Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 07, pp. 1–18.

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

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO 3D

KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS

WITH UNBOUNDED DELAYS

LE THI THUY

Abstract. In this article we consider a 3D Kelvin Voigt Brinkman Forch-

heimer equations involving unbounded delays in a bounded domain Ω ⊂ R3.

First, we show the existence and uniqueness of weak solutions by using the
Galerkin approximations method and the energy method. Second, we prove

the existence and uniqueness of stationary solutions by employing the Brouwer

fixed point theorem. Finally, we study the stability of stationary solutions via
the direct classical approach and the construction of a Lyapunov function. We

also give a sufficient condition for the polynomial stability of the stationary

solution for a special case of unbounded variable delay.

1. Introduction

Let Ω be a bounded domain in R3 with a smooth boundary ∂Ω. We consider
the 3D Kelvin-Voigt-Brinkman-Forchheimer equations with delays in Ω,

∂t(u− α2∇u)− ν∆u+ (u · ∇)u+∇p+ f(u) = g(t, ut) + h(t)

in (0, T )× Ω,

div u = 0 in (0, T )× Ω,

u(x, t) = 0 in (0, T )× ∂Ω,

u(θ, x) = φ(θ, x), in (−∞, 0]× Ω,

(1.1)

where ν > 0 is the kinematic viscosity, α > 0 is a scale parameter with dimension of
length, u = u(x, t) = (u1, u2, u3) is the velocity field of the fluid, p is the pressure,
h is a nondelayed external force field, g is another external force term and contains
hereditary characteristic ut, where ut is the function defined on (−∞, 0] by ut(θ) =
u(t+ θ), θ ∈ (−∞, 0], φ is the initial datum on the interval.

The nonlinearity f ∈ C1(R3,R3) satisfies the following conditions:

f ′(u)v · v ≥ (−K + κ|u|β−1)|v|2, ∀u, v ∈ R3,

|f ′(u)| ≤ Cf (1 + |u|β−1), ∀u ∈ R3,
(1.2)

where K,κ,Cf , are some positive constants, β ≥ 1 (in the case of β > 3 to show
the uniqueness of solutions) and u · v stands for standard inner product in R3. A

2020 Mathematics Subject Classification. 35B41, 35K65, 35D05.
Key words and phrases. Kelvin-Voigt-Brinkman-Forchheimer equation; delay equation;

stationary solution; local stability; asymptotically stable; polynomial stable.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted November 3, 2020. Published January 17, 2022.

1



2 L. T. THUY EJDE-2022/07

typical example for f is

f(u) = au+ b|u|β−1u, β ∈ [1; +∞), (1.3)

where a ∈ R and b > 0 are the Darcy and Forchheimer coefficients respectively.
The case α ≡ 0 and G ≡ 0 has been studied by Zelik and Kalantarov [15].

The case of the so-called subcritical growth rate of the nonlinearity f (for β ≤ 3 in
(1.3)) has been considered in the literature. The aim in [15] is to remove this growth
restriction and verify the global existence, uniqueness and dissipativity of smooth
solutions of the Brinkman-Forchheimer equations for a large class of nonlinearity f
with an arbitrary growth exponent β > 3.

Note that the case f ≡ 0 and G ≡ 0 corresponds to the classical Navier-Stokes-
Voigt problem. The existence, long-time behavior and regularity of solutions to
the 3D Navier-Stokes-Voigt equations without delays in bounded domains and un-
bounded domains satisfying the Poincare’s inequality have been studied by many
mathematicians [2, 3, 6, 9, 10, 13, 14, 22, 24, 25]. There are many results involving
PDEs in fluid mechanics with delays [1, 7, 8, 18, 19, 20] and many results about
asymptotic behavior to PDEs [21, 23]. However, all the results with finite delay
(constant delays, bounded variable delay or bounded distributed delay) has been
studied in the phase spaces C([−h, 0);X) and L2(−h, 0;X) with a suitable Banach
space X, or infinite distributed delay in Cγ(X), where

Cγ(X) =
{
ϕ ∈ C((−∞, 0];X) : lim

θ→−∞
eγθϕ(θ) exists in X

}
(γ > 0)

is the Banach space endowed with the norm

‖ϕ‖γ = sup
θ∈(−∞,0]

eγθ‖ϕ(θ)‖X .

In this paper, following [16] we continue studying the system (1.1) with unbounded
variable delays in the space

BCL−∞(X) =
{
ϕ ∈ C((−∞, 0];X) : lim

θ→−∞
ϕ(θ) exists in X

}
which is a Banach space equipped with the norm

‖ϕ‖BCL−∞(X) = sup
θ∈(−∞,0]

‖ϕ(θ)‖X .

