > 0, We have

|〈
(
Aε −A)u, ψ〉−1,1|

≤
∣∣ ∫

Ω

(
h∗ε∇Ωεh

∗−1
ε u

)
(x) · [

(
h∗ε∇Ωεh

∗−1
ε ψ

)
(x)−

(
∇Ωψ

)
(x)
]
dx
∣∣

+
∣∣ ∫

Ω

(
h∗ε∇Ωεh

∗−1
ε u−∇Ωu

)
(x) ·

(
∇Ωψ

)
(x) dx

∣∣
+
∣∣ ∫

Ω

(h∗ε∇Ωεh
∗−1
ε u)(x) · (h∗ε∇Ωεh

∗−1
ε Jhε)(x) · 1

Jhε
ψ(x) dx

∣∣.
(4.6)

Now, writing |v|p = (
∑n
i=1 |vi|p)1/p, 1 ≤ p <∞, |v|∞ = sup(|vi|, i = 1, 2, . . . , n)

for the p-norm of the vector v = (v1, v2, . . . , vn) ∈ Rn, we observe that

|h∗ε∇Ωεh
∗−1
ε u (x)|p =

(∑
i

∣∣h∗ε ∂

∂yi
h∗−1
ε u (x)

∣∣p)1/p

=
(∑

i

(∑
j

∣∣bεi,j(x)
∂u

∂xj
(x)
∣∣)p)1/p

≤
[∑

i

(∑
j

|bεi,j |q(x)
)p/q(∑

j

(∣∣ ∂u
∂xj

∣∣)p(x)
)]1/p



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 11

≤
[∑

i

(∑
j

|bεi,j |q(x)
)p−1]1/p

|∇u(x)|p

≤ ‖bε‖∞
[∑

i

np−1
]1/p
|∇u(x)|p

≤ n‖bε‖∞|∇u(x)|p
≤ B(ε)|∇u(x)|p ,

|h∗ε∇Ωεh
∗−1
ε u (x)−∇Ωu(x)|p

=
(∑

i

∣∣h∗ε ∂

∂yi
h∗−1
ε u (x)− ∂u

∂xi
(x)
∣∣p)1/p

=
(∑

i

(∑
j

∣∣∣(bεi,j(x)− δi,j)
∂u

∂xj
(x)
∣∣∣)p)1/p

≤
[∑

i

(∑
j

|bεi,j − δi,j |q(x)
)p/q(∑

j

(∣∣ ∂u
∂xj

∣∣)p(x)
)]1/p

≤
[∑

i

(∑
j

|bεi,j − δi,j |q(x)
)p−1]1/p

|∇u(x)|p

≤ ‖bε − δ‖∞
[∑

i

np−1
]1/p
|∇u(x)|p

≤ n‖bε − δ‖∞|∇u(x)|p
≤ η(ε)|∇u(x)|p ,

1

Jhε(x)
|h∗ε∇Ωεh

∗−1
ε Jhε(x)|∞ =

1

Jhε(x)
sup
i

{∣∣h∗ε ∂

∂yi
h∗−1
ε Jhε (x)

∣∣}
=

1

Jhε(x)
sup
i

{∑
j

∣∣bεi,j(x)
∂Jhε
∂xj

(x)
∣∣}

=
1

Jhε(x)
‖bε‖∞

∑
j

∣∣∂Jhε
∂xj

(x)
∣∣

≤ 1

Jhε(x)
‖bε‖∞|∇Jhε(x)|1 ≤ n‖bε‖∞|∇Jhε(x)|∞

≤ 1

Jhε(x)
B(ε)|∇Jhε(x)|∞ ≤

1

Jhε(x)
B(ε)‖∇Jhε‖∞

≤ µ(ε) ,

1

Jhε(x)

∣∣h∗ε∇Ωεh
∗−1
ε Jhε(x)ψ(x)

∣∣
q

=
1

Jhε(x)

(∑
i

∣∣h∗ε ∂

∂yi
h∗−1
ε Jhε (x) · ψ(x)

∣∣q)1/q

≤ 1

Jhε(x)

∣∣h∗ε∇h∗−1
ε Jhε (x)

∣∣
∞

(∑
i

∣∣ψ(x)
∣∣q)1/q

≤ nµ(ε)ψ(x) ,



12 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

where ‖bε‖∞ := sup{|bεi,j(x)|, 1 ≤ i, j ≤ n, x ∈ Ω}, ‖bε − δ‖∞ := sup{|bεi,j −
δi,j |(x)|, 1 ≤ i, j ≤ n, x ∈ Ω}, B(ε) → n and η(ε), µ(ε) → 0 as ε → 0, by
hypotheses (H1) and (H2).

In a similar way, we obtain

|h∗ε∇Ωεh
∗−1
ε ψ(x)|p ≤ B(ε)|∇ψ(x)|p,

|h∗ε∇Ωεh
∗−1
ε ψ(x)−∇ψ(x)|p ≤ η(ε)|∇ψ(x)|p.

It follows that

|〈(Aε −A)u, ψ〉−1,1|

≤
∫

Ω

∣∣(h∗ε∇Ωεh
∗−1
ε u

)
(x)
∣∣
p

∣∣(h∗ε∇Ωεh
∗−1
ε ψ

)
(x)−

(
∇Ωψ

)
(x)
∣∣
q
dx

+

∫
Ω

|
(
h∗ε∇Ωεh

∗−1
ε u−∇Ωu

)
(x)|p

∣∣(∇Ωψ
)
(x)
∣∣
q
dx

+

∫
Ω

∣∣(h∗ε∇Ωεh
∗−1
ε u)(x)

∣∣
p

∣∣ 1

Jhε(x)
(h∗ε∇Ωεh

∗−1
ε )(x)ψ(x)

∣∣
q
dx

≤ B(ε)
[ ∫

Ω

|∇u(x)|pp dx
]1/p

η(ε)
[ ∫

Ω

|∇ψ(x)|qq dx
]1/q

+ η(ε)
[ ∫

Ω

|∇u(x)|pp dx
]1/p[ ∫

Ω

|∇ψ(x)|qq dx
]1/q

+B(ε)n · µ(ε)
[ ∫

Ω

|∇u(x)|pp dx
]1/p[ ∫

Ω

|ψ(x)|qq dx
]1/q

≤ ((1 +B(ε))η(ε) + nβ(ε))µ(ε)) ‖u‖W 1,p(Ω)‖ψ‖W 1,q(Ω)

≤ K(ε)‖u‖W 1,p(Ω)‖ψ‖W 1,q(Ω)

with limε→0+ K(ε) = 0 (independently of u). We conclude that

‖ (Aε −A)u‖W 1,q(Ω)′ ≤ K(ε)‖u‖W 1,p(Ω) ≤ τ(ε)‖Au‖W 1,q(Ω)′ (4.7)

with limε→0+ τ(ε) = 0, (and τ(ε) does not depend on u). �

4.3. Existence and continuity of the linear semigroup. Using well known
facts about the “unperturbed operator” A and Theorem 4.4, one can now establish
existence and continuity of the linear semigroup, based on the following results.

Lemma 4.5. Suppose A is a sectorial operator with ‖(λ − A)−1‖ ≤ M
|λ−a| for all

λ in the sector Sa,φ0 = {λ : φ0 ≤ | arg(λ − a)| ≤ π, λ 6= a}, for some a ∈ R
and 0 ≤ φ0 < π/2. Suppose also that B is a linear operator with D(B) ⊃ D(A)
and ‖Bx − Ax‖ ≤ ε‖Ax‖ + K‖x‖, for any x ∈ D(A), where K and ε are positive

constants with ε ≤ 1
4(1+LM) , K ≤

√
5

20M

√
2L−1
L2−1 , for some L > 1.

Then B is also sectorial. More precisely, if b = L2

L2−1a−
√

2L
L2−1 |a|, φ = max

{
φ0,

π
4

}
and M ′ = 2M

√
5, then

‖(λ−B)−1‖ ≤ M ′

|λ− b|
,

in the sector Sb,φ = {λ | φ ≤ |arg(λ− b)| ≤ π, λ 6= b}.

For a proof of the above lemma, see [16, p. 346].

Remark 4.6. Observe that b can be made arbitrarily close to a by taking L
sufficiently large. In particular, if a > 0 then b > 0.



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 13

Theorem 4.7. Suppose that A is as in Lemma 4.5, Λ a topological space and
{Aγ}γ∈Λ is a family of operators in X with Aγ0 = A satisfying the following con-
ditions:

(1) D(Aγ) ⊃ D(A), for all γ ∈ Λ;
(2) ‖Aγx − Ax‖ ≤ ε(γ)‖Ax‖ + K(γ)‖x‖ for any x ∈ D(A), where K(γ) and

ε(γ) are positive functions with limγ→γ0 ε(γ) = 0 and limγ→γ0 K(γ) = 0.

