Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 11, pp. 1–20.

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

DOI: 10.58997/ejde.2025.11

LONG-TIME DYNAMICS AND UPPER-SEMICONTINUITY OF

ATTRACTORS FOR A POROUS-ELASTIC SYSTEM WITH

NONLINEAR LOCALIZED DAMPING

MAURO L. SANTOS, MIRELSON M. FREITAS, RONAL Q. CALJARO

Abstract. In this article we consider a one-dimensional porous-elastic sys-
tem with nonlinear localized damping acting in an arbitrarily small region of

the interval under consideration. We prove the existence of a smooth global

attractor with finite fractal dimension and the existence of exponential attrac-
tors via quasi-stability theory recently proposed by Chueshov and Lasiecka.

We also prove the continuity of the attractors with respect to two parameters

in a residual dense set. Finally, we prove that the family of global attractors is
upper-semicontinuous with respect to small perturbations of external forces.

These aspects were not previously considered for porous-elastic system with
localized damping.

1. Introduction

The study of mathematical models of vibrating flexible structures have been
considerably stimulated in recent years by an increasing number of questions of
practical concern. Research on stabilization of distributed parameter systems has
largely focused on the stabilization of dynamic models of individual structural mem-
bers such as strings, membranes, and beams. See [15] and references therein.

On the other hand, localized frictional damping has been studied by several
authors in one or more space dimension, (see [3, 5, 13, 21, 33, 34]). The main result
of the above articles is that localized frictional damping produces exponential decay
in time of the solution. A more general result occurs in one-dimensional space where
the solution always decays exponentially to zero for any localized frictional damping
active over an open subset of the domain.

Motivated by the above, this article is devoted to the study of the porous-elastic
system with nonlinear arbitrary localized elastic damping and nonlinear arbitrary

2020 Mathematics Subject Classification. 35B40, 35B41, 37L30, 35L75.
Key words and phrases. Porous-elastic system; nonlinear localized damping; quasi-stability;

global attractor; upper-semicontinuity.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted April 12, 2024. Published February 3, 2025.

1



2 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

localized porous dissipation given by

ρutt − µuxx − bϕx + a1(x)g1(ut) + f1(u, ϕ) = ϵ1h1 in (0, L)× (0,∞),

Jϕtt − δϕxx + bux + ξϕ+ a2(x)g2(ϕt) + f2(u, ϕ) = ϵ2h2 in (0, L)× (0,∞),

u(0, t) = u(L, t) = ϕ(0, t) = ϕ(L, t) = 0, t > 0,

(u(x, 0), ϕ(x, 0)) = (u0(x), ϕ0(x)), in (0, L),

(ut(x, 0), ϕt(x, 0)) = (u1(x), ϕ1(x)), in (0, L),

(1.1)

where the variables u and ϕ represent the displacement of a solid elastic material
and the volume fraction, respectively. Here ρ, µ, J , δ, b and ξ are the constitutive
coefficients whose physical meaning is well known. The constitutive coefficients, in
one-dimensional case, satisfy

ξ > 0, δ > 0, µ > 0, ρ > 0, J > 0, µξ ≥ b2. (1.2)

The functions g1(ut) and g2(ϕt) represent the nonlinear damping terms, ϵ1 and ϵ2
are positive constants small enough, f1 and f2 are nonlinear source terms, a1 and
a2 are smooth, nonnegative functions responsible by the localized damping effect,
h1 and h2 represent external forces.

Quintanilla [25] studied the system (1.1) when ϵ1 = ϵ2 = 0, g1 = 0 and g2(s) = τs
with a2(x) = 1. He used the Hurtwitz theorem to prove that the system lacks
exponential decay when ρ

µ ̸= J
δ . In Magaña and Quintanilla [20] considered the

system:

ρutt − µuxx − bϕx − γuxxt = 0 in (0, L)× (0,∞),

Jϕtt − δϕxx + bux + ξϕ+ τϕt = 0 in (0, L)× (0,∞),

u(0, t) = u(L, t) = ϕx(0, t) = ϕx(L, t) = 0, t > 0,

(u(x, 0), ϕ(x, 0)) = (u0(x), ϕ0(x)), in (0, L)

(ut(x, 0), ϕt(x, 0)) = (u1(x), ϕ1(x)), in (0, L).

(1.3)

They proved that the system (1.3) is exponentially stable using the semigroup
arguments due to Liu and Zheng [16]. Also, they proved that when τ = 0 the
system is not exponentially stable. Muñoz Rivera and Quintanilla [22] proved that
when τ = 0 the energy is controlled by a rate decay of the type 1

t . Moreover, using
a result on [24], they improved the polynomial rate of decay by taking more regular
initial data. Santos et al. [28] proved that system (1.3) with τ = 0 lacks exponential
decay independent of any relation between the coefficients of the wave propagation,
and it decays as 1√

t
. In addition they also proved that this rate is optimal. On the

other hand, Santos and Almeida Júnior [27] studied the porous-elastic system

ρutt − µuxx − bϕx + γ(x)(ut + ϕt) = 0 in Ω× (0,∞),

Jϕtt − δϕxx + bux + ξϕ+ γ(x)(ut + ϕt) = 0 in Ω× (0,∞),

(u(x, 0), ϕ(x, 0)) = (u0(x), ϕ0(x)), in Ω,

(ut(x, 0), ϕt(x, 0)) = (u1(x), ϕ1(x)), in Ω

(1.4)

where the localized damping involves the sum of displacement velocity of a solid
elastic material and the volume fraction velocity. Note that, Ω = (0, L) and ω =
(L1, L2) with 0 ≤ L1 < L2 ≤ L and γ ∈ L∞(Ω) is a nonnegative function satisfying

∃ γ0 > 0; γ(x) ≥ γ0, a.e. x ∈ ω. (1.5)



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 3

The main contribution in [27] has been providing a necessary and sufficient condi-
tion for the strong stability and the exponential decay of the porous-elastic system
with the rank-one localized damping where the boundary of the damping region
must contain at least one of the end points of the spatial domain. Other problems
associated with porous elastic systems can be found in references [23, 30, 31, 32].

Feireisl and Zuazua [8] proved the existence of the global attractor with critical
semilinear term. The finite fractal dimension and regularity of global attractors
for the critical case has been considered by Chueshov, Lasiecka and Toundykov
[6]. Finally, very recently, Ma and Huertas [17] proved the continuity of attractors
with respect to a parameter forcing in a residual dense set and the existence of
generalized exponential attractors.