The main novelty of this article is that we are interested in the problem with
unbounded delays. The stability of stationary solutions to the 3D Kelvin-Voigt-
Brinkman-Forchheimer equations with unbounded delays, has apparently not been
studied previously.

We will discuss the existence and uniqueness of the weak solution and stationary
solution. Moreover, stability will be established for the stationary solution. The
existence and the uniqueness of solution is proved by using the classic Galerkin
approximation and the energy method. The existence of stationary solution is
established by employing a corollary of the Brouwer fixed point theorem. The sta-
bility of stationary solution is shown by using the direct classical method, Lyapunov
functions and giving a sufficient condition for a special case of unbounded variable
delays.

The rest of this paper is organized as follows. In section 2, we will set up some
spaces and lemmas which will be used in the later sections. Section 3 will be
devoted to the existence and uniqueness of solutions of the model. In section 4, we
will study the existence, uniqueness and stability of stationary solutions.



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 3

2. Preliminaries

We consider the space

V = {u ∈ (C∞0 (Ω))3 : div u = 0}.

Let H be the closure of V in (L2(Ω))3 with the norm | · |, and inner product (·, ·)
defined by

(u, v) =

3∑
j=1

∫
Ω

uj(x)vj(x)dx for u, v ∈ (L2(Ω))3.

We also denote V the closure of V in (H1
0 (Ω))3 with norm ‖ · ‖, and the associated

scalar product ((·, ·)) defined by

((u, v)) =

3∑
i,j=1

∫
Ω

∂uj
∂xj

∂vj
∂xi

dx for u, v ∈ (H1
0 (Ω))3.

We use ‖ · ‖∗ for the norm in V ′ and 〈·, ·〉V,V ′ for the dual pairing between V and
V ′. We recall the Stokes operator A : V → V ′ by 〈Au, v〉 = ((u, v)). Denote by
P the Helmholtz-Leray orthogonal projection in (H1

0 (Ω))3 onto the space V . Then
Au = −P∆u, for all u ∈ D(A) = (H2(Ω))3 ∩ V . The Stokes operator A is a
positive self-adjoint operator with compact inverse. Hence there exists a complete
orthonormal set of eigenfunctions {wj}∞j=1 ⊂ H such that Awj = λjwj and

0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λj → +∞ as t→∞.

We have the following Poincaré inequalities

‖u‖2 ≥ λ1|u|2 ∀u ∈ V,
|u|2 ≥ λ1‖u‖2∗ ∀u ∈ H.

(2.1)

From (2.1), we have

|u|2 ≥ d0(|u|2 + α2‖u‖2) ∀u ∈ V,

where d0 = λ1/(1 + α2λ1). Furthermore, for α > 0, the operator I + α2A has a
compact inverse (I + α2A)−1 : D(A)′ → H with the estimate

‖(I + α2A)−1u‖ ≤ α−2‖u‖∗ ∀u ∈ V ′.

We define the trilinear form b on V × V × V by

b(u, v, w) =

3∑
i,j=1

∫
Ω

ui
∂vj
∂xi

wjdx, ∀u, v, w ∈ V,

and B : V ×V → V ′ by 〈B(u, v), w〉 = b(u, v, w). We can write B(u, v) = P [(u·∇)v].
It is easy to check that if u, v, w ∈ V , then b(u, v, w) = −b(u,w, v), and in particular,

b(u, v, v) = 0, ∀u, v ∈ V. (2.2)

Using Hölder’s inequality and Ladyzhenskaya’s inequality, we can choose the best
positive constant c0 such that

|b(u, v, w)| ≤ c0‖u‖‖v‖|w|1/2‖w‖1/2, ∀u, v, w ∈ V. (2.3)

From (2.3) and using Poincaré’s inequality (2.1), we obtain

|b(u, v, w)| ≤ c0λ−1/4
1 ‖u‖‖v‖‖w‖, ∀u, v, w ∈ V. (2.4)



4 L. T. THUY EJDE-2022/07

We will assume that f ∈ L2(0, T ;V ′). For the function g : [0, T ]×BCL−∞(H)→
(L2(Ω))3,we have the following assumptions:

(A1) For any ξ ∈ BCL−∞(H), the mapping [0, T ] 3 t 7→ g(t, ξ) ∈ (L2(Ω))3 is
measurable.

(A2) g(·, 0) = 0.
(A3) There exists a constant Lg > 0 such that, for any t ∈ [0, T ] and all ξ, η ∈

BCL−∞(H),

|g(t, ξ)− g(t, η)| ≤ Lg‖ξ − η‖BCL−∞(H).

Some examples of g satisfying (A1)–(A3) can be seen in [16].
We can rewrite the 3D Kelvin-Voigt-Brinkman-Forchheimer equations (1.1) in

the functional form

∂t(u+ α2Au) + νAu+B(u, u) + Pf(u) = Pg(t, ut) + Ph(t),

in (0, T )× Ω,

u(θ) = φ(θ), θ ∈ (−∞, 0].