Then, there exists a neighborhood V of γ0 such that Aγ is sectorial if γ ∈ V and
the family of (linear) semigroups e−tAγ satisfies

‖e−tAγ − e−tA‖ ≤ C(γ)e−bt

‖A
(
e−tAγ − e−tA

)
‖ ≤ C(γ)

1

t
e−bt

‖Aα
(
e−tAγ − e−tA

)
‖ ≤ C(γ)

1

tα
e−bt, 0 < α < 1

(4.8)

for t > 0, where b is as in Lemma 4.5, and C(γ)→ 0 as γ → γ0.

For a proof of the above lemma, see [16, p. 349].

Theorem 4.8. The operators Aε given by (4.5) in the space X = (W 1,q(Ω))′, with
domain W 1,p(Ω), 1 < p < ∞, 1

p + 1
q = 1, are sectorial operators with sectors and

constant in the sectorial inequality independent of ε, for ε0 sufficiently small. The
family of analytic linear semigroups e−tAε generated by Aε in the “base space” X,
satisfies (4.8).

The first assertion of the above theorem follows from Theorem 4.5, and the
second from Theorem 4.7.

5. The abstract problem in a scale of Banach spaces

Our goal in this section is to pose the problem (1.1) in a convenient abstract
setting. We proved in Theorem 4.1 that, if ε is small, the operator Aε in Lp(Ω)
defined by (4.1) with domain given in (4.2) is sectorial and, in Theorem 4.8 that

the same is true for its extension Ãε to (W 1,q(Ω))′(Ω).

It is then well-known that the domains Xα
ε (resp. X̃α

ε ), α ≥ 0 of the frac-

tional powers of Aε (resp. Ãε) are Banach spaces, X0
ε = Lp(Ω), (resp. X̃0

ε =

(W 1,q(Ω))′(Ω)), X1
ε = D(Aε) = W 2,p(Ω), (resp. X̃1

ε = D(Ãε) = W 1,p(Ω)), Xα
ε ,

(X̃α
ε ) is compactly embedded in Xβ

ε , (X̃β
ε ) when 0 ≤ α < β < 1, and Xα

ε = W 2α,p,
when 2α is an integer number.

Since X
1/2
ε = X̃1

ε , it follows easily that X
α− 1

2
ε = X̃α

ε , for 1
2 ≤ α ≤ 1 and, by an

abuse of notation, we will still write X
α− 1

2
ε instead of X̃α

ε , for 0 ≤ α ≤ 1
2 so we

may denote by {Xα
ε , − 1

2 ≤ α ≤ 1} = {Xα
ε , 0 ≤ α ≤ 1} ∪ {X̃α

ε , 0 ≤ α ≤ 1}, the
whole family of fractional power spaces. We will denote simply by Xα the fractional
power spaces associated to the unperturbed operator A.

For any −1/2 ≤ β ≤ 0, we may now define an operator in these spaces as the

restriction of Ãε. We then have the following result.

Theorem 5.1. For any −1/2 ≤ β ≤ 0 and ε sufficiently small, the operator (Aε)β
in Xβ

ε , obtained by restricting Ãε, with domain Xβ+1
ε is a sectorial operator.



14 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

Proof. Writing β = − 1
2 + δ, for some 0 ≤ δ ≤ 1/2, we have (Aε)β = Ã−δε ÃεÃ

δ
ε .

Since Ãδε is an isometry from Xβ
ε to X

− 1
2

ε = (W 1,q(Ω))′, the result follows easily. �

We can now pose the problem (2.1) as an abstract problem in the scale of Banach
spaces {Xβ

ε ,−1/2 ≤ β ≤ 0}.

ut = −(Aε)βu+ (Hε)βu, t > t0 ;

u(t0) = u0 ∈ Xη
ε ,

(5.1)

where

(Hε)β = H(·, ε) := (Fε)β + (Gε)β : Xη
ε → Xβ

ε , ε > 0, 0 ≤ η ≤ β + 1, (5.2)

(i) (Fε)β = F (·, ε) : Xη
ε → Xβ

ε is given by

〈F (u, ε),Φ〉β,−β =

∫
Ω

f(u)Φ dx, for any Φ ∈ (Xβ
ε )′, (5.3)

(ii) (Gε)β = G(·, ε) : Xη
ε → Xβ

ε is given by

〈G(u, ε),Φ〉β,−β =

∫
∂Ω

g(γ(u))γ(Φ)
∣∣J∂Ωhε
Jhε

∣∣ dσ(x), for any Φ ∈ (Xβ
ε )′, (5.4)

where γ is the trace map and J∂Ωhε is the determinant of the Jacobian
matrix of the diffeomorphism hε : ∂Ω −→ ∂hε(Ω).

We will choose β, small enough in order that Xβ+1
ε does not incorporate the

boundary conditions, that is, the closure of the subset defined by smooth functions
with Neumann boundary condition is the whole space. It is not difficult to show,
integrating by parts, that a regular enough solution of (5.1), must satisfy (2.1) (see
[6, 13]).

6. Local well-posedness

To prove local well-posedness for the abstract problem, without assuming growth
conditions in the nonlinearities, we want to have two somewhat conflicting require-
ments for our phase space Xη

ε : we need it to be continuously embedded in L∞ and
we do not want it to incorporate the boundary conditions. To this end, we need
to choose η and p big enough so that the inclusion holds and, on the other hand,
we need η small enough so that the normal derivative does not have a well defined
trace. To achieve both requirements we will henceforth assume that

p and η are such that the inclusion Xη
ε ↪→ L∞(Ωε) holds, for some

µ ≥ 0 and η < 1/2.
(6.1)

It is easy to check that (6.1) holds, for instance, if p = 2n, and 1/4 < η < 1/2.
Also, the last inequality is automatically satisfied if we choose our base space Xβ

ε =

X
−1/2
ε = (W 1,q(Ω))′, where q and p are conjugate exponents, since we must have

η − β < 1.

Lemma 6.1. Suppose that p and η are such that (6.1) holds and f is locally Lip-

schitz. Then, the operator (Fε)η : Xη
ε→X

−1/2
ε given by (5.3) is well defined and

Lipschitz in bounded sets.



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 15

Proof. Suppose u ∈ Xη
ε . From (6.1), it follows that u ∈ L∞(Ω) and, therefore,

if Lf is the Lipschitz constant of f on the interval [−‖u‖∞, ‖u‖∞], it follows that

|f(u(x)− f(0)| ≤ Lf |u(x)|, for any x ∈ Ω. If Φ ∈ (X
−1/2
ε )′ = W 1,q(Ω), then∣∣〈(Fε)η(u),Φ〉β,−β

∣∣ ≤ ∫
Ω

|f(u)||Φ| dx

≤ Lf
∫

Ω

|u||Φ| dx+

∫
Ω

|f(0)||Φ| dx

≤ Lf‖u‖Lp(Ω) · ‖Φ‖Lq(Ω) + ‖f(0)‖Lp(Ω) · ‖Φ‖Lq(Ω)

Since W 1,q(Ω) ⊂ Lq(Ω) and Xη
ε ⊂ Lp(Ω) with stronger norms, we have∣∣〈(Fε)η(u),Φ〉β,−β

∣∣ ≤ Lf‖u‖Lp(Ω)‖Φ‖W 1,q(Ω) + ‖f(0)‖Lp(Ω)‖Φ‖W 1,q(Ω),

so (Fε)η is well defined and

‖(Fε)η(u)‖(W 1,q(Ω))′ ≤ Lf‖u ‖Lp(Ω) + ‖f(0)‖Lp(Ω) (6.2)

≤ Lf‖u ‖Xηε + ‖f(0)‖Lp(Ω) (6.3)

where Lf is the Lipschitz constant of f on the interval [−‖u‖∞, ‖u‖∞]
Alternatively, if Mf = Mf (u) := sup{|f(x)| x ∈ [−‖u‖∞, ‖u‖∞]}, it follows that∣∣〈(Fε)η(u),Φ〉β,−β

∣∣ ≤ ∫
Ω

|f(u)||Φ| dx

≤Mf |Ω|1/p‖Φ‖Lq(Ω)

≤Mf |Ω|1/p‖Φ‖W 1,q(Ω).

Thus

‖(Fε)η(u)‖(W 1,q(Ω))′ ≤Mf |Ω|1/p. (6.4)

Suppose now that u1, u2 belong to a bounded set B ∈ Xη
ε . From (6.1) it

follows now that u1, u2 belong to a ball of radius R = supu∈B ‖u‖∞ in L∞(Ω)
and, therefore, if L is the Lipschitz constant of f in the interval [−R,R], we have

|f(u1(x)) − f(u2(x)))| ≤ L|u1(x) − u2(x)|, for any x ∈ Ω. If Φ ∈ (X
−1/2
ε )′ =

W 1,q(Ω), we obtain∣∣〈(Fε)η(u1)− (Fε)η(u2),Φ〉β,−β
∣∣ =

∣∣ ∫
Ω

[f(u1)− f(u2)] Φ dx
∣∣

≤
∫

Ω

L|u1 − u2||Φ| dx

≤ Lf‖u1 − u2‖Lp(Ω) · ‖Φ‖Lq(Ω)

≤ Lf‖u1 − u2‖Xηε · ‖Φ‖W 1,q(Ω).