In [9], the long-time behavior of porous-elastic systems with nonlinear damp-
ing and source terms was investigated for the first time. Considering two globally
defined nonlinear dampings and arbitrary source terms, the authors show the exis-
tence of local and global mild solutions, uniqueness of mild solutions, and continuous
dependence of initial data. Under some restrictions on the parameters, they also
proved that every mild solution to system blows up in finite time, provided the ini-
tial energy is negative and the sources are more dominant than the damping in the
system. Additional results are obtained via potential well theory. They proved the
existence of a unique global mild solution with initial data coming from the ”good”
part of the potential well. For such a global solution, we prove that the total energy
of the system decays exponentially or algebraically, depending on the behavior of
the dissipation in the system near the origin. To our knowledge, the study of global
attractors for porous-elastic systems with nonlinear localized damping has not been
discussed in the literature. This paper aims to fill this gap.

The purpose of this article is to obtain the existence and upper-semicontinuity
of a global attractor for porous-elastic systems subject to a nonlinear localized
damping and nonlinear source terms placed in both equations, with a minimal
support for the damping. The contributions of the paper are:

(i) The existence of attractors with finite fractal dimension using quasi-stability
methods by Chueshov and Lasiecka [7]. Observe that the present result was not
previously considered for porous-elastic systems with nonlinear localized damping
and nonlinear source terms,

(ii) Stability estimates (see Theorem 4.3) independent of ϵ1 and ϵ2. The standard
multipliers method leads to terms of the energy level which cannot be directly
absorbed (this is not the case when one of the damping functions is supported on
the entire domain). In order to handle this, special weight functions are introduced,
which eliminate undesirable terms of higher order while contributing lower-order
terms,

(iii) The continuity of global attractors, containing residual continuity and upper
semicontinuity with respect to the parameters ϵi ∈ [0, 1], i = 1, 2.

This article is organized as follows: Section 2 presents assumptions, notations
and well-posedness results. In Section 3, we summarize the main results. Section
4 is devoted to prove the existence of attractors and their properties. In the first
subsection we prove that the system is gradient by using a unique continuation
property proposed by Ma et al. [18]. The second subsection is devoted to prove the
stabilizability inequality and quasi-stability of the system. In the third subsection,
we prove the Theorem 3.1. More precisely, the existence of finite fractal global



4 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

attractors with smoothness properties and the existence of a generalized fractal
exponential attractor. In the last subsection the upper-semicontinuity of attractor
with respect to α := (ϵ1, ϵ2) is proved (see Theorem 5.4).

2. Assumptions and Preliminary Results

We use throughout this paper the standard Lebesgue spaces Lp(0, L), p ≥ 1,
with the norm denoted by ∥ · ∥p. We denote by ⟨·, ·⟩ the inner product in L2(0, L).
Let us consider the Hilbert spaces

H := H1
0 (0, L)×H1

0 (0, L)× L2(0, L)× L2(0, L),

V := H2(0, L) ∩H1
0 (0, L)×H2(0, L) ∩H1

0 (0, L)×H1
0 (0, L)×H1

0 (0, L)
(2.1)

with inner product in H given by

⟨U, V ⟩H := ρ⟨φ,Φ⟩+ J⟨ψ,Ψ⟩+ µ⟨ux, vx⟩+ δ⟨ϕx, wx⟩+ ξ⟨ϕ,w⟩
+ b⟨ux, w⟩+ b⟨ϕ, vx⟩.

(2.2)

for U = (u, ϕ, φ, ψ), V = (v, w,Φ,Ψ) ∈ H.

Remark 2.1. Since, by hypothesis µξ ≥ b2, using the same ideas in Raposo et al.
[26] we see that (2.2) defines an inner product on H and that the associated norm
∥ · ∥H is equivalent to the usual one. In particular, there exists γ0 > 0 such that

∥ux∥22 + ∥ϕx∥22 ≤ γ0
(
µ∥ux∥22 + δ∥ϕx∥22 + ξ∥ϕ∥22 + 2b⟨ux, ϕ⟩

)
. (2.3)

Using the Poincaré’s inequality and (2.3), there exists a constant γ1 > 0 such that

∥u∥22 + ∥ϕ∥22 ≤ γ1
(
µ∥ux∥22 + δ∥ϕx∥22 + ξ∥ϕ∥22 + 2b⟨ux, ϕ⟩

)
. (2.4)

If we denote z = (u, ϕ, ut, ϕt) and z0 = (u0, ϕ0, u1, ϕ1) then system (1.1) can be
rewritten as

dz

dt
= (A+ B)z + F(z), for t > 0,

z(0) = z0 ∈ H,
(2.5)

where

A(u, ϕ, φ, ψ) =
(
φ,ψ,

µ

ρ
uxx +

b

ρ
ϕx,

δ

J
ϕxx − b

J
ux − ξ

J
ϕ
)
,

for (u, ϕ, φ, ψ) ∈ D(A) = V;

B(u, ϕ, φ, ψ) =
(
0, 0,−1

ρ
a1(x)g1(φ),

1

J
a2(x)g2(ψ)

)
, for (u, ϕ, φ, ψ) ∈ H,

F(u, ϕ, φ, ψ) =
(
0, 0,

1

ρ

(
ϵ1h1 − f1(u, ϕ)

)
,
1

J

(
ϵ2h2 − f2(u, ϕ)

))
,

for (u, ϕ, φ, ψ) ∈ H.

We are ready to state the result about the existence of solutions. To this end we
introduce the following assumptions:

(i) There exists a function F ∈ C2(R2) such that

∇F = (f1, f2), (2.6)

and for i = 1, 2:

|∇fi(u, v)| ≤ β0
(
1 + |u|θ−1 + |v|θ−1

)
, ∀u, v ∈ R, (2.7)



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 5

with fi(0, 0) = 0, β0 > 0 and θ ≥ 1. Moreover, we assume that there exist
constants β1 ≥ 0 and mF > 0 such that

F (u, v) ≥ −β1(|u|2 + |v|2)−mF , ∀u, v ∈ R, (2.8)

∇F (u, v) · (u, v)− F (u, v) ≥ −β1(|u|2 + |v|2)−mF , ∀u, v ∈ R, (2.9)

where 0 ≤ β1 <
1

2γ1
.

(ii) The functions gi ∈ C1(R), i = 1, 2, are monotonically increasing with
gi(0) = 0 and there exist constant mi,Mi > 0 such that

mi ≤ g′i(s) ≤Mi, ∀s ∈ R. (2.10)

(iii) The functions ai ∈ C∞(0, L), i = 1, 2, are nonnegative and satisfy

ai(x) ≥ ai > 0 in Ii, i = 1, 2, and (α1, α2) = I1 ∩ I2 ̸= ∅. (2.11)

where I1, I2 are open intervals contained in [0, L].
(iv) The external forces h1, h2 belong to L2(0, L).

Observe that (2.10) implies the monotonicity property, i.e.

(gi(u)− gi(v))(u− v) ≥ mi|u− v|2, ∀u, v ∈ R. (2.12)

Remark 2.2. The localizing functions allows us to consider damping mechanisms
acting in an arbitrarily small region of the string.