(2.5)

3. Existence and uniqueness of weak solutions

We first give the definition of a weak solution.

Definition 3.1. Given an initial datum φ ∈ BCL−∞(H) with φ(0) ∈ V , a weak
solution u to (1.1) in the interval (−∞, T ], T > 0, is a function u ∈ C((−∞, T ];H)∩
L2(0, T ;V )∩Lβ+1(0, T ;Lβ+1(Ω)) with u(θ) = φ(θ), θ ≤ 0 and du

dt ∈ L
2(0, T ;V ′) +

L(β+1)/β(0, T ;L(β+1)/β(Ω)) such that, for all v ∈ V , and a.e. t ∈ (0, T ),

d

dt
((u(t), v) + α2((u(t), v))) + ν((u(t), v)) + b(u(t), u(t), v) + 〈f(u), v〉

= 〈h(t), v〉+ (g(t, ut), v).

Now we show the existence of weak solutions.

Theorem 3.2. Consider h ∈ L2(0, T ;V ′), g : [0, T ]×BCL−∞(H)→ H satisfying
(A1)–(A3) and φ ∈ BCL−∞(H) with φ(0) ∈ V are given. Then there exists a
unique weak solution to (1.1).

Proof. Existence. We split the proof of the existence into several steps.

Step 1: A Galerkin scheme. Let {vj}∞j=1 be the basis consisting of eigenfunc-
tions of the Stokes operator A, which is orthonormal in H and orthogonal in V .
Denote Vm = span{v1, . . . , vm} and consider the projector Pmu =

∑m
j=1(u, vj)vj .

Define also

um(t) =

m∑
j=1

γm,j(t)vj ,

where the coefficients γm,j are required to satisfy

d

dt
((um(t), vj) + α2((um(t, vj))) + ν((um(t), vj))

+ b(um(t), um(t), vj) + 〈f(um), vj〉
= 〈h(t), vj〉+ (g(umt ), vj),

(3.1)

for j = 1, . . . ,m, and the initial condition um(θ) = Pmφ(θ) for θ ∈ (−∞, 0].
This system of ordinary functional differential equations with infinite delay in

the unknowns (γm,1(t), . . . , γm,m(t)) fulfills the conditions for the existence and



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 5

uniqueness of local solutions (see [11], [12]). Hence, we conclude that the approx-
imate solutions um to (3.1) exists uniquely and locally on [0, t∗) with 0 ≤ t∗ ≤ T .
Next, we will obtain a priori estimates and ensure that the solutions um exist in
whole interval [0, T ].

Step 2: A priori estimates. Multiplying (3.1) by γm,j(t), j = 1, . . . ,m, summing
up and using (2.2), we obtain

1

2

d

dt
(|um(t)|2 + α2‖um(t)‖2) + ν‖um(t)‖2 +

∫
Ω

f(um)umdx

= 〈h(t), um(t)〉+ (g(umt ), um(t)).

Using the inequality f(u) · u ≥ −K + κ|u|β+1, the Cauchy inequality and noting
that |um(t)| ≤ ‖umt ‖BCL−∞(H), we obtain

1

2

d

dt
(|um(t)|2 + α2‖um(t)‖2) + ν‖um(t)‖2 + κ

∫
Ω

|um|β+1dx

≤ K|Ω|+ ‖h(t)‖∗‖um(t)‖+ Lg‖umt ‖BCL−∞(H)|um(t)|

≤ K|Ω|+ ν

2
‖um(t)‖2 +

‖h(t)‖2∗
2ν

+ Lg‖umt ‖2BCL−∞(H),

and hence

d

dt
(|um(t)|2 + α2‖um(t)‖2) + ν‖um(t)‖2 + 2κ

∫
Ω

|um|β+1dx

≤ 2K|Ω|+ ‖h(t)‖2∗
ν

+ 2Lg‖umt ‖2BCL−∞(H).

Integrating from 0 to t, we obtain

|um(t)|2 + α2‖um(t)‖2 + ν

∫ t

0

‖um(s)‖2ds+ 2κ

∫ t

0

∫
Ω

|um|β+1dxds

≤ 2K|Ω|t+ |um(0)|2 + α2‖um(0)‖2 +
1

ν

∫ t

0

‖h(s)‖2∗ds

+ 2Lg

∫ t

0

‖ums ‖2BCL−∞(H)ds.

(3.2)

In particular, for any t > 0,

sup
−t<θ≤0

|um(t+ θ)|2 + α2‖um(t)‖2

≤ 2K|Ω|t+ ‖φ‖2BCL−∞(H) + α2‖φ(0)‖2 +
1

ν

∫ t

0

‖h(s)‖2∗ds

+ 2Lg

∫ t

0

(
‖ums ‖2BCL−∞(H) + α2‖um(s)‖2

)
ds.