Thus

‖(Fε)η(u1)− (Fε)η(u2)‖(W 1,q(Ω))′ ≤ Lf‖u1 − u2‖Lp(Ω) (6.5)

≤ Lf‖u1 − u2‖Xηε . (6.6)

This concludes the proof. �

Lemma 6.2. Suppose that p and η are such that (6.1) holds and g is locally Lip-
schitz. Then, if ε0 is sufficiently small, the operator (Gε)η= G : Xη

ε→(W 1,q(Ω))′

given by (5.4) is well defined, for 0 ≤ ε < ε0 and bounded in bounded sets.



16 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

Proof. Suppose u ∈ Xη
ε . From (6.1) it follows that u ∈ L∞(Ω) and, therefore,

if Lg is the Lipschitz constant of g in the interval [−‖u‖∞, ‖u‖∞], it follows that
|g(γ(u)(x)− g(0)| ≤ Lg|γ(u)(x)|, for any x ∈ ∂Ω.

If u ∈ Xη
ε and Φ ∈ (X

−1/2
ε )′ = W 1,q(Ω), we have∣∣〈G(u, ε),Φ〉β,−β
∣∣

≤
∫
∂Ω

|g(γ(u))||γ(Φ)|
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)

≤ ‖µ‖∞
∫
∂Ω

Lg|γ(u)||γ(Φ)|+ |g(0)||γ(Φ)|dσ(x)

≤ ‖µ‖∞
(
Lg‖γ(u)‖Lp(∂Ω)‖γ(Φ)‖Lq(∂Ω) + ‖g(0)‖Lp(∂Ω)‖γ(Φ)‖Lq(∂Ω)

)
where µ(x, ε) = |J∂Ωhε

Jhε
|, and ‖µ‖∞ = sup{|µ(x, ε)| : x ∈ ∂Ω, 0 ≤ ε ≤ ε0} is finite by

hypothesis (H1).
By the imbedding and trace theorems,

‖γ(Φ)‖Lq(∂Ω) ≤ K1‖Φ‖W 1,q(Ω), ‖γ(u)‖Lp(∂Ω) ≤ K2‖u‖Xηε ,

for some constants K1, K2. Thus∣∣〈G(u, ε),Φ〉β,−β
∣∣

≤ ‖µ‖∞
(
LgK1‖γ(u)‖Lp(∂Ω)‖Φ‖W 1,q(Ω) +K1‖g(0)‖Lp(∂Ω) · ‖Φ‖W 1,q(Ω)

)
proving that (Gε)β is well defined and

‖G(u, ε)‖(W 1,q(Ω))′ ≤ ‖µ‖∞
(
LgK1‖γ(u)‖Lp(∂Ω) +K1‖g(0)‖Lp(∂Ω)

)
(6.7)

≤ ‖µ‖∞
(
LgK1K2‖u‖Xηε +K1‖g(0)‖Lp(∂Ω)

)
. (6.8)

Alternatively, if Mg = Mg(u) := sup{|g(x)| : x ∈ [−‖u‖∞, ‖u‖∞]}, it follows that∣∣〈G(u, ε),Φ〉β,−β
∣∣ ≤ ∫

∂Ω

|g(γ(u))||γ(Φ)|
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)

≤ ‖µ‖∞Mg

∫
∂Ω

|γ(Φ)|dσ(x)

≤ ‖µ‖∞Mg|∂Ω|1/p‖γ(Φ)‖Lq(∂Ω)

≤ ‖µ‖∞Mg|∂Ω|1/pK1‖Φ‖W 1,q(Ω).

Thus

‖G(u, ε)‖(W 1,q(Ω))′ ≤ ‖µ‖∞Mg|∂Ω|1/pK1. (6.9)

�

Lemma 6.3. Suppose the hypotheses of Lemma 6.2 hold. Then the operator
G(u, ε)= G(u) : Xη

ε × [0, ε0] → (W 1,q(Ω))′ given by (5.4) is uniformly continuous
in ε, for u in bounded sets of Xη

ε and locally Lipschitz continuous in u, uniformly
in ε.

Proof. We first show that (Gε)β is locally Lipschitz continuous in u ∈ Xη
ε . Suppose

that u1, u2 belong to a bounded set B ∈ Xη
ε . From (6.1), the Trace Theorem and

the hypotheses, it follows now that γ(u1), γ(u2) belong to a ball of some radius R in
L∞(∂Ω) and, therefore, if Lg is the Lipschitz constant of g in the interval [−R,R],



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 17

we have |g(γ(u1)(x))− g(γ(u2)(x)))| ≤ Lg|γ(u1)(x)− γ(u2)(x)|, for any x ∈ ∂Ω. If
Φ ∈ (W 1,q(Ω))′ and ε ∈ [0, ε0], we obtain∣∣〈G(u1, ε)−G(u2, ε),Φ〉β,−β

∣∣ ≤ ∫
∂Ω

|g(γ(u1))− g(γ(u2))| |γ(Φ)|
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)

≤
∫
∂Ω

Lg|γ(u1)− γ(u2)| |γ(Φ)|
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)

≤ Lg‖µ‖∞
∫
∂Ω

|γ(u1)− γ(u2)| |γ(Φ)| dσ(x)

≤ Lg‖µ‖∞‖γ(u1)− γ(u2)‖Lp(∂Ω)‖γ(Φ)‖Lq(∂Ω)

≤ Lg‖µ‖∞K1K2‖u1 − u2‖Xηε ‖Φ‖W 1,q(Ω),

where K1,K2 are the norms of the trace mappings. Therefore,

‖G(u1, ε)−G(u2, ε)‖(W 1,q(Ω))′ ≤ Lg‖µ‖∞K1‖γ(u1)− γ(u2)‖Lp(∂Ω) (6.10)

≤ Lg‖µ‖∞K1K2‖u1 − u2‖Xηε (6.11)

so (Gε)β is locally Lipschitz in u.
Now, if u ∈ Xη

ε , Φ ∈ (W 1,q(Ω))′ and ε1, ε2 ∈ [0, ε0], we have∣∣〈G(u, ε1)−G(u, ε2),Φ〉β,−β
∣∣

≤
∫
∂Ω

|γ(g(u))| |γ(Φ)|
∣∣∣∣∣J∂Ωhε1
Jhε1

∣∣− ∣∣J∂Ωhε2
Jhε2

∣∣∣∣∣ dσ(x)

≤ ‖µε1 − µε2‖∞
∫
∂Ω

|g(γ(u))||γ(Φ)| dσ(x)

≤ ‖µε1 − µε2‖∞
∫
∂Ω

(Lg|γ(u)|+ |g(0)|) |γ(Φ)|dσ(x)

≤ ‖µε1 − µε2‖∞
(
Lg‖γ(u)‖Lp(∂Ω) · ‖γ(Φ)‖Lq(∂Ω) + ‖g(0)‖Lp(∂Ω)‖γ(Φ)‖Lq(∂Ω)

)
≤ ‖µε1 − µε2‖∞

(
LgK1K2‖u‖η‖Φ‖W 1,q(Ω) +K1‖g(0)‖Lp(∂Ω)‖Φ‖W 1,q(Ω)

)
where ‖µε1 − µε2‖∞ = sup

{∣∣J∂Ωhε1
Jhε1

∣∣− ∣∣J∂Ωhε2
Jhε2

∣∣ : x ∈ ∂Ω
}
→ 0 as |ε1 − ε2| → 0, by

hypothesis (H1) and K1,K2 are trace constants given by the Trace Theorem. It
follows that

‖G(u, ε1)−G(u, ε2)‖(W 1,q(Ω))′

≤ ‖µε1 − µε2‖∞
(
LgK1K2‖u‖η +K1‖g(0)‖Lp(∂Ω)

)
.

(6.12)

Alternatively, if Mg = Mg(u) := sup{|g(x)| : x ∈ [−‖u‖∞, ‖u‖∞]},∣∣〈G(u, ε1)−G(u, ε2),Φ〉β,−β
∣∣

≤
∫
∂Ω

|γ(g(u))||γ(Φ)|
∣∣∣∣∣J∂Ωhε1
Jhε1

∣∣− ∣∣J∂Ωhε2
Jhε2

∣∣∣∣∣ dσ(x)

≤ ‖µε1 − µε2‖∞
∫
∂Ω

|g(γ(u))||γ(Φ)|dσ(x)

≤ ‖µε1 − µε2‖∞Mg

∫
∂Ω

|γ(Φ)| dσ(x)

≤ ‖µε1 − µε2‖∞Mg|∂Ω|1/p‖γ(Φ)‖Lp(∂Ω)

≤ ‖µε1 − µε2‖∞Mg|∂Ω|1/pK1‖Φ‖W 1,q(Ω).