Theorem 2.3. If (i)–(iii) hold, then:
(a) If initial data z0 ∈ H, then (2.5) has a unique mild solution z(t) ∈ C([0,∞),H),
with z(0) = z0, given by

z(t) = e(A+B)tz0 +

∫ t

0

e(t−τ)(A+B)F(z(τ))dτ.

(b) If z1(t) and z2(t) are two mild solutions of (2.5) then there exists a positive
constant C0 = C(z1(0), z2(0)), such that

∥z1(t)− z2(t)∥H ≤ eC0T ∥z1(0)− z2(0)∥H, ∀t ∈ [0, T ]. (2.13)

Proof. It is easy to see that the operator A + B is a maximal monotone operator.
In addition, by (2.7), F is a locally Lipschitz continuous on H. Therefore, applying
the theory of maximal nonlinear monotone operators (see [2, 4]) items(a)-(b) follow.
The continuous dependence (b) is also obtained by using standard computations in
the difference of solutions. □

Next result gives us a relation between mild and strong solutions for (2.5). It
says that every mild solution can be obtained as limit of strong solutions.

Lemma 2.4. Let z0 = (u0, ϕ0, u1, ϕ1) ∈ H be given and z = (u, ϕ, ut, ϕt) ∈
C(R+;H) the respective mild solution of (2.5). Then, there exist a sequence of
strong solutions {zn} of (2.5), such that

lim
n→∞

zn = z in C(R+;H).

Hence the mild solution is a strong solution.

Proof. Given z0 ∈ H, we take a sequence of initial data z0n ∈ D(A) such that
z0n → z0 in H. The difference wn(t) = zn(t)− z(t) can be estimated as

∥wn(t)∥ ≤ ∥et(A+B)(z0n − z0)∥+ L

∫ t

0

∥wn(τ)∥ dτ.



6 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

By Gronwall’s lemma, wn(t) → 0 uniformly in t ∈ R+, hence zn(t) → z(t) in
C(R+;H). The proof is complete. □

The following lemma shows the dissipative property of system (1.1).

Lemma 2.5. The energy functional associated with the strong solution of system
(1.1) satisfies

d

dt
E(t) = −

∫ L

0

(a1(x)g1(ut)ut + a2(x)g2(ϕt)ϕt) dx ≤ 0, ∀t > 0, (2.14)

where

E(t) = E(t) +

∫ L

0

F (u, ϕ) dx−
∫ L

0

(ϵ1h1u+ ϵ2h2ϕ) dx,

E(t) =
1

2
∥(u, ϕ, ut, ϕt)∥2H .

Moreover, there exist positive constants C0, C1 independent of ϵ1 and ϵ2 such that

C0∥(u, ϕ, ut, ϕt)∥2H − C1 ≤ E(t) ≤ C2

(
1 + ∥(u, ϕ, ut, ϕt)∥θ+1

H
)
, ∀t ≥ 0. (2.15)

Proof. A straightforward computation yields (2.14) by multiplying the first and
second equations in (1.1) by ut and ϕt, respectively. It follows from (2.8) and (2.4)
that ∫ L

0

F (u, ϕ) dx ≥ −β1(∥u∥22 + ∥ϕ∥22)− LmF ≥ −β1γ1∥z∥2H − LmF ,

and therefore,

E(t) ≥
(1
2
− β1γ1

)
∥(u, ϕ, ut, ϕt)∥2H − LmF −

∫ L

0

(ϵ1h1u+ ϵ2h2ϕ) dx.

Now letting

C0 =
1

4

(
1− 2β1γ1

)
> 0, (2.16)

and using the estimate∫ L

0

(ϵ1h1u+ ϵ2h2ϕ) dx ≤ C0

γ1

(
∥u∥22 + ∥ϕ∥22

)
+

γ1
4C0

(
∥h1∥22 + ∥h2∥22

)
, (2.17)

the first (or left) inequality in (2.15) is obtained with

C1 = LmF +
γ1
4C0

(
∥h1∥22 + ∥h2∥22

)
.

Now, using the embedding H1
0 (0, L) ↪→ L∞(0, L) and (2.7), we deduce that∫ L

0

F (u, ϕ) dx ≤ C2(1 + ∥ux∥θ+1
2 + ∥ϕx∥θ+1

2 ).

So, using this estimative, we have

E(t) ≤ C2∥(u, ϕ, ut, ϕt)∥θ+1
H + C2(1 + ∥(u, ϕ, ut, ϕt)∥θ+1

H )

This implies the second inequality in (2.15) holds. The proof is complete. □



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 7

3. Main results

First, we observe that the system (1.1) defines a dynamical system (H, Sα(t)),
where H is given in (2.1), and Sα(t) : H → H is the strongly continuous semigroup
given by

Sα(t)z0 = (u(t), ϕ(t), ut(t), ϕt(t)) t ≥ 0. (3.1)

where (u(t), ϕ(t), ut(t), ϕt(t)) is the unique mild solution of the system (1.1) with
the initial data z0 = (u0, ϕ0, u1, ϕ1) ∈ H and α = (ϵ1, ϵ2) ∈ Λ = [0, 1]× [0, 1].

The main result for long-time dynamics is given by the following theorem whose
proof will be provided in the next section.

Theorem 3.1. Suppose that assumptions of Theorem 2.3 hold and α = (ϵ1, ϵ2) ∈ Λ.
Then

(i) The dynamical system (H, Sα(t)) is quasi-stable (uniformly in α) on any
bounded positively invariant set B ⊂ H.

(ii) The dynamical system (H, Sα(t)) possesses a unique compact global attractor
Aα ⊂ H, which is characterized by the unstable manifold Aα = M+(Nα) of the set
of stationary solutions

Nα =
{
(u, ϕ, 0, 0) ∈ H : −µuxx − bϕx + f1(u, ϕ) = ϵ1h1

− δϕxx + bux + ξϕ+ f2(u, ϕ) = ϵ2h2
}

(iii) The dynamical system (H, Sα(t)) has a bounded absorbing set B independent
of α. In particular,

Aα ⊂ B, ∀α ∈ Λ.

(iv) The attractor Aα has finite fractal and Hausdorff dimension dimf
HAα.

(v) The global attractor Aα is bounded in

V = (H2(0, L) ∩H1
0 (0, L))

2 × (H1
0 (0, L))

2.

Moreover, every trajectory z = (u, ϕ, ut, ϕt) in Aα satisfies

∥(u, ϕ)∥2(H2∩H0)2
+ ∥(ut, ϕt)∥2(H1

0 )
2 + ∥(utt, ϕtt)∥2(L2)2 ≤ R2

1, (3.2)

for some constant R1 > 0 independent of α.
(vi) The dynamical system (H, Sα(t)) possesses a generalized fractal exponential

attractor. More precisely, for any δ ∈ (0, 1], there exists a generalized exponential

attractor Aexp
α,δ ⊂ H, with finite fractal dimension in the extended space H̃−δ, defined

as interpolation of

H̃0 := H, quadand H̃−1 := (L2(0, L))2 × (H−1(0, L))2.