Since

‖umt ‖2BCL−∞(H) + α2‖um(t)‖2

= max
{

sup
−t<θ≤0

|um(t+ θ)|2 + α2‖um(t)‖2; sup
θ≤−t

|um(t+ θ)|2 + α2‖um(t)‖2
}

≤ max
{

sup
−t<θ≤0

|um(t+ θ)|2 + α2‖um(t)‖2; ‖φ‖2BCL−∞(H) + α2‖um(t)‖2
}
,



6 L. T. THUY EJDE-2022/07

we obtain

‖umt ‖2BCL−∞(H) + α2‖um(t)‖2

≤ 2K|Ω|t+ 2‖φ‖2BCL−∞(H) + α2‖φ(0)‖2 +
1

ν

∫ t

0

‖h(s)‖2∗ds

+ 2Lg

∫ t

0

(
‖ums ‖2BCL−∞(H) + α2‖um(s)‖2

)
ds.

By the Gronwall inequality we have

‖umt ‖2BCL−∞H) + α2‖um(t)‖2

≤ e2Lgt
(

2K|Ω|T + ‖φ‖2BCL−∞(H) + α2‖φ(0)‖2 +
1

ν

∫ t

0

(‖h(s)‖2∗)ds
)
.

Then we obtain the following estimates: for anyR > 0 such that ‖φ‖BCL−∞(H) ≤ R,
there exists a constant C depending on ν, Lg, f , such that

‖umt ‖2BCL−∞(H) + α2‖um(t)‖2 ≤ C(T,R), ∀t ∈ [0, T ], ∀m ≥ 1. (3.3)

In particular,

{um} is uniformly bounded in L∞(0, T ;BCL−∞(H)) ∩ L∞(0, T ;V ).

From (3.2) and the uniform estimates above, we obtain

ν

∫ t

0

∫
Ω

‖um(s)‖2dxds+ 2κ

∫ t

0

∫
Ω

|um(s)|β+1dxds

≤ |um(0)|2 + α2‖um(0)‖2 +K|Ω|
∫ t

0

ds+
1

ν

∫ t

0

‖h(s)‖2∗ds

+ 2Lg

∫ t

0

‖ums ‖2BCL−∞(H)ds

≤ |um(0)|2 + α2‖um(0)‖2 +

∫ t

0

(
1

ν
‖h(s)‖2∗ + 2LgC(T,R) +K|Ω|

)
ds.

Then {um} is uniformly bounded in L2(0, T ;V )∩Lβ+1(0, T ;Lβ+1(Ω)). Using (1.2),
we obtain that |f(u)| ≤ C(1 + |u|β) with C depending on Cf . Hence,∫ t

0

∫
Ω

|f(u)|
β+1
β dxdt ≤ C

∫ t

0

∫
Ω

(1 + |u|β)
β+1
β dxdt

≤ C
∫ t

0

∫
Ω

(1 + |u|β+1)dxdt.

From the boundedness of {um} in Lβ+1(0, T ;Lβ+1(Ω)), we obtain {f(x, um)} is

bounded in L(β+1)/β(0, T ;L(β+1)/β(Ω)). Now, we prove the boundedness of {du
m

dt }.
We have

d

dt
(um(t) + α2Aum(t)) =− νAum(t)− PmB(um, um)− Pf(um)

+ Pmh(t) + Pmg(t, umt ).
(3.4)



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 7

From (2.4), (3.3) and (3.4), we obtain∥∥ d
dt

(um + α2Aum)
∥∥
∗

≤ ν‖Aum‖∗ + ‖B(um, um)‖∗ + ‖f(um)‖L(β+1)/β(Ω) + ‖h(t)‖∗ + ‖g(t, umt )‖∗
≤ ν‖um‖+ c0λ

−1/4
1 ‖um‖+ ‖f(um)‖L(β+1)/β(Ω) + ‖h(t)‖∗ + λ

−1/2
1 |g(t, umt )|

≤ ν‖um‖+ c0λ
−1/4
1 ‖um‖+ ‖f(um)‖L(β+1)/β(Ω)

+ ‖h(t)‖∗ + Lgλ
−1/2
1 ‖um‖BCL−∞(H)

≤ C(T,R), ∀m ≥ 1.

This implies that d
dt (u

m + α2Aum) is uniformly bounded in

L2(0, T ;V ′) + L(β+1)/β(0, T ;L(β+1)/β(Ω)).

Then {du
m

dt } is uniformly bounded in L2(0, T ;V ′) + L(β+1)/β(0, T ;L(β+1)/β(Ω)).