18 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

It follows that

‖G(u, ε1)−G(u, ε2)‖(W 1,q(Ω))′ ≤ ‖µε1 − µε2‖∞Mg|∂Ω|1/pK1. (6.13)

�

Corollary 6.4. Suppose the hypotheses of Lemmas 6.1 and 6.2 hold. Then the
map (H(u, ε))η := (Fε(u))η + (G(u, ε))η : Xη

ε × [0, ε0]→ (W 1,q(Ω))′ is well defined,
bounded in bounded sets uniformly in ε, uniformly continuous in ε for u in bounded
sets of Xη

ε and locally Lipschitz continuous in u uniformly in ε.

Proof. From (6.2), (6.3), (6.7) and (6.8), we obtain

‖(Hε)η(u)‖(W 1,q(Ω))′

≤ Lf‖u‖Lp(Ω) + LgK1‖γ(u)‖Lp(∂Ω) + ‖f(0)‖Lp(Ω) +K1‖g(0)‖Lp(∂Ω) (6.14)

≤ (Lf + LgK1K2) ‖u‖Xηε + ‖f(0)‖Lp(Ω) +K1‖g(0)‖Lp(∂Ω), (6.15)

where Lf and Lg are Lipschitz constants of f and g in the interval [−‖u‖∞, ‖u‖∞],
respectively.

Alternatively, if Mf = Mf (u) := sup{|f(x)| : x ∈ [−‖u‖∞, ‖u‖∞]}, Mg =
Mg(u) := sup{|g(x)| : x ∈ [−‖u‖∞, ‖u‖∞]}, from (6.4) and (6.13) we obtain

‖(Hε)η(u)‖(W 1,q(Ω))′ ≤Mf |Ω|1/p + ‖µ‖∞Mg|∂Ω|1/pK1. (6.16)

From (6.5), (6.6), (6.10) and (6.11), we have

‖H(u1, ε)−H(u2, ε)‖(W 1,q(Ω))′

≤ Lg‖µ‖∞K1‖γ(u1)− γ(u2)‖Lp(∂Ω) + Lf‖u1 − u2‖Lp(Ω) (6.17)

≤ (Lg‖µ‖∞K1K2 + Lf ) ‖u1 − u2‖Xηε . (6.18)

From (6.12),

‖H(u, ε1)−H(u, ε2)‖(W 1,q(Ω))′

≤ ‖µε1 − µε2‖∞
(
LgK1K2‖u‖η +K1‖g(0)‖Lp(∂Ω)

)
.

(6.19)

Alternatively, from (6.13), we have

‖H(u, ε1)−H(u, ε2)‖(W 1,q(Ω))′ ≤ ‖µε1 − µε2‖∞Mg|∂Ω|1/pK1. (6.20)

In the estimates above K1 and K2 are the norms of the trace mappings. �

Theorem 6.5. Suppose the hypotheses of Corollary 6.4 hold. Then, for any (t0, u0)
in R × Xη

ε , problem (5.1) has a unique solution u(t, t0, u0, ε) with initial value
u(t0) = u0.

Proof. From Theorem 5.1 it follows that (Aε)β is a sectorial operator in (W 1,q(Ω))′,

with domain X
1/2
ε = W 1,p(Ω), if ε is small enough. The result follows then from

Corollary 6.4 and the results in [10, 16]. �

7. Global existence and boundedness of the semigroup

We will use Tε(t)u0 for the (local) solution of problem (5.1) given by Theorem 6.5,
with initial condition u0 in some fractional power space of Aε. We now want to show
that these solutions are globally defined if an additional (dissipative) hypothesis on
f and g is assumed. We use the hypotheses



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 19

There exist constants c0 and d0 such that

lim sup
|u|→∞

f(u)

u
≤ c0 , lim sup

|u|→∞

g(u)

u
≤ d0 (7.1)

and the first eigenvalue µ1(ε) of the problem

−h∗ε∆Ωεh
∗
ε
−1∆u+ (a− c0)u = µu in Ω

h∗ε
∂u

∂NΩ
h∗ε
−1 = d0 u on ∂Ω

(7.2)

is positive for ε sufficiently small.

Remark 7.1. Observe that if hypothesis (7.2) hold for ε = 0, then this is also true
for ε small, since the eigenvalues change continuously with ε by (4.7).

Remark 7.2. The arguments bellow are a slight modification of the ones in [13],
but we include them here for the sake of completeness. Similar arguments were
used in [4] in a somewhat different setting.

For using comparison results, we start by defining the concepts of sub- and
super-solutions.

Definition 7.3. Suppose Ω is a C1,α, domain for some α ∈ (0, 1), L is a uniformly
elliptic second order differential operator in Ω, u0 ∈ Cα(Ω), T > 0 and ū : Ω ⊂
R→Rn (u respectively) a function which is continuous in [0, T ] × Ω̄, continuously
differentiable in t and twice continuously differentiable in x for (t, x) ∈ (0, T ] × Ω.
Then u (respectively, u) is a super-solution (sub-solution) of the problem

ut = Lu+ f(u), in (0, T ]× Ω,

∂u

∂N
= g(u), on ∂Ω

u(0) = u0.

(7.3)

if it satisfies
ut ≥ Lu+ f(u), in (0, T ]× Ω,

∂u

∂N
≥ g(u), on ∂Ω

u(0) ≥ u0.

(7.4)

(and respectively with the ≥ sign replaced by the ≤ sign).

The following is a basic result for our arguments.

Theorem 7.4 ([15]). If f is locally Lipschitz and ū and u are respectively a super
and sub-solution of problem (7.3), satisfying

u ≤ ū, in Ω× (0, T ),

then there exists a solution u of (7.3) such that

u ≤ u ≤ ū, in Ω× (0, T ).

Let ϕε be the first positive normalized eigenfunction of (7.2) and let
mε = minx∈Ω̄ ϕε(x). We know that mε > 0. For each θ > 0 ∈ R, define

Σεθ = {u ∈ Xη
ε : |u(x)| ≤ θϕε(x), for all x ∈ Ω̄}.



20 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

From the dissipative hypothesis (7.1) on f and g, we know that there exists ξ ∈ R,
such that

f(s)

s
≤ c0 and

g(s)

s
≤ d0,

for all s with |s| ≥ ξ. To simplify notation, we take the ε = 0, in the proofs below,
since the argument is the same for any ε such that (7.2) is true (see Remark 7.1).

Lemma 7.5. In addition to the hypotheses of Theorem 6.5, suppose hat (7.1) and
(7.2) hold. Then, if θmε ≥ ξ and ε is small enough, the set Σεθ is a positively
invariant set for T (t).

Proof. Let

Σ1
θ = {u ∈ Xη : u(x) ≤ θϕ(x), for all x ∈ Ω̄} ,

Σ2
θ = {u ∈ Xη : u(x) ≥ −θϕ(x), for all x ∈ Ω̄}.

Since Σθ = Σ1
θ ∩Σ2

θ it is sufficient to show that Σ1
θ and Σ2

θ are positively invariant.
Let u0 ∈ Σ1

θ, and suppose, for contradiction, that there exists t0 ∈ [0, tmax[ and
x0 ∈ Ω̄ such that

T (t0)u0(x0) > θϕ(x0).

Consider v̄(t) = e−µ(t−t0)θϕ, where µ is the eigenvalue associated with ϕ. We have

∂v̄

∂t
= (∆v̄ − av̄ + c0v̄) ≥ ∆v̄ − av̄ + f(v̄)

∂v̄

∂N
= d0v̄ ≥ g(v̄),

for all t ∈]0, t0].
Thus v̄ is a super-solution for problem (2.1). It follows from Theorem 7.4 that

T (t)u0 ≤ v̄(t), in Ω̄ for all t ∈ [0, t0[.

In particular, T (t0)u0(x0) ≤ θϕ(x0) and we reach a contradiction.
To prove that Σ2

θ is positively invariant we proceed in a similar way, using now
that v = −v̄ is a sub-solution for the problem (2.1). �

Lemma 7.6. Suppose the hypotheses of Lemma 7.5 hold. If θmε ≥ ξ, and η ≤ α <
1
2 , there exists a constant R = R(θ, η), and T > 0 independent of ε, such that the
orbit of any bounded subset V of Xη

ε ∩ Σεθ under Tε(t) is in the ball of radius R of
Xα
ε , for t > T . In particular, the solutions with initial condition in Xη

ε ∩ Σθ are
globally defined.