4. Proofs of main results

4.1. Gradient system and stationary solutions. We recall that a dynamical
system (H,S(t)) is gradient if it possesses a strict Lyapunov functional. That is, a
functional Φ : H → R is a strict Lyapunov function for a system (H,S(t)) if,

(i) the map t→ Φ(S(t)z) is non-increasing for each z ∈ H,
(ii) if Φ(S(t)z) = Φ(z) for some z ∈ H and for all t, then z is a stationary point

of S(t), that is, S(t)z = z.



8 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

Lemma 4.1. Suppose that assumptions (i)–(iii) hold. Then the dynamical system
(H, Sα(t)) is gradient, that is, there exists a strict Lyapunov function Φ defined in
H. In addition,

Φ(z) → ∞ if and only if ∥z∥H → ∞. (4.1)

Proof. Let us define the function Φ : H → R by

Φ(Sα(t)z) =
1

2
∥(u(t), ϕ(t), ut(t), ϕt(t))∥2H +

∫ L

0

F (u(t), ϕ(t)) dx

−
∫ L

0

(ϵ1h1u+ ϵ2h2ϕ) dx.

(4.2)

From (2.14) we have

d

dt
Φ(Sα(t)z) = −

∫ L

0

(a1(x)g1(ut)ut + a2(x)g2(ϕt)ϕt) dx ≤ 0, ∀t ≥ 0, (4.3)

which shows that t 7→ Φ(Sα(t)z) is a non-increasing function.
Now suppose that Φ(Sα(t)z) = Φ(z) for all t ≥ 0. Then (4.3) implies that∫ L

0

(a1(x)g1(ut)ut + a2(x)g2(ϕt)ϕt) dx = 0, t ≥ 0.

Then using (2.11) and (2.10), we can deduce for all T > 0 that

ut = ϕt = 0 a.e. in (α1, α2)× (0, T ),

a1(x)g1(ut) = a2(x)g2(ϕt) = 0 a.e. in (0, L)× (0, T ).

This means that z(t) = (u(t), ϕ(t), ut(t), ϕt(t)) is a solution of

ρutt − µuxx − bϕx + f1(u, ϕ) = ϵ1h1 in (0, L)× (0, T ),

Jϕtt − δϕxx + bux + ξϕ+ f2(u, ϕ) = ϵ2h2 in (0, L)× (0, T ),

ut = ϕt = 0 in (α1, α2)× (0, T ).

(4.4)

Taking the derivative of (4.4) with respect to the variable t in distributional sense
and defining v = ut and w = ϕt yields

ρvtt − µvxx − bwx + p1(x, t)v + q1(x, t)w = 0 in (0, L)× (0, T ),

Jwtt − δwxx + bvx + ξw + p2(x, t)v + q2(x, t)w = 0 in (0, L)× (0, T ),

v = w = 0 in (α1, α2)× (0, T ).

(4.5)

where pi = ∂ufi(u, ϕ), qi = ∂vfi(u, ϕ) for i = 1, 2. From assumption (2.7) we can
deduce that pi, qi ∈ L2(0, T ;L2(0, L)). Using the unique continuation property in
[18, Theorem 3.2.]), we conclude that v = w = 0 in (0, L)× (0, T ). Therefore,

ut = ϕt = 0 in (0, L)× (0, T ).

Therefore z = (u0, ϕ0, 0, 0) is a stationary solution of Sα(t). This proves that Φ is
a strict Lyapunov function.

Now, by the second inequality in (2.15), we have

Φ(z) ≤ C2(1 + ∥z∥θ+1
H ).

Considering the last estimate and taking Φ(z) → +∞ we have ∥z∥H → +∞. On
the other hand, by the first inequality in (2.15) we obtain

∥z∥2H ≤ 1

C0
(Φ(z) + C1) ,



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 9

from here we conclude that ∥z∥H → +∞ implies Φ(z) → +∞, proving (4.1). The
proof is complete. □

Lemma 4.2. Suppose that assumptions (i)–(iii) hold. Then the set Nα of the
stationary points of (H, Sα(t)) is bounded in H uniformly in α ∈ Λ.

Proof. Let z ∈ Nα be arbitrary. We know that z = (u, ϕ, 0, 0) and z satisfies the
system

−µuxx − bϕx + f1(u, ϕ) = ϵ1h1,

−δϕxx + bux + ξϕ+ f2(u, ϕ) = ϵ2h2.
(4.6)

Multiplying the first equation in (4.6) by u and the second by ϕ, respectively, taking
the sum and integrating over (0, L), we obtain

µ∥ux∥22 + δ∥ϕx∥22 + ξ∥ϕ∥22 + 2b⟨ux, ϕ⟩

= −
∫ L

0

∇F (u, ϕ) · (u, ϕ) dx+

∫ L

0

(ϵ1h1u+ ϵ2h2ϕ) dx.
(4.7)

Hence, using (2.4), (2.8), and (2.9), we obtain

−
∫ L

0

∇F (u, ϕ) · (u, ϕ) dx

≤ 2β1γ1
(
µ∥ux∥22 + δ∥ϕx∥22 + ξ∥ϕ∥22 + 2b⟨ux, ϕ⟩

)
+ 2LmF .

(4.8)

Combining (4.7) and (4.8) on account of (2.16) yields

4C0

(
µ∥ux∥22 + δ∥ϕx∥22 + ξ∥ϕ∥22 + 2b⟨ux, ϕ⟩

)
≤ 2LmF +

∫ L

0

(ϵ1h1u+ ϵ2h2ϕ) dx.
(4.9)

Hence, using the estimate (2.17), we deduce

3C0∥z∥2H ≤ 2mFL+
γ1
4C0

(
∥h1∥22 + ∥h2∥22

)
, (4.10)

which shows that the set Nα is bounded in H uniformly in α ∈ Λ. The proof is
complete. □

4.2. Uniform stabilizability inequality. The following theorem plays an im-
portant role to prove the existence of a global attractor and its properties. We
usually call it the stabilizability estimate. An important fact is that this estimate
is independent of the parameter α = (ϵ1, ϵ2) ∈ Λ = [0, 1]× [0, 1].

Theorem 4.3. Suppose that assumptions (i)–(iii) hold. Let B ⊂ H be a bounded
positively invariant set and let Sα(t)z

i = (ui(t), ϕi(t), uit(t), ϕ
i
t(t)), i = 1, 2, be

mild solutions of (1.1) with initial conditions zi ∈ B. Then, there exist constants
ϑB , ηB , CB > 0, depending on B yet independent of α, such that

E(t) ≤ ϑBE(0)e−ηBt + CB sup
s∈[0,t]

(
∥u(s)∥22θ + ∥ϕ(s)∥22θ

)
, (4.11)

for all t ≥ 0, where u = u1 − u2 and ϕ = ϕ1 − ϕ2.