Step 3. Approximation in BCL−∞(H) of the initial datum. We will show
that

Pmφ→ φ in BCL−∞(H). (3.5)

Assume on the contrary that (3.5) is not true. Then there exists ε > 0 and a
subsequence, relabeled similarly, such that

‖Pmφ(θm)− φ(θm)‖ > ε, ∀m. (3.6)

One can assume that θm → −∞, otherwise if θm → θ, then Pmφ(θm)→ φ(θ), since
‖Pmφ(θm) − φ(θ‖ ≤ ‖Pmφ(θm) − Pmφ(θ)‖ + ‖Pmφ(θ) − φ(θ)‖ → 0 as m → +∞.
But with θm → −∞ as m→ +∞, we obtain that

‖Pmφ(θm)− φ(θm)‖ = ‖Pmφ(θm))− Pmx‖+ ‖Pmx− x‖+ ‖x− φ(θm)‖ → 0,

where x = limθ→−∞ φ(θ). This contradicts (3.6), so (3.5) holds.

Step 4: Compactness results. We obtain

um ⇀∗ u weakly in L∞(0, T ;V ),

um ⇀ u in L2(0, T ;V ) ∩ Lβ+1(0, T ;Lβ+1(Ω)),

f(um) ⇀ χ in L(β+1)/β(0, T ;L(β+1)/β(Ω)),

dum

dt
⇀

du

dt
in L2(0, T ;V ′) + L(β+1)/β(0, T ;L(β+1)/β(Ω)).

Since {um} is uniformly bounded in L2(0, T ;V ) and {du
m

dt } is uniformly bounded

in L2(0, T ;V ′) + L(β+1)/β(0, T ;L(β+1)/β(Ω)) and by using the Aubin-Lions com-
pactness lemma, we deduce that um → u strongly in L2(0, T ; (L2(Ω))3). Thus, we
have (up to a subsequence)

um → u a.e. in ΩT .

From the continuity of f , we obtain that f(um)→ f(u) a.e. in ΩT . Because of the
uniqueness of the limit, we have f(u) ≡ χ.

Next, we show that
um → u in C([0, T ];H), (3.7)

by applying the Arzelà-Ascoli lemma (following the same method in [17]). We have

sup
θ≤0
|um(t+ θ)− u(t+ θ)|



8 L. T. THUY EJDE-2022/07

≤ max
{

sup
θ≤−t

|Pmφ(θ + t)− φ(θ + t)|, sup
−t≤θ≤0

|um(t+ θ)− u(t+ θ)|
}

≤ max
{
|Pmφ− φ|BCL−∞(H), sup

−t≤θ≤0
|um(t+ θ)− u(t+ θ)|

}
→ 0.

Then (3.5) and (3.7) imply that

umt → ut in BCL−∞(H), ∀t ∈ [0, T ].

Therefore, taking into account (A3), we have

g(·, um· )→ g(·, u·) in L2(0, T ;H).

Finally, we can pass to the limit in (3.1) and conclude that u solves (1.1).

Uniqueness. Let u, v be two weak solutions of problem (2.5) with the same
initial condition. Setting w = u− v, we have

1

2

d

dt
(|w|2 + α2‖w‖2) + ν‖w‖2 + b(u, u, w)− b(v, v, w)

+

∫
Ω

(f(u1)− f(u2))(u1 − u2)dx

= (g(ut)− g(vt), w).

(3.8)

It is known (see [5]) that there exists two nonnegative constants α = α(β) and Cf
such that∫

Ω

(f(u)− f(v))(u− v)dx ≥ −Cf |u− v|2 + α

∫
Ω

(|u|β−1 + |v|β−1)|u− v|2dx. (3.9)

Using that f satisfies (3.9), then (3.8) becomes

d

dt
(|w|2 + α2‖w‖2) + 2ν‖w‖2 + α

∫
Ω

(|u|β−1 + |v|β−1)|u− v|2dx

≤ Cf |w|2 + 2

∫
Ω

|((w · ∇)u) · w|dx+ 2

∫
Ω

|g(ut)− g(vt)| · |w(t)|dx.

By Holder’s inequality and Young’s inequality, we have

2

∫
Ω

|((w · ∇)u) · w|dx ≤ 2|u||w||∇w| ≤ ν|∇w|2 + C|u|2|w|2,

where C = C(ν). Assuming that β − 1 > 2 and using Young’s inequality again, we
obtain

2

∫
Ω

|((w · ∇)u) · w|dx ≤ ν|∇w|2 + α(|u|β−1 + |v|β−1)|w|2 + C|w|2,

where C = C(ν, α).
Taking (A3) into account and using Young’s inequality, we obtain

d

dt
(|w(t)|2 + α2‖w‖2) + ν‖w‖2 ≤ Cf |w|2 + C|w|2 + 2Lg‖wt‖BCL−∞(H)|w(t)|

≤ (Cf + C)|w|2 +
νλ1

2
|w|2 +

Lg
λ1
‖w‖2BCL−∞(H)

≤ (Cf + C)|w|2 +
ν

2
‖w‖2 +

Lg
λ1
‖w‖2BCL−∞(H).