Proof. Lemma 7.5 implies that Tε(t)u0 ∈ Σεθ, for all t ∈ [0, tmax[ so

‖Tε(t)u0‖∞ ≤ θ‖ϕ‖∞.
Applying the variation of constants formula, we obtain (see [10])

‖T (t)u0‖α ≤Mt−(α−η)e−δt‖u0‖η

+M

∫ t

0

(t− s)−(α+ 1
2 )e−δ(t−s)‖(Hε)η(T (s)u0)‖X−1/2 ds,

where M, δ > 0 are constants depending only on the decay of the linear semigroup
eAεt, and can be chosen independently of ε. By (6.16),

‖(Hε)η(T (s)u0)‖X−1/2 ≤Mf |Ω|1/p + ‖µ‖∞Mg|∂Ω|1/pK1,



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 21

where Mf = Mf (u) := sup{|f(x)| : x ∈ I}, Mg = Mg(u) := sup{|g(x)| : x ∈
I}, with [−θ‖ϕε‖∞, θ‖ϕε‖∞] ⊂ I, for all ε sufficiently small. Thus, writing K =
Mf |Ω|1/p + ‖µ‖∞Mg|∂Ω|1/pK1, we obtain

‖Tε(t)u0‖α ≤Mt−(α−η)e−δt‖u0‖η +KM

∫ t

0

(t− s)−(α+ 1
2 )e−δ(t−s) ds

≤Mt−(α−η)e−δt‖u0‖η +KM
Γ( 1

2 − α)

δ
1
2−α

,

for all t ∈ [0, tmax[.
Therefore ‖Tε(t)u0‖α is bounded by a constant for any t > 0. Since Xα is

compactly embedded in Xη, if α > η, it follows that the solution is globally defined.

Also, if T is such that t−(α−η)e−δt‖u0‖η ≤ K
Γ( 1

2−α)

δ
1
2
−α , then ‖Tε(t)u0‖α belongs to

the ball of Xα of radius R(θ) = 2KM
Γ( 1

2−α)

δ
1
2
−α , for t ≥ T . �

8. Existence of global attractors

The first step to show the existence of global attractors will be to obtain a “con-
traction property” of the sets Σθ, similar to the property for rectangles, considered
by Smoller [18].

Lemma 8.1. Suppose that the hypotheses of Lemma 7.5 hold and θ̄ ∈ R satisfy
θ̄mε > ξ. Then, for any θ there exists a t̄, which can be chosen independently of ε,
such that

Tε(t)Σ
ε
θ ⊂ Σεθ̄,

for all t ≥ t̄.

Proof. Let u ∈ Σθ. We can suppose without loss of generality that θ ≥ θ̄. Let
v̄ = e−tµεθϕ, v = −v̄. As in Lemma 7.5, we can prove that v̄ and v are super- and
sub-solutions respectively. Thus, using Theorem 7.4 and the uniqueness of solution,
we have

v ≤ Tε(t)u ≤ v̄.
Therefore Tε(t)u enters Σθ̄ after a time depending only on θ, and on the first
eigenvalue µε of Aε (and not on the particular solution u ∈ Σθ). Since µε is bigger
than a constant µ, for ε sufficiently small, and Σθ̄ is positively invariant, the result
follows. �

Theorem 8.2. Suppose that the hypotheses of Lemma 7.5 hold. Then problem
(5.1) has a global attractor Aε in Xη

ε . Furthermore Aε ⊂ Σεθ if θmε ≥ ξ.

Proof. Let V be a bounded subset of Xη, and θ̄ ∈ R be such that θ̄m ≥ ξ. If u is any
element of Xη, it follows from the continuity of the embedding Xη ↪→ C0(Ω̄) that
u ∈ Σθ, for some θ and then, applying Lemma 8.1, we conclude that T (t)u ∈ Σθ̄,
for t big enough. From Lemma 7.6, it follows that V enters and remains in a ball
of Xα, with α > η of radius R(α, θ̄), which does not depend on V . Since this
ball is a compact set of Xα, the existence of a global compact attractor A follows
immediately. Furthermore, since Σθ̄ is positively invariant by Lemma 7.5 it also
follows that A ⊂ Σθ̄, as claimed. �

Corollary 8.3. Suppose that the hypotheses of Lemma 7.5 hold. If ε0 is sufficiently
small, the attractor Aε is uniformly bounded in L∞(Ω), for 0 ≤ ε ≤ ε0.



22 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

Proof. From (4.4) and results in [11], it follows that the first eigenvalue and eigen-
function of Aε are continuous in W 1,p(Ω) and, therefore, also in L∞(Ω), Thus the
sets Σεθ are uniformly bounded in L∞(Ω) and the result follows from Theorem
8.2. �

9. Upper semicontinuity of the family of global attractors

Recall that a family of subsets Aλ of a metric space (X, d) is said to be upper-
semicontinuous at λ = λ0 if δ(Aλ,Aλ0) → 0 as λ → λ0, where δ(A,B) =
supx∈A d(x,B) = supx∈A infy∈B d(x, y) and lower-semicontinuous if δ(Aλ0 ,Aλ)→
0 as λ→ λ0.

To prove the upper semicontinuity of the family of attractors Aε, given by The-
orem 8.2 in the (fixed) fractional space Xη, 0 < η < 1

2 , we will need two main
ingredients: the uniform boundedness of the family and the continuity of the non-
linear semigroup Tε with respect to ε. This is the content of the next two results. In
view of the uniform boundedness of the solutions, proved in Corollary 8.3 we may
suppose, without loss of generality, the following hypothesis on the nonlinearites.

f and g are globally bounded, and globally Lipschtiz with con-
stants Lf and Lg respectively.

(9.1)

Lemma 9.1. Suppose that the hypotheses of Lemma 7.5 and (9.1) hold. If ε0 is
sufficiently small, the family of attractors Aε given by Theorem 8.2 is uniformly
bounded in the (fixed) fractional space Xη, 0 < η < 1/2, for 0 ≤ ε ≤ ε0.

Proof. Let b be the exponential rate of decay of the linear semigroup generated by
Aε, for ε small, given by Theorem 4.7. Let u ∈ Aε. By the variation of constants
formula, Lemma 4.5 and Theorem 4.7, we obtain

‖Tε(t)(u)‖η

≤ ‖eAε(t)u‖η +

∫ t

0

‖eAε(t−s)Hε(Tε(s)u)‖η ds

≤ ‖eA(t)u‖η + ‖
(
eAε(t) − eA(t)

)
u‖η +

∫ t

0

‖eA(t−s)Hε(Tε(s)u)‖η ds

+

∫ t

0

‖
(
eAε(t−s) − eA(t−s)

)
Hε(Tε(s)u)‖η ds

≤
(
Ce−at + C(ε)e−bt

) 1

tη+ 1
2

‖u‖+

∫ t

0

Ce−a(t−s) 1

(t− s)η + 1
2

‖Hε(Tε(s)u)‖ds

+

∫ t

0

Ce−b(t−s)
1

(t− s)η+ 1
2

‖Hε(Tε(s)u)‖ds.

By (6.16),

‖(Hε)η(T (s)u0)‖X−1/2 ≤Mf |Ω|1/p + ‖µ‖∞Mg|∂Ω|1/pK1

≤ ‖f‖∞|Ω|1/p + ‖µ‖∞‖g‖∞|∂Ω|1/pK1,

where K1 is a constant of the trace mapping. Thus

‖Tε(t)(u)‖η ≤ C ′e−bt
1

tη+ 1
2

‖u‖∞

+ C ′′
(
‖f‖∞|Ω|1/p + ‖µ‖∞‖g‖∞|∂Ω|1/p

)∫ t

0

e−b(t−s)
1

(t− s)η + 1
2

ds,



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 23

where the constants C ′ and C ′′ do not depend on ε.
Since the right hand side is uniformly bounded for u ∈ Aε and t > 0, and the

attractors are invariant, the result follows immediately. �

Lemma 9.2. Suppose that the hypotheses of Lemma 9.1 hold. Then the map

(u, ε) ∈ Xη × [0, ε0] 7→ Tεu ∈ Xη

is continuous at ε = 0, uniformly for u in bounded sets and 0 < t ≤ T <∞.