Proof. For u = u1 − u2 and ϕ = ϕ1 − ϕ2, the following notation is adopted

Fi(u, ϕ) = fi(u
1, ϕ1)− fi(u

2, ϕ2), G1(ut) = g1(u
1
t )− g1(u

2
t ),

G2(ϕt) = g2(ϕ
1
t )− g2(ϕ

2
t ).



10 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

Then, (u, ϕ, ut, ϕt) solves the system

ρutt − µuxx − bϕx + a1(x)G1(ut) + F1(u, ϕ) = 0 in (0, L)× (0,∞),

Jϕtt − δϕxx + bux + ξϕ+ a2(x)G2(ϕt) + F2(u, ϕ) = 0 in (0, L)× (0,∞),

u(0, t) = u(L, t) = ϕ(0, t) = ϕ(L, t) = 0, t > 0,

(u(x, 0), ϕ(x, 0)) = (u0(x), ϕ0(x)), in (0, L),

(ut(x, 0), ϕt(x, 0)) = (u1(x), ϕ1(x)), in (0, L).

(4.12)

Take |Σ| = α2 − α1. Let us consider ϵ0, small enough, such that 0 < ϵ0 <
|Σ|
2 and

we define the auxiliary function, as in [13],

hλ(x) =


(λ− 1)x, x ∈ [0, α1 + ϵ0),

λ(x− α1 − ϵ0) +
α1−α2+2ϵ0

L (α1 + ϵ0), x ∈ [α1 + ϵ0, α2 − ϵ0],

(λ− 1)(x− L), x ∈ (α2 − ϵ0, L],

(4.13)

with λ := L−(α2−α1−2ϵ0)
L ∈ (0, 1) and 0 ≤ α1 < α2 ≤ L.

Multiplying the first and second equations of the system (4.12) by uxhλ and
ϕxhλ, respectively, and integrating by parts, we have∫ T

0

∫ L

0

(ρ
2
u2t +

µ

2
u2x +

J

2
ϕ2t +

δ

2
ϕ2x +

ξ

2
ϕ2 + buxϕ

)
h′λ dx dt

= −
[
ρ

∫ L

0

utuxhλ dx
]T
0
−

[
J

∫ L

0

ϕtϕxhλ dx
]T
0

+ ξ

∫ T

0

∫ L

0

ϕ2h′λ dx dt+ b

∫ T

0

∫ L

0

uxϕh
′
λ dx dt

+

∫ T

0

∫ L

0

(a1(x)G1(ut)ux + a2(x)G2(ϕt)ϕx)hλ dxdt

+

∫ T

0

∫ L

0

(F1(u, ϕ)ux + F2(u, ϕ)ϕx)hλ dx dt.

(4.14)

Observing that

h′λ(x) =

{
λ, x ∈ (α1 + ϵ0, α2 − ϵ0),

(λ− 1), x ∈ [0, α1 + ϵ0) ∪ (α2 − ϵ0, L],
(4.15)

from the above equality we have

(1− λ)

∫ T

0

E(t) dt

=

[∫ L

0

(ρutux + Jϕtϕx)hλ dx

]T

0

− ξ

∫ T

0

∫ L

0

ϕ2h′λ dx dt− b

∫ T

0

∫ L

0

uxϕh
′
λ dx dt

+
1

2

∫ T

0

∫ α2−ϵ0

α1+ϵ0

(ρu2t + Jϕ2t ) dx dt+
1

2

∫ T

0

∫ α2−ϵ0

α1+ϵ0

(µu2x + δϕ2x) dx dt

+
ξ

2

∫ T

0

∫ α2−ϵ0

α1+ϵ0

ϕ2 dx dt+ b

∫ T

0

∫ α2−ϵ0

α1+ϵ0

uxϕdx dt (4.16)



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 11

−
∫ T

0

∫ L

0

(a1(x)G1(ut)ux − a2(x)G2(ϕt)ϕx)hλ dx dt

−
∫ T

0

∫ L

0

(F1(u, ϕ)ux + F2(u, ϕ)ϕx)hλ dx dt. (4.17)

Let us estimate the right-hand side of (4.17). Using the equivalence between the
norm of the energy and the usual norm in H, we obtain[ ∫ L

0

(ρutux + ϕtϕx)hλ dx
]T
0
≤ C(E(0) + E(T )). (4.18)

On the other hand, since L2θ(0, L) ↪→ L2(0, L) we obtain

−ξ
∫ T

0

∫ L

0

ϕ2h′λ dx ≤ C

∫ T

0

∥ϕ∥22θ dt, (4.19)

and for ϵ > 0,

−b
∫ T

0

∫ L

0

uxϕh
′
λ dx dt ≤ ϵ

∫ T

0

E(t) dt+ Cϵ

∫ T

0

∥ϕ∥22θ dt. (4.20)

Using (2.10) and applying Young’s inequality, we obtain

−
∫ T

0

∫ L

0

a1(x)G1(ut)uxhλ dx dt ≤M1

∫ T

0

∫ L

0

a1(x)|ut||uxhλ| dx dt

≤ Cϵ

∫ T

0

∫ L

0

a1(x)|ut|2 dx dt+ ϵ

∫ T

0

E(t) dt

≤ Cϵ

∫ T

0

∫ L

0

a1(x)G1(ut)ut dx dt+ ϵ

∫ T

0

E(t) dt.

Analogously,

−
∫ T

0

∫ L

0

a2(x)G2(ϕt)ϕxhλ dx dt ≤ Cϵ

∫ T

0

∫ L

0

a2(x)G2(ϕt)ϕt dx dt+ ϵ

∫ T

0

E(t) dt.

Then the two inequalities above imply

−
∫ T

0

∫ L

0

(a1(x)G1(ut)ux + a2(x)G2(ϕt)ϕx)hλ dxdt

≤ Cϵ

∫ T

0

∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dx dt+ ϵ

∫ T

0

E(t) dt.

(4.21)

Using (2.7), we have∫ T

0

∫ L

0

(F1(u, ϕ)ux + F2(u, ϕ)ϕx)hλ dx dt

≤ CB

∫ T

0

(∥u∥2θ + ∥ϕ∥2θ)(∥ux∥2 + ∥ϕx∥2) dt

≤ CB,ϵ

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt+ ϵ

∫ T

0

E(t) dt.

(4.22)



12 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

Inserting the estimates (4.18)-(4.22) into (4.17) with ϵ > 0 small enough, we have∫ T

0

E(t) dt ≤ C(E(T ) + E(0)) +
1

2

∫ T

0

∫ α2−ϵ0

α1+ϵ0

(ρu2t + Jϕ2t ) dx dt

+
1

2

∫ T

0

∫ α2−ϵ0

α1+ϵ0

(µu2x + δϕ2x) dx dt+
ξ

2

∫ T

0

∫ α2−ϵ0

α1+ϵ0

ϕ2 dx dt

+ C

∫ T

0

∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dx dt

+ CB

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt.