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 9

Thus, we have

|w(t)|2 + α‖w‖2 ≤ (Cf + C)

∫ t

0

|w(s)|2ds+
Lg
λ1

∫ t

0

‖ws‖2BCL−∞(H)ds.

Since w(θ) = 0 for θ ≤ 0, then taking the maximum in [0, t] for any t ∈ [0, T ], we
have

‖wt‖2BCL−∞(H) ≤ (Cf + C +
Lg
λ1

)

∫ t

0

‖ws‖2BCL−∞(H)ds.

We complete the proof of uniqueness after applying the Gronwall inequality. �

4. Existence and stability of stationary solutions

4.1. Existence of weak stationary solutions. To study the existence and the
other properties of stationary solutions, we need to impose some extra assumptions.
Firstly, we assume that h is independent of time, i.e., h(t) ≡ h ∈ V ′. Denote by i
the trivial immersion i : H → BCL−∞(H) given by i(u) = ũ with ũ(t) = u for all
t ≤ 0. We now require that g satisfy

(A4) g(s, ξ) = g(t, ξ) for any s, t ∈ R+ and ξ ∈ i(H).

If (A2)–(A4) hold, we trivially have that g̃ : H → (L2(Ω))3 defined by g̃(u) =
g(0, i(u)), i.e., g̃ = g|R+×i(H), is of course autonomous, Lipschitz (with the same
Lipschitz constant Lg) and g̃(0) = 0.

Hence, the stationary equation to (2.5) is the following form which does not
contain a delay term:

νAu+B(u, u) + Pf(u) = Ph+ P g̃(u). (4.1)

Let us consider the definition of stationary solutions to problem (1.1).

Definition 4.1. A weak stationary solution to (1.1) is an element u∗ ∈ V such
that

ν((u∗, v)) + b(u∗, u∗, v) + 〈f(u∗), v〉 = 〈h, v〉+ (g̃(u∗), v), ∀v ∈ V.

Theorem 4.2. Assume (A2)–(A4) hold and h ∈ V ′. If 2Lg < νλ1 then problem
(1.1) admits at least one stationary solution u∗ satisfying

‖u∗‖ ≤
(λ1(2νK|Ω|+ ‖h‖2∗)

ν(λ1ν − 2Lg)

)1/2

, (4.2)

and

ν > Cfλ
−1
1 + Lgλ

−1
1 + 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

, (4.3)

then the stationary solution of (1.1) is unique.

Proof. Existence. Estimate (4.2) can be obtained by taking into account that,
any stationary solution u∗, if it exists, should satisfy

ν((u∗, u∗)) + 〈f(u∗), v〉 = 〈h, u∗〉+ (g̃(u∗), u∗).

Using inequality f(u) · u ≥ −K + κ|u|β+1 again, we have

ν‖u∗‖2 + κ

∫
Ω

|u∗|β+1dx ≤ K|Ω|+ ‖h‖∗‖u∗‖+ Lgλ
−1
1 ‖u∗‖2.

Hence we obtain the desired estimate.



10 L. T. THUY EJDE-2022/07

To show the existence, let {vj}∞j=1 be the basis of V consisting of eigenfunctions
of the operator A. For each m ≥ 1, let us define Vm = span{v1, . . . , vm} and an
approximate stationary solution um of (1.1) by

um =

m∑
i=1

γmivi,

ν((um, vi)) + b(um, um, vi) + 〈f(um), vi〉 = 〈h, vi〉+ (g̃(um), vi),

i = 1, . . . ,m.

(4.4)

To prove the existence of um, we define the operators Rm : Vm → Vm by

[Rmu, v] := ν((u, v)) + b(u, u, v) + 〈f(u), v〉 − 〈h, v〉 − (g̃(u), v), ∀u, v ∈ Vm.

For all u ∈ Vm, we have

[Rmu, u] = ν((u, u)) +

∫
Ω

f(u)udx− 〈h, u〉 − (g̃(u), u)

≥ ν‖u‖2 + κ

∫
Ω

|u|β+1dx−K|Ω| − ‖h‖∗‖u‖ − Lgλ−1
1 ‖u‖2

=
(ν

2
− Lgλ−1

1

)
‖u‖2 + κ

∫
Ω

|u|β+1dx−K|Ω| − 1

2ν
‖h‖2∗.