Proof. Using the variation of constants formula, (6.16), (6.20) and (6.18), we obtain

‖Tε(t)(u)− T (t)(u)‖η
≤ ‖eAε(t)u− eA(t)u‖η

+

∫ t

0

‖
(
eAε(t−s) − eA(t−s))Hε(Tε(s)u)‖η ds

+

∫ t

0

‖eA(t−s) (Hε(Tε(s)u)−H(Tε(s)u)) ‖ηds

+

∫ t

0

‖eA(t−s) (H(Tε(s)u)−H(T (s)u)) ‖η ds

≤ C(ε)e−bt
1

tη+ 1
2

‖u‖+

∫ t

0

C(ε)e−b(t−s)
1

(t− s)η+ 1
2

‖Hε(Tε(s)u)‖d s

+

∫ t

0

Ce−b(t−s)
1

(t− s)η+ 1
2

‖Hε(Tε(s)u)−H(Tε(s)u)‖ds

+

∫ t

0

Ce−b(t−s)
1

(t− s)η+ 1
2

‖H(Tε(s)u)−H(T (s)u)‖ds

≤ C(ε)e−bt
1

tη+ 1
2

‖u‖

+

∫ t

0

C(ε)e−b(t−s)
1

(t− s)η+ 1
2

(
‖f‖∞|Ω|1/p + ‖µ‖∞‖g‖∞|∂Ω|1/pK1

)
d s

+

∫ t

0

Ce−b(t−s)
1

(t− s)η+ 1
2

‖
(
‖µε − 1‖∞Mg|∂Ω|1/pK1

)
ds

+

∫ t

0

Ce−b(t−s)
1

(t− s)η+ 1
2

‖ (Lg‖µ‖∞K1K2 + Lf ) ‖Tε(s)u− T (s)u‖Xηε ‖ds .

Writing

A(ε) := C(ε)‖u‖+ tη+ 1
2

∫ t

0

C(ε)ebs
1

(t− s)η+ 1
2

×
(
‖f‖∞|Ω|1/p + ‖µ‖∞‖g‖∞|∂Ω|1/pK1

)
ds

+ tη+ 1
2

∫ t

0

Cebs
1

(t− s)η+ 1
2

(
‖µε − 1‖∞Mg|∂Ω|1/pK1

)
ds

and B := C
(
Lg‖µ‖∞K1K2 + Lf

)
, we obtain

ebt‖Tε(t)(u)− T (t)(u)‖η ≤ A(ε)t−(η+ 1
2 ) +B

∫ t

0

t−(η+ 1
2 )ebs‖Tε(s)u− T (s)u‖Xηε ds.



24 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

From the singular Gronwall’s inequality, it follows that

‖Tε(t)(u)− T (t)(u)‖η ≤ A(ε)Me−btt−(η+ 1
2 ),

for 0 < t ≤ T , where the constant M depends on B, η and T , for u in a bounded
set of Xη

ε . �

Theorem 9.3. Suppose that the hypotheses of Lemma 9.1 hold. Then the family
of attractors Aε, given by Theorem 8.2 is upper semicontinuous with respect to ε at
ε = 0.

Proof. From Lemma 9.1 there exists a bounded set B ⊂ Xη such that ∪0≤ε≤ε0Aε ⊂
B. Given δ > 0, there exists tδ > 0 such that T (tδ)(B) ⊂ A

δ
2
0 , where A

δ
2
0 is the

δ
2 -neighborhood of A0.

From Lemma 9.2, there exists ε̄ > 0 such that ‖Tε(tδ)u−T (tδ)u‖η ≤ δ
2 , for every

u ∈ B and 0 ≤ ε ≤ ε̄. It follows that Tε(tδ)B ⊂ Aδ0. In particular, Tε(tδ)Aε ⊂ Aδ0.
Since Aε is invariant under Tε, we conclude that Aε ⊂ Aδ0, for 0 ≤ ε ≤ ε̄, thus
proving the claim. �

From the semicontinuity of attractors, we can easily prove the corresponding
property for the equilibria.

Corollary 9.4. Suppose the hypotheses of Theorem 9.3 hold. Then the family of
sets of equilibria {Eε | 0 ≤ ε ≤ ε0}, of problem (5.1) is upper semicontinuous in Xη.

Proof. The result is well-known, but we sketch a proof here for completeness. Sup-
pose un ∈ An, with limn→∞ εn = 0. We choose an arbitrary subsequence and still
call it (un), for simplicity. It is enough to show that, there exists a subsequence
(unk), which converges to a point u0 ∈ E0. Since (un)→ A0, there exists (vn) ∈ A0

with ‖un−vn‖η → 0. Since A0 is compact, there exists a subsequence (vnk), which
converges to a point u0 ∈ A0, so also (unk) → A0. Now, since the flow Tε(t) is
continuous in ε, for any t > 0 we have

unk → u0 ⇔ Tεnk (t)unk → T0(t)u0 ⇔ unk → T0(t)u0.

Thus, by uniqueness of the limit, T0(t)u0 = u0, for any t > 0, so u0 ∈ E0. �

10. Lower semicontinuity

For having lower semicontinuity we need to assume the following additional prop-
erties for the nonlinearities:

f and g belong to C1(R,R) and have bounded derivatives. (10.1)

Lemma 10.1. Suppose that η and p are such that (6.1) holds and f satisfies (10.1).
Then the operator F : Xη×R→X−1/2 given by (5.3) is Gateaux differentiable with
respect to u, with Gateaux differential ∂F

∂u (u, ε)w given by

〈∂F
∂u

(u, ε)w,Φ〉−1/2,1/2 =

∫
Ω

f ′(u)wΦ dx , (10.2)

for all w ∈ Xη and Φ ∈ X1/2.

Proof. Observe first that F (u, ε) is well-defined, since the conditions of Lemma 6.1
are met.



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 25

It is clear that ∂F
∂u (u, ε) is linear. We now show that it is bounded. In fact for

all u,w ∈ Xη and Φ ∈ X−1/2 = W 1,q(Ω) we have∣∣〈∂F
∂u

(u, ε)w,Φ〉−1/2,1/2

∣∣ ≤ ∫
Ω

|f ′(u)||w||Φ|dx

≤ ‖f ′‖∞
∫

Ω

|w||Φ| dx

≤ ‖f ′‖∞‖w‖Lp(Ω)‖Φ‖Lq(Ω) dx

≤ ‖f ′‖∞‖w‖Xη‖Φ‖X1/2 dx ,

where ‖f ′‖∞ = sup{f ′(x)|x ∈ R}. This proves boundedness.
Now, for all u,w ∈ Xη and Φ ∈ X1/2 we have∣∣1

t
〈F (u+ tw, ε)− F (u, ε)− t∂F

∂u
(u, ε)w,Φ〉−1/2,1/2

∣∣
≤ 1

|t|

∫
Ω

∣∣[ f(u+ tw)− f(u)− tf ′(u)w ] Φ
∣∣ dx

≤ 1

|t|

(∫
Ω

∣∣f(u+ tw)− f(u)− tf ′(u)w
∣∣pdx)1/p

‖Φ‖X1/2

≤
(∫

Ω

∣∣(f ′(u+ t̄w)− f ′(u))w
∣∣p dx︸ ︷︷ ︸

(I)

)1/p

‖Φ‖X1/2 ,

where 0 ≤ t̄ ≤ t. Since f ′ is bounded and continuous, the integrand of (I) is
bounded by an integrable function and goes to 0 as t → 0. Thus, the integral (I)
goes to 0 as t → 0, from Lebesgue’s Dominated Convergence Theorem. It follows
that

lim
t→0

F (u+ tw, ε)− F (u, ε)

t
=
∂F

∂u
(u, ε)w in X−1/2,

for all u,w ∈ Xη; so F is Gateaux differentiable with Gateaux differential given by
(10.2). �

Now we want to prove that the Gateaux differential of F (u, ε) is continuous in
u. Let us denote by B(X,Y ) the space of linear bounded operators from X to Y .
We will need the following result, whose simple proof is omitted.

Lemma 10.2. Suppose X,Y are Banach spaces and Tn : X → Y is a sequence
of linear operators converging strongly to the linear operator T : X → Y . Suppose
also that X1 ⊂ X is a Banach space, the inclusion i : X1 ↪→ X is compact and let

T̃n = Tn ◦ i and T̃ = T ◦ i. Then T̃n → T̃ uniformly for x in a bounded subset of
X1 (that is, in the or norm of B(X1, Y )).

Lemma 10.3. Suppose that η and p are such that (6.1) holds and f satisfies (10.1).
Then the Gateaux differential of F (u, ε), with respect to u is continuous in u, that
is, the map u 7→ ∂F

∂u (u, ε) ∈ B(Xη, X−1/2) is continuous.

Proof. Let un be a sequence converging to u em Xη, and choose 0 < η̃ < η, such
that the hypotheses still hold. Then, for any Φ ∈ X1/2 and w ∈ X η̃ we have∣∣〈(∂F

∂u
(un, ε)−

∂F

∂u
(u, ε)

)
w,Φ〉−1/2,1/2

∣∣



26 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

≤
∫

Ω

∣∣( f ′(u)− f ′(un)
)
wΦ
∣∣ dx

≤
(∫

Ω

∣∣(f ′(u)− f ′(un)
)
w
∣∣pdx)1/p(∫

Ω

|Φ
∣∣qdx)1/q

≤
(∫

Ω

∣∣(f ′(u)− f ′(un)
)
w
∣∣pdx︸ ︷︷ ︸

(I)

)1/p

‖Φ‖X1/2 .