(4.23)

Now, let us consider a cut-off function η ∈ C∞
0 (0, L) such that

η(x) =


1, x ∈ [α1 + ϵ0, α2 − ϵ0],

0, x ∈ [0, α1) ∪ (α2, L]

0 ≤ η(x) ≤ 1, x ∈ [0, L].

(4.24)

So multiplying the first and second equations of (4.12) by uη and ϕη, respectively,
and integrating by parts, we obtain∫ T

0

∫ L

0

(ρu2t + Jϕ2t + µu2x + δϕ2x + ξϕ2)η dx dt

= −
[ ∫ L

0

(ρutu+ Jϕtϕ)η dx
]T
0
+

∫ T

0

∫ L

0

(2ρu2t + 2Jϕ2t )η dx dt

+
1

2

∫ T

0

∫ L

0

(µu2 + δϕ2)ηxx dx dt+ b

∫ T

0

∫ L

0

(ϕxu− uxϕ)η dx dt

−
∫ T

0

∫ L

0

(G1(ut)u+G2(ϕt)ϕ)η dx dt

−
∫ T

0

∫ L

0

(F1(u, ϕ)u+ F2(u, ϕ)ϕ)η dx dt.

(4.25)

Consequently, by calculations to the ones before, we infer that∫ T

0

∫ L

0

(ρu2t + Jϕ2t + µu2x + δϕ2x + ξϕ2)η dx dt

≤ C(E(T ) + E(0)) + C

∫ T

0

∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dxdt

+ ϵ

∫ T

0

E(t) dt+ CB

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt.

Substituting the last estimate in (4.23) with ϵ > 0 small enough (4.17) and using
the fact that η has support contained in [α1, α2], we obtain∫ T

0

E(t) dt ≤ C(E(T ) + E(0))

+ C

∫ T

0

∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dx dt

+ CB

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt.

(4.26)



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 13

Next, multiplying the first and second equations in (4.12) by ut and ϕt, and
integrate by parts over [0, L]× [s, T ] so that∫ T

s

∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dx dt

= E(s)− E(T )−
∫ T

0

∫ L

0

(F1(u, ϕ)ut + F2(u, ϕ)ϕt) dx dt.

(4.27)

For each ϵ > 0, we have∫ T

0

∫ L

0

(F1(u, ϕ)ut + F2(u, ϕ)ϕt) dx dt

≤ CB

∫ T

0

(∥u∥2θ + ∥ϕ∥2θ)(∥ut∥2 + ∥ϕt∥2) dt

≤ CB,ϵ

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt+ ϵ

∫ T

0

E(t) dt.

(4.28)

Now we use (4.27) and (4.28) to obtain∫ T

0

∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dt

≤ E(0) + E(T ) + ϵ

∫ T

0

E(t) dt+ CB,ϵ

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt.
(4.29)

Next, we combine estimates (4.26) and (4.29) for ϵ > 0 small enough to obtain∫ T

0

E(t) dt ≤ C(E(T ) + E(0)) + CB

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt. (4.30)

Now, integrate the energy equality (4.27) with respect to s so that

TE(T ) =

∫ T

0

E(t) dt−
∫ T

0

∫ T

s

(a1(x)G1(ut)ut + a2(x)G2(ϕt)ϕt) dt ds

−
∫ T

0

∫ T

s

∫ L

0

(F1(u, ϕ)ut + F2(u, ϕ)ϕt) dx dt ds.

By (4.28) and that a1(x)G1(ut)ut + a2(x)G2(vt)vt ≥ 0, the following is immediate,

TE(T ) ≤ 2

∫ T

0

E(t) dt+ CB,T

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt. (4.31)

Substituting (4.30) in (4.31) yields

TE(T ) ≤ C(E(T ) + E(0)) + CB,T

∫ T

0

(∥u∥22θ + ∥ϕ∥22θ) dt.

We choose T > 2C to deduce that

E(T ) ≤ γTE(0) + CB,T sup
s∈[0,T ]

(∥u(s)∥22θ + ∥ϕ(s)∥22θ), (4.32)

where

γT =
C

T − C
< 1.



14 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

Then a standard argument (see [19, Lemma 4.6]) shows that there exist ϑB , ηB , CB >
0 such that

E(t) ≤ ϑBE(0)e−ηBt + CB sup
σ∈[0,t]

(∥u(s)∥22θ + ∥ϕ(s)∥22θ), ∀t ≥ 0.

The proof is complete. □

Proof of Theorem 3.1. (i) Consider a bounded positively invariant set B ⊂ H with
respect to Sα(t), and call it Sα(t)z

i = (ui(t), ϕi(t), uit(t), ϕ
i
t(t)) for z

i ∈ B, i = 1, 2.
Set also u = u1 − u2, ϕ = ϕ1 − ϕ2, as before. It follows from (2.13) that

∥Sα(t)z
1 − Sα(t)z

2∥H ≤ a(t)∥z1 − z2∥H (4.33)

with a(t) = eC0T . Now let X = H1
0 (0, L)×H1

0 (0, L), and define the semi-norm

nX(u, v) := (∥u∥22θ + ∥ϕ∥22θ)1/2

Since the embedding (in 1D) H1
0 (0, L) ↪→ L2θ(0, L) is compact, we know that nX

is a compact semi-norm on X.
By (4.11) we deduce that

∥Sα(t)z
1 − Sα(t)z

2∥2H ≤ b(t)∥z1 − z2∥2H + c(t) sup
s∈[0,t]

[nX(u(s), ϕ(s))]
2
, (4.34)

where b(t) = ϑBe
−ηBt and c(t) = CB . Clearly,

b(t) ∈ L1(R+) and lim
t→∞

b(t) = 0.

Since B ⊂ H is bounded, we know that c(t) is locally bounded on [0,∞). We
now have that the dynamical system (H, Sα(t)) is quasi-stable on any bounded
positively invariant set B ⊂ H by [7, Definition 7.9.2].

(ii) Since (H, Sα(t)) is quasi-stable, applying [7, Proposition 7.9.4], we have that
(H, Sα(t)) is asymptotically smooth. Thus, noting Lemmas 4.1 and 4.2 and using
[7, Corollary 7.5.7], we know that (H, Sα(t)) has a compact global attractor given
by Aα = M+(Nα).

(iii) Let Φ be the Lyapunov functional given in (4.2). By (2.15) and [7, Remark
7.5.8], we obtain

sup
z∈Aα

∥z∥2H ≤
supz∈Aα

Φ(z) + C1

C0
≤

supz∈N Φ(z) + C1

C0

≤
C2(1 + supz∈N ∥z∥θ+1

H ) + C1

C0
.