It follows that ((Rmu, u)) ≥ 0 for ‖u‖X = ‖u‖+ ‖u‖Lβ+1 = k sufficiently large, we
obtain

k =
(2K|Ω|ν + ‖h‖2∗
ν(λ1ν − 2LG)

)1/2

+
(2K|Ω|ν + ‖h‖2∗

2νκ

)1/(β+1)

,

where 2Lg < νλ1. So, there exists a solution um ∈ Vm satisfying Rm(um) = 0.
Replacing vi by um in (4.4) and taking (2.2) into account, we obtain

ν‖um‖2 + 〈f(um), um〉 = 〈h, um〉+ (g̃(um), um).

This implies

ν‖um‖2 + κ

∫
Ω

|um|β+1dx ≤ K|Ω|+ ‖h‖∗‖um‖+ Lgλ
−1
1 ‖um‖2.

Hence (ν
2
− Lg
λ1

)
‖um‖2 + κ‖um‖β+1

Lβ+1 ≤ K|Ω|+
1

2ν
‖h‖2∗.

We extract from {um} a subsequence {um′}, which converges weakly in V ∩Lβ+1 to
some limit u. Since the domain Ω is bounded, the injection of V into H is compact.
Thus,

um
′
→ u weakly in V and strongly in H,

up to a subsequence. Using the same method in the step compactness results in
the Theorem 3.2, we pass to the limit in (4.4) using the sequence m′ and find that
u is a stationary solution of (1.1).

Uniqueness. Suppose that u∗ and v∗ are two stationary solutions of (4.1).
Then

ν((u∗ − v∗, v)) + b(u∗, u∗, v)− b(v∗, v∗, v) +

∫
Ω

(f(u∗)− f(v∗))vdx

= (g̃(u∗)− g̃(v∗), v)



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 11

for all v ∈ V . Choosing v = u∗ − v∗, we have

ν((u∗ − v∗, u∗ − v∗)) +

∫
Ω

(f(u∗)− f(v∗))(u∗ − v∗)dx

= b(u∗, u∗, u∗ − v∗)− b(v∗, v∗, u∗ − v∗) + (g̃(u∗)− g̃(v∗), u∗ − v∗).

Using estimate (2.4), we obtain

|b(u∗, u∗, u∗ − v∗)− b(v∗, v∗, u∗ − v∗)| ≤ 2c0λ
−1/4
1 ‖u∗‖‖u∗ − v∗‖2.

From
∫

Ω
(f(u)− f(v))(u− v)dx ≥ −Cf |u− v|2, we have

ν‖u∗ − v∗‖2 ≤ Cf |u∗ − v∗|2 + 2c0λ
−1/4
1 ‖u∗‖‖u∗ − v∗‖2 + Lgλ

−1
1 ‖u∗ − v∗‖2.

Using estimate (4.2) we deduce that

(ν − Cfλ−1
1 − Lgλ

−1
1 − 2c0λ

−1/4
1 ‖v‖)‖u− v‖2 ≤ 0,

and hence the uniqueness follows from condition (4.3). �

4.2. Stability results.

Definition 4.3. A stationary u∗ to (1.1) is stable if for any ε > 0 there exists δ > 0
such that if φ ∈ BCL−∞(H) satisfies ‖φ− i(u∗)‖BCL−∞(H) ≤ δ, then the solution
u(·;φ) to (1.1) exists for all t ≥ 0 and satisfies |u(t;φ)− u∗| < ε for any t ≥ 0.

We consider the case of g(t, ut) = G(u(t − ρ(t))), where G : H → (L2(Ω))3 is a
measurable function satisfying G(0) = 0, and assume that there exists Lg > 0 such
that

|G(u)−G(v)| ≤ Lg|u− v|, ∀u, v ∈ H. (4.5)

Consider a function ρ(t) ∈ C1(R+,R+) with ρ∗ = maxt∈[0,T ] ρ
′(t) < 1. The system

(2.5) becomes

d

dt
(u+ α2Au) = −νAu−B(u, u)− Pf(u) + Ph+ PG(u(t− ρ(t))), (4.6)

with initial condition u(θ) = φ(θ), θ ∈ (−∞, 0]. We then have the following stability
results.

4.2.1. Local stability via a direct approach.

Theorem 4.4. Assume that h ∈ V ′ and (4.5) hold. If νλ1 > 2Lg then there exists
at least one stationary solution u∗ to (4.1) satisfying (4.2). Moreover, if

ν > Cfλ
−1
1 + 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
(2− ρ∗)Lgλ−1

1

2(1− ρ∗)
, (4.7)

then the stationary solution u∗ is unique, and there exists C = C(ρ∗, Lg) such that
the solution u to (4.6) satisfies

|u(t)− u∗|2 + α2‖u(t)− u∗‖2

≤ C
(
|φ(0)− u∗|2 + α2‖φ(0)− u∗‖2 + ‖φ− u∗‖2L2(−ρ(0),0;H)

)
,

for all φ ∈ BCL−∞(H).