Now, the integrand in (I) is bounded by the integrable function ‖f ′‖p∞wp and
goes to 0 a.e. as un → u in Xη. Therefore the sequence of operators ∂F

∂u (un, ε) con-

verges strongly in the space B(X η̃, X−1/2) to the operator ∂F
∂u (u, ε). From Lemma

10.2 the convergence holds in the norm of B(Xη, X−1/2), since Xη is compactly
embedded in X η̃. �

Lemma 10.4. Suppose that η and p are such that (6.1) holds and g satisfies (10.1).
Then the operator G : Xη×R→X−1/2 given by (5.4) is Gateaux differentiable with
respect to u, with Gateaux differential

〈∂G
∂u

(u, ε)w,Φ〉−1/2,1/2 =

∫
∂Ω

g′(γ(u))γ(w)γ(Φ)
∣∣J∂Ωhε
Jhε

∣∣ dσ(x) , (10.3)

for all w ∈ Xη and Φ ∈ X1/2.

Proof. Observe first that G(u, ε) is well-defined, since the conditions of Lemma 6.2
are met. It is clear that ∂G

∂u (u, ε) is linear. We now show that it is bounded. In

fact, for all u,w ∈ Xη and Φ ∈ X1/2, we have∣∣〈∂G
∂u

(u, ε)w,Φ〉−1/2,1/2

∣∣ =
∣∣ ∫
∂Ω

g′(γ(u))γ(w)γ(Φ)
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)
∣∣

≤ ‖µ‖∞‖g′‖∞
∫
∂Ω

|γ(w)| |γ(Φ)| dσ(x)

≤ ‖µ‖∞‖g′‖∞‖γ(w)‖Lp(∂Ω)‖γ(Φ)‖Lq(∂Ω)

≤ K1K2‖µ‖∞‖g′‖∞‖w‖η‖Φ‖X1/2 ,

where ‖g′‖∞ = sup{g′(x) : x ∈ R}, ‖µ‖∞ = sup{|µ(x, ε)| : x ∈ ∂Ω} = sup{J∂Ωhε
Jhε

(x) :

x ∈ ∂Ω} and K1, K2 are embedding constants. This proves boundedness.
Now, for all u,w ∈ Xη and Φ ∈ X1/2, we have∣∣1
t
〈G(u+ tw, ε)−G(u, ε)− t∂G

∂u
(u, ε)w,Φ〉−1/2,1/2

∣∣
≤ 1

|t|

∫
∂Ω

|[g(γ(u+ tw))− g(γ(u))− tg′(γ(u))]γ(w)| |γ(Φ)|
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)

≤ K1‖µ‖∞
1

|t|

{∫
∂Ω

|[g(γ(u+ tw))− g(γ(u))− tg′(γ(u))]γ(w)|p dσ(x)
}1/p

‖Φ‖X1/2

≤ K1‖µ‖∞
{∫

∂Ω

|[ g′(γ(u+ t̄w))− g′(γ(u))]γ(w)|p dσ(x)︸ ︷︷ ︸
(I)

}1/2

‖Φ‖X1/2 ,

where K1 is an embedding constant given by Trace Theorem and 0 ≤ t̄ ≤ t.
Since g′ is bounded and continuous, the integrand of (I) is bounded by an in-
tegrable function and goes to 0 as t → 0. Thus, the integral (I) goes to 0



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 27

as t → 0, from Lebesgue’s Dominated Convergence Theorem. It follows that

limt→0
G(u+tw,ε)−G(u,ε)

t = ∂G
∂u (u, ε)w in X−1/2, for all u,w ∈ Xη; so G is Gateaux

differentiable with Gateaux differential given by (10.3). �

Lemma 10.5. Suppose that η and p are such that (6.1) holds and g satisfies (10.1).
Then the Gateaux differential of G(u, ε), with respect to u is continuous in u (that
is, the map u 7→ ∂G

∂u (u, ε) ∈ B(Xη, X−1/2) is continuous) and uniformly continuous
in ε for u in bounded sets of Xη and 0 ≤ ε ≤ ε0 < 1.

Proof. Let 0 ≤ ε ≤ ε0, un be a sequence converging to u em Xη, and choose
0 < η̃ < η, still satisfying the hypotheses. Then, for any Φ ∈ X1/2 and w ∈ X η̃, we
have∣∣〈(∂G

∂u
(un, ε)−

∂G

∂u
(u, ε)

)
w,Φ〉−1/2,1/2

∣∣
≤
∫
∂Ω

|(g′(γ(u))− g′(γ(un)))γ(w)γ(Φ)|
∣∣J∂Ωhε
Jhε

∣∣ dσ(x)

≤ ‖µε‖∞
{∫

∂Ω

|(g′(γ(u))− g′(γ(un))γ(w)|p dσ(x)
}1/p{∫

∂Ω

|γ(Φ)|q dσ(x)
}1/q

≤ K1‖µε‖∞
{∫

∂Ω

|(g′(γ(u))− g′(γ(un))γ(w)|p dσ(x)︸ ︷︷ ︸
(I)

}1/p

‖Φ‖X1/2 ,

where K1 is the constant for continuity of the trace map from X1/2 into L2(∂Ω),
as in Lemma 6.2.

Now, the integrand in (I) is bounded by the integrable function ‖g′‖2∞|γ(w)|2
and approaches 0 a.e. as un → u in Xη. Therefore the sequence of operators
∂G
∂u (un, ε) converges strongly in the space B(X η̃, X−1/2) to the operator ∂G

∂u (u, ε).

From Lemma 10.2 the convergence holds in the norm of B(Xη, X−1/2), since Xη is
compactly embedded in X η̃ (see [10]).

Finally, if 0 ≤ ε1 ≤ ε2 < ε0, for any Φ ∈ X1/2 and w ∈ Xη, we have∣∣〈(∂G
∂u

(u, ε1)− ∂G

∂u
(u, ε2)

)
w,Φ〉−1/2,1/2

∣∣
≤
∫
∂Ω

|g′(γ(u))γ(w)γ(Φ)| |µε1 − µε2 | dσ(x)

≤ ‖µε1 − µε2‖∞
{∫

∂Ω

|g′(γ(u))γ(w)|p dσ(x)
}1/p{∫

∂Ω

|γ(Φ)|q dσ(x)
}1/q

≤ K1K2‖g′‖∞‖‖w‖Xη‖Φ‖X1/2‖µε1 − µε2‖∞,

where K2 is the constant for the continuity of the trace map from Xη into Lq(∂Ω),
as before. This proves uniform continuity in ε. �

Lemma 10.6. Suppose that η and p are such that (6.1) holds and f and g satisfy
(10.1). Then the map (Hε)−

1
2 = (F ε)−

1
2 + (Gε)−

1
2 : Xη × R 7→ X−1/2 given

by (5.2) is continuously Fréchet differentiable with respect to u and the derivative
∂G
∂u is uniformly continuous with respect to ε, for u in bounded sets of Xη and
0 ≤ ε ≤ ε0 < 1.

The above lemma follows from Lemmas 10.3, 10.5 and [17, Proposition 2.8]. We
now prove lower semicontinuity for the equilibria.



28 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

Theorem 10.7. If f and g satisfy the conditions of Theorem 7.5 and (10.1), then
the equilibria of (5.1) with ε = 0 are all hyperbolic and 1/4 < η < 1/2, then the
family of sets of equilibria {Eε : 0 ≤ ε < ε0} of (5.1) is lower semicontinuous in
Xη at ε = 0.

Proof. A point e ∈ Xη is an equilibrium of (5.1) if and only if it is a root of the
map Z : W 1,p(Ω)× R→ X−1/2 given by

(u, ε) 7→ (Aε)−1/2(u) + (Hε)−1/2(u) .

By Lemma 10.6 the map (Hε)−1/2 : Xη → X−1/2 is continuously Fréchet differ-
entiable with respect to u and by Lemmas 6.3 and 6.1 it is also continuous in ε if
η = 1

2 − δ, with δ > 0 is sufficiently small. Therefore, the same holds if η = 1/2.

The map Aε = −h∗ε∆Ωεh
∗
ε + aI is a bounded linear operator from W 1,p(Ω) to

X−1/2. It is also continuous in ε since it is analytic as a function of hε ∈ Diff1(Ω)
and hε is continuous in ε.

Thus, the map Z is continuously differentiable in u and continuous in ε. The
derivative of ∂Z

∂u (e, 0) is an isomorphism by hypotheses. Therefore, the Implicit
Function Theorem apply, implying that the zeroes of Z(·, ε) are given by a contin-
uous function e(ε). This proves the claim. �

To prove the lower semi continuity of the attractors, we also need the continuity
of local unstable manifolds at equilibria.