Hence, by (4.10), we conclude that there exists a constant R > 0 independent of α
such that

sup
z∈Aα

∥z∥2H ≤ R.

Therefore, the closed ball B = B(0, R0) in H of center zero and radius R0 > R is a
bounded absorbing independent of α ∈ Λ.

(iv) From the above, (H, Sα(t)) is quasi-stable on the attractor Aα. Thus, using
in [7, Theorem 7.9.6 ], we know that the attractor Aα has finite fractal dimension

dimf
HAα.



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 15

(v) Since the system (H, Sα(t)) is quasi-stable on the attractor Aα with c∞ =
supt∈R+ c(t) = CAα < ∞, it follows from [7, Theorem 7.9.8] that any complete
trajectory z = (u, ϕ, ut, ϕt) in Aα has the following regularity properties

vt, ϕt ∈ L∞(R, H1
0 (0, L)) ∩ C(R, L2(0, L)),

vtt, ptt ∈ L∞(R, L2(0, L)).

Thus, since Aα ⊂ B for all α ∈ Λ by (iii), there exists CB > 0 such that

∥(ut, ϕt)∥2H1
0×H1

0
+ ∥(utt, ϕtt)∥2L2×L2 ≤ CB.

Hence, using (1.1) and noting that the nonlinear terms are continuous, we conclude
there exists a constant C ′

B > 0 such that

∥(u, ϕ)∥2H2∩H1
0
≤ C ′

B.

Therefore (3.2) holds. Since the global attractors Aα are also characterized by

Aα = {z(0) : z is a bounded full trajectory of Sα(t)},

we conclude the Aα is bounded in H1.
(vi) Let B be the bounded absorbing of (H, Sα(t)) given by (iii). Hence the

system (H, Sα(t)) is quasi-stable on B. For the solution z(t) with initial data
z0 = z(0) ∈ B, there exists CB > 0 such that for any 0 ≤ t ≤ T ,

∥zt(t)∥H̃−1
≤ CB

which leads to

∥Sα(t1)z0 − Sα(t2)z0∥H̃−1
≤

∫ t2

t1

∥zt(τ)∥H̃−1
dτ ≤ CB|t1 − t2| (4.35)

for each 0 ≤ t1 < t2 ≤ T . From (4.35), we conclude that for any z0 ∈ B, the map

t 7→ Sα(t)z0 is Hölder continuous in the extended space H̃ with the exponent δ = 1.
Then, the existence of a generalized exponential attractor, whose fractal dimension

is finite, is immediate in H̃−1.
Following the similar arguments in [19, Theorem 5.1], the existence of exponential

attractors is obtained in H̃−δ with δ ∈ (0, 1). The proof of Theorem 3.1 is complete.
□

5. Continuity and upper-semicontinuity of attractors

Let X be a complete metric space and Aλ be a family of global attractors for a
semigroup Sλ(t) on X, where λ belongs to a complete metric space Λ.

Definition 5.1. We say that the global attractor Aλ is

• Upper semicontinuous at λ0 ∈ Λ if

lim
λ→λ0

distX(Aλ,Aλ0
) = 0.

• Lower semicontinuous at λ0 ∈ Λ if

lim
λ→λ0

distX(Aλ0
,Aλ) = 0.



16 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

• Continuous at λ0 ∈ Λ if it

lim
λ→λ0

dX(Aλ,Aλ0) = 0,

where dX(A,B) = max{distX(A,B),distX(B,A)} denotes the denotes the
Hausdorff metric in X.

Note that upper semicontinuity is typically easier to obtain than lower semiconti-
nuity and the key ingredient of the proof are the a priori estimates on the attractor
and no knowledge on the attractor structure is needed. On the other hand, the
lower semicontinuity of attractors need a careful description of the structure of the
attractor for the limit equation, which is then transferred to the attractors under
perturbation (see [12]).

We use the recent results in [14] on the continuity of attractors with respect to
a parameter, where the results were obtained as a extension of the previous results
in [1]. Let Sλ(t) be a family of parametrized semigroups defined on X, where λ
belongs to a complete metric space Λ.

The result in [14, Theorem 5.2] provides sufficient conditions for the continuity
of global attractors on a residual dense subset.

Theorem 5.2. Suppose that

(1) Sλ(t) has a global attractor Aλ for every λ ∈ Λ,
(2) There is a bounded subset D of X such that Aλ ⊂ D for every λ ∈ Λ,
(3) For t > 0, Sλ(t)x is continuous in λ, uniformly for x in bounded subsets of

X.

Then Aλ is continuous on J where J is a “residual” set dense in Λ.

Theorem 5.3. Under the assumptions of Theorem 3.1, there exists a set J dense
in Λ = [0, 1] × [0, 1] such that Aα, where α = (ϵ1, ϵ2) ∈ Λ, is continuous at α0 =
(ϵ01, ϵ

0
2) ∈ J , that is,

lim
α→α0

dH(Aα,Aα0
) = 0, ∀α0 ∈ J. (5.1)

Proof. We shall apply the Theorem 5.2 with Λ = [0, 1]×[0, 1]. Theorem 3.1 indicates
that (1) holds. The property (2) follows promptly from Theorem 3.1 (iii).

Now, we shall prove the condition (3). Let D be a bounded set of H. Given
α1 = (ϵ1, ϵ2), α2 = (ϵ′1, ϵ

′
2) ∈ Λ and z ∈ D, let us denote

Sαi
(t)z = (ui(t), ϕi(t), uit(t), ϕ

i
t(t)), i = 1, 2,

u = u1 − u2, ϕ = ϕ1 − ϕ2.

Then z(t) = (u(t), ϕ(t), ut(t), ϕt(t)) satisfies the system

ρutt − µuxx − bϕx + a1(x)G1(ut) + F1(u, ϕ) = (ϵ1 − ϵ′1)h1,

Jϕtt − δϕxx + bux + ξϕ+ a2(x)G2(ϕt) + F2(u, ϕ) = (ϵ2 − ϵ′2)h2,
(5.2)

where

Fi(u, ϕ) = fi(u
1, ϕ1)− fi(u

2, ϕ2), i = 1, 2;

G1(ut) = g1(u
1
t )− g1(u

2
t ), G2(ϕt) = g2(ϕ

1
t )− g2(ϕ

2
t ).



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 17

Multiplying the first equation in (5.2) by ut, the second by ϕt, respectively, and
using integration by parts, we obtain

1

2

d

dt
∥U∥2H = −

∫ L

0

(F1(u, ϕ)ut + F2(u, ϕ)ϕt) dx

−
∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(vt)vt) dx

+

∫ L

0

(
(ϵ1 − ϵ′1)h1ut + (ϵ2 − ϵ′2)h2ϕt

)
dx.