Theorem 10.8. Suppose that η and p are such that (6.1) holds and f and g satisfy
(10.1), u0 is an equilibrium of (5.1) with ε = 0, and for each ε > 0 sufficiently small,
let uε be the unique equilibrium of (5.1), whose existence is asserted by Corollary
9.4 and Theorem 10.7. Then, for ε and δ sufficiently small, there exists a local
unstable manifold Wu

loc(uε) of uε, and if we denote Wu
δ (uε) = {w ∈ Wu

loc(uε) :
‖w − uε‖η < δ}, then

−1

2

(
Wu
δ (uε),W

u
δ (u0)

)
and − 1

2

(
Wu
δ (u0),Wu

δ (uε)
)

approach zero as ε→ 0, where − 1
2 (O,Q) = supo∈O infq∈Q ‖q−o‖Xη for O, Q ⊂ Xη.

Proof. Let Hε(u) = H(u, ε) be the map defined by (5.2) and uε a hyperbolic equi-
librium of (5.1). Since H(u, ε) is differentiable by Lemma 10.6, it follows that

Hε(uε + w, ε) = Hε(uε, ε) +Hu(uε, ε)w + r(w, ε)

= Aεuε +Hu(uε, ε)w + r(w, ε),

with r(w, ε) = o(‖w‖Xη ), as ‖w‖Xη → 0. The claimed was proved in [16], assuming
the following properties of Hε:

(a) ‖r(w, 0)− r(w, ε)‖X−1/2 ≤ C(ε), with C(ε)→ 0 when ε→ 0, uniformly for
w in a neighborhood of 0 in Xη.

(b) ‖r(w1, ε) − r(w2, ε)‖X−1/2 ≤ k(ρ)‖w1 − w2‖η, for ‖w1‖η ≤ ρ, ‖w2‖η ≤ ρ,
with k(ρ)→ 0 when ρ→ 0+ and k(∗) is non decreasing.

Property (a) follows easily from the fact that both H(u, ε) and Hu(u, ε) are
uniformly continuous in ε for u in bounded sets of Xη, by Lemmas 6.3, 6.1 and
10.6.

It remains to prove property (b). If w1, w2 ∈ Xη and ε ∈ [0, ε0], with 0 < ε0 < 1
small enough, we have

‖r(w1, ε)− r(w2, ε)‖X−1/2



EJDE-2020/97 CONTINUITY OF ATTRACTORS FOR C1 PERTURBATIONS 29

= ‖H(uε + w1, ε)−H(uε, ε)−Hu(uε, ε)w1

−H(uε + w2, ε) +Hε(uε, ε) +Hu(uε, ε)w2‖X−1/2

≤ ‖F (uε + w1, ε)− F (uε, ε)− Fu(uε, ε)w1 (10.4)

− F (uε + w2, ε) + F (uε, ε) + Fu(uε, ε)w2‖X−1/2

+ ‖G(uε + w1, ε)−G(uε, ε)−Gu(uε, ε)w1 (10.5)

−G(uε + w2, ε) +G(uε, ε) +Gu(uε, ε)w2‖X−1/2 .

We first estimate (10.4). Since f ′ is bounded by (10.1), we have∣∣∣〈F (uε + w1, ε)− F (uε, ε)− Fu(uε, ε)w1 − F (uε + w2, ε)

+ F (uε, ε) + Fu(uε, ε)w2,Φ〉−1/2,1/2

∣∣∣
≤
∫

Ω

|[f(uε + w1)− f(uε)− f ′(uε)w1 − f(uε + w2) + f(uε) + f ′(uε)w2]Φ| dx

=

∫
Ω

|[f ′(uε + ξx)− f ′(uε)](w1(x)− w2(x))Φ| dx

≤ K1

{∫
Ω

|[f ′(uε + ξx)− f ′(uε)]p(w1(x)− w2(x))p| dx
}1/p

‖Φ‖X1/2 ,

≤ K1K2

{∫
Ω

|[f ′(uε + ξx)− f ′(uε)]p| dx
}1/p

‖w1 − w2‖Xη‖Φ‖X1/2 ,

where K1 is the embedding constant of X1/2 into Lq(Ω), K2 is the embedding
constant of Xη in L∞(Ω) and w1(x) ≤ ξx ≤ w2(x) or w2(x) ≤ ξx ≤ w1(x).
Therefore,

‖F (uε + w1, ε)− F (uε, ε)− Fu(uε, ε)w1 − F (uε + w2, ε)

+ F (uε, ε) + Fu(uε, ε)w2‖X−1/2

≤ K1K2

{∫
Ω

[f ′(uε + ξx)− f ′(uε)]pdx
}1/p

‖w1 − w2‖η .

Now the integrand above is bounded by 2p‖f ′‖2p∞ and approaches 0 a.e. as ρ → 0,
since ‖w1‖η ≤ ρ, ‖w2‖η ≤ ρ and w1(x) ≤ ξx ≤ w2(x). Thus, the integral approaches
0 by Lebesgue’s bounded convergence Theorem.

We now estimate (10.5):∣∣∣〈G(uε + w1, ε)−G(uε, ε)−Gu(uε, ε)w1 −G(uε + w2, ε) +G(uε, ε)

+Gu(uε, ε)w2,Φ〉−1/2,1/2

∣∣∣
≤
∫
∂Ω

∣∣∣[g(γ(uε + w1))− g(γ(uε))− g′(γ(uε))w1

− g(γ(uε + w2)) + g(γ(uε)) + g′(γ(uε))w2]γ(Φ)γ
(∣∣J∂Ωhε

Jhε

∣∣)∣∣∣ dσ(x)

=

∫
∂Ω

∣∣∣[g′(γ(uε + ξx))− g′(γ(uε))]γ(w1(x)− w2(x))γ(Φ)γ
(∣∣J∂Ωhε

Jhε

∣∣)∣∣∣ dσ(x)

≤ K1

{∫
∂Ω

[ (g′(γ(uε + ξx))− g′(γ(uε)))]
p[γ(w1(x)− w2(x))]p



30 P. S. BARBOSA, A. L. PEREIRA EJDE-2020/97

×
[
γ
(∣∣J∂Ωhε

Jhε

∣∣)]pdσ(x)
}1/p

‖Φ‖X1/2

≤ K1K2µε

{∫
∂Ω

[(g′(γ(uε + ξx))− g′(γ(uε)))]
pdσ(x)

}1/p

‖w1 − w2‖Xη‖Φ‖X1/2 ,

where µε = |J∂Ωhε
Jhε
| is bounded, uniformly in ε and w1(x) ≤ ξx ≤ w2(x) or w2(x) ≤

ξx ≤ w1(x).
Now the integrand above is bounded by 2p‖g′‖ p∞‖ and approaches o 0 a.e. as

ρ → 0, since ‖w1‖η ≤ ρ, ‖w2‖η ≤ ρ and w1(x) ≤ ξx ≤ w2(x). Thus, the integral
approaches 0 by Lebesgue’s dominated convergence Theorem. �

We are now in a position to prove the main result of this section.

Theorem 10.9. Assume the hypotheses of Theorem 10.7 hold. Then the family
of attractors {Aε : 0 ≤ ε ≤ ε0}, of problem (5.1), whose existence is guaranteed by
Theorem 8.2 is lower semicontinuous in Xη.

Proof. The system generated by (5.1) is gradient for any ε and its equilibria are
all hyperbolic for ε in a neighborhood of 0. Also, the equilibria are continuous in ε
by Theorem 10.7, the linearization is continuous in ε as shown during the proof of
Theorem 10.7 and the local unstable manifolds of the equilibria are continuous in
ε, by Theorem 10.8. The result follows then from [16, Theorem 3.10]. �

Acknowledgments. A. L. Pereira was partially supported by FAPESP-Brazil
grant 2016/02150-8.

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10.1007/978-1-4612-0645-3.

Pricila S. Barbosa
Universidade Tecnológica Federal do Paraná, Paraná, Brazil

Email address: pricilabarbosa@utfpr.edu.br

Antônio L. Pereira

Instituto de Matemática e Estat́ıstica, Universidade de São Paulo, São Paulo, Brazil

Email address: alpereir@ime.usp.br


	1. Introduction
	2. Reduction to a fixed domain
	3. Basic assumptions and examples on domain perturbations
	4. Linear semigroup
	4.1. Strong form in Lp spaces
	4.2. Weak form in Lp spaces
	4.3. Existence and continuity of the linear semigroup

	5. The abstract problem in a scale of Banach spaces
	6. Local well-posedness
	7. Global existence and boundedness of the semigroup
	8. Existence of global attractors
	9. Upper semicontinuity of the family of global attractors
	10. Lower semicontinuity
	Acknowledgments

	References