(5.3)

Using (2.7), Hölder’s inequality and the embedding H1
0 (0, L) ↪→ L∞(0, L), we de-

duce that ∫ L

0

F1(u, ϕ)ut dx

≤ C(1 + ∥Sσ1
(t)z∥θ−1

H + ∥Sσ2
(t)z∥θ−1

H )(∥u∥2 + ∥ϕ∥2)∥ut∥2
≤ C(1 + ∥Sσ1(t)z∥θ−1

H + ∥Sσ2(t)z∥θ−1
H )(∥ux∥2 + ∥ϕx∥2)∥ut∥2.

(5.4)

Using that E(t) is a non-increasing function and (2.15), we find that for i = 1, 2,

∥Sσi
(t)z0∥p−1

H ≤ E(0) + C1

C0
≤
C2(1 + ∥z∥p+1

H ) + C1

C0
≤ CD, ∀z ∈ D.

Inserting the above estimate into (5.4) and using Young’s inequality, we see that∫ L

0

F1(u, ϕ)ut dx ≤ CD(∥ux∥2 + ∥ϕx∥2)∥ut∥2

≤ CD(∥ux∥22 + ∥ϕx∥22) + ρ∥ut∥22.
Analogously, ∫ L

0

F2(u, ϕ)ϕt dx ≤ CD(∥ux∥22 + ∥ϕx∥22) + J∥ϕt∥22.

Adding the last two estimates and using (2.3), we conclude that∫ L

0

(F1(u, ϕ)ut + F2(u, ϕ)ϕt) dx ≤ CD∥z∥2H. (5.5)

By the monotonicity property (2.12), we obtain

−
∫ L

0

(a1(x)G1(ut)ut + a2(x)G2(vt)vt) dx ≤ 0. (5.6)

In addition,∫ L

0

(
(ϵ1 − ϵ′1)h1ut + (ϵ2 − ϵ′2)h2ϕt

)
dx

≤ 1

4
(ρ∥ut∥22 + J∥ϕt∥22) +

1

ρ
|ϵ1 − ϵ′1|2∥h1∥2 +

1

J
|ϵ2 − ϵ′2|2∥h2∥22

≤ 1

4
∥z∥2H +

1

ρ
|ϵ1 − ϵ′1|2∥h1∥2 +

1

J
|ϵ2 − ϵ′2|2∥h2∥22.

(5.7)

Substituting the estimates (5.5)-(5.7) into (5.3), we obtain

d

dt
∥z∥2H ≤ CD∥z∥2H +

1

ρ
|ϵ1 − ϵ′1|2∥h1∥2 +

1

J
|ϵ2 − ϵ′2|2∥h2∥22. (5.8)



18 M. L. SANTOS, M. M. FREITAS, R. Q. CALJARO EJDE-2025/11

Applying Gronwall’s inequality to (5.8) and using that ∥z(0)∥2H = 0, we conclude
that

∥z(t)∥2H ≤ C
(
eCt − 1

) (
|ϵ1 − ϵ′1|2∥h1∥2 + |ϵ2 − ϵ′2|2∥h2∥22

)
, t > 0.

This implies

∥Sα1
(t)z − Sα2

(t)z∥H ≤
√
C (eCt − 1) (|ϵ1 − ϵ′1|2∥h1∥2 + |ϵ2 − ϵ′2|2∥h2∥22), t > 0.

Therefore (3) holds. As a conclusion, by applying Theorem 5.2, there exists a dense
set J ⊂ Λ such that (5.1) holds. The proof is complete. □

The next result deals with the upper-semicontinuity of the attractor with respect
to parameter α.

Theorem 5.4. Suppose that assumptions (i)–(iii) hold. Then, the attractor Aα is
s upper semicontinuous with respect to the pair α = (ϵ1, ϵ2) in Λ = [0, 1]× [0, 1], i.e.

lim
α→α0

distH(Aα,Aα0) = 0, ∀α0 = (ϵ01, ϵ
0
2) ∈ Λ. (5.9)

Proof. We proceed by contradiction as in [10, 11]. Suppose that (5.9) does not
hold. Then, there exist an ϵ > 0 and a sequence αn = (ϵn1 , ϵ

n
2 ) → α0 such that

distH(Aαn
,Aα0

) ≥ ϵ > 0, ∀n ∈ N.
Thus, there exists a sequence {zn0 } ∈ Aαn

by the compactness of Aα such that

distH(zn0 ,Aα0) ≥ ϵ > 0, ∀n. (5.10)

Let zn(t) = (un(t), ϕn(t), unt (t), ϕ
n
t (t)) be a full trajectory from the attractor Aαn

such that zn(0) = zn0 . We know by the Theorem 3.1-(iv) that

{zn} is uniformly bounded in L∞(R;V). (5.11)

Since V is compactly embedded into H, using Simon’s Compactness Theorem (see
[29]), we obtain a subsequence {znk} and z ∈ C([−T, T ];H) such that

lim
k→∞

max
t∈[−T,T ]

∥znk(t)− z(t)∥H = 0. (5.12)

By (5.11) and (5.12), we conclude that supt∈R ∥z(t)∥H <∞.
Using the same argument as in the proof of property (3) in Theorem 5.2, we can

see that
z(t) = (u(t), ϕ(t), ut(t), ϕt(t))

solves (in distributional sense) the limiting equations (α = α0)

ρutt − µuxx − bϕx + a1(x)g1(ut) + f1(u, ϕ) = ϵ01h1,

Jϕtt − δϕxx + bux + ξϕ+ a2(x)g2(ϕt) + f2(u, ϕ) = ϵ02h2.

Thus, z(t) is a bounded full trajectory for the limiting semi-flow Sα0(t). Finally,
the limit (5.12) implies

znk
0 → z(0) ∈ Aα0

,

which is contradict (5.10). The proof is complete. □

Acknowledgments. We would like to thank the anonymous referees for construc-
tive comments that improved the final version of our paper. M. L. Santos wants to
thank CNPq for financial support through the projects: CNPq Grant 308056/2021-
3 and CNPq Grant 444331/2024-7 (Control and Numerical Analysis of a Nonlinear
Marine Riser Model). M. M. Freitas was supported by CNPq grant 313081/2021-2.



EJDE-2025/11 ATTRACTORS FOR A POROUS-ELASTIC SYSTEM 19

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Mauro L. Santos

PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street 01,

Belém–PA, 66075-110, Brazil
Email address: ls@ufpa.br

Mirelson M. Freitas
Department of Mathematics, University of Braśılia, Braśılia-DF, 70910-900, Brazil

Email address: mirelson.freitas@unb.br

Ronal Q. Caljaro

PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street 01,

Belém–PA, 66075-110, Brazil
Email address: ronalquispecaljaro@gmail.com


	1. Introduction
	2. Assumptions and Preliminary Results
	3. Main results
	4. Proofs of main results
	4.1. Gradient system and stationary solutions
	4.2. Uniform stabilizability inequality

	5. Continuity and upper-semicontinuity of attractors
	Acknowledgments

	References