12 L. T. THUY EJDE-2022/07

Proof. We first see that all assumptions of Theorem 4.2 are satisfied (note that if ν
satisfies (4.7) then condition (4.3) is obtained). Hence, the existence and uniqueness
of stationary solution u∗ satisfying (4.2) are established. If we write u(t) in the form
u(t) = u∗ + v(t), then from (2.5), v(t) satisfies

d

dt
(v+α2Av) + νAv+B(u, u)−B(u∗, u∗) + f(u)− f(u∗) = G(u(t)−ρ(t))−G(u∗).

Multiplying this equation by v, integrating over Ω and using (2.2), we obtain

d

dt
(|v(t)|2 + α2‖v(t)‖2) + 2

∫
Ω

(f(u)− f(u∗))(u− u∗)dx

= −2ν‖v(t)‖2 + 2(b(u, u, v(t))− b(u∗, u∗, v(t))) + 2((G(u(t− ρ(t)))−G(u∗), v(t)).

Using Cauchy’s inequality, the Lipschitz condition on G and the following estimate
which is obtained as in (2.2),

|b(u, u, v(t))− b(u∗, u∗, v(t))| ≤ 2c0λ
−1/4
1 ‖u∗‖‖v(t)‖2,

Using
∫

Ω
(f(u)− f(u∗))(u− u∗)dx ≥ −Cf |u− u∗|2, we obtain

d

dt
(|v(t)|2 + α2‖v(t)‖2)

≤ −2ν‖v(t)‖2 + 4c0λ
−1/4
1 ‖u∗‖‖v(t)‖2 + 2Cf |v(t)|2 + Lg|v(t− ρ(t))|2

+ Lg|v(t)|2

≤ Lg|v(t− ρ(t))|2 +
(

4c0λ
−1/4
1 ‖u∗‖+ 2Cfλ

−1
1 + Lgλ

−1
1 − 2ν

)
‖v(t)‖2.

(4.8)

Using (4.2) and integrating the above inequality from 0 to t, we obtain

|v(t)|2 + α2‖v(t)‖2

≤ |v(0)|2 + α2‖v(0)‖2 + Lg

∫ t

0

|v(s− ρ(s))|2ds

+
(
Lgλ

−1
1 + 2Cfλ

−1
1 + 4c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

− 2ν
)∫ t

0

‖v(s)‖2ds.

Letting τ(s) = s− ρ(s), we obtain∫ t

0

|v(s− ρ(s))|2ds =
1

1− ρ′

∫ t

−ρ(0)

|v(τ)|2dτ ≤ 1

1− ρ∗

∫ t

−ρ(0)

|v(τ)|2dτ

=
1

1− ρ∗

∫ 0

−ρ(0)

|v(τ)|2dτ +
1

1− ρ∗

∫ t

0

|v(τ)|2dτ.

We then have

|v(t)|2 + α2‖v(t)‖2

≤ |v(0)|2 + α2‖v(0)‖2 +
Lg

1− ρ∗

∫ 0

−ρ(0)

|v(τ)|2dτ +
Lg

1− ρ∗

∫ t

0

|v(τ)|2dτ

+
(
Lgλ

−1
1 + 2Cfλ

−1
1 + 4c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

− 2ν
)∫ t

0

‖v(s)‖2ds

≤ |v(0)|2 + α2‖v(0)‖2 +
Lg

1− ρ∗

∫ 0

−ρ(0)

|v(τ)|2dτ



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 13

+
(
Lgλ

−1
1 + 2Cfλ

−1
1 + 4c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
Lgλ

−1
1

(1− ρ∗)
− 2ν

)
×
∫ t

0

‖v(s)‖2ds

≤ |v(0)|2 + α2‖v(0)‖2 +
Lg

1− ρ∗

∫ 0

−ρ(0)

|v(τ)|2dτ

+
(

2Cfλ
−1
1 + 4c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
(2− ρ∗)Lgλ−1

1

(1− ρ∗)
− 2ν

)
×
∫ t

0

‖v(s)‖2ds.

If

ν > Cfλ
−1
1 + 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
(2− ρ∗)Lgλ−1

1

2(1− ρ∗)
,

then

|v(t)|2 ≤ |v(0)|2 +
Lg

1− ρ∗

∫ 0

−ρ(0)

|v(τ)|2dτ.

We can choose C = max{1, Lg
1−ρ∗ }, and the proof is complete. �

4.2.2. Asymptotical stability via the construction of Lyapunov functionals.

Theorem 4.5. Suppose that f ∈ V ′ and (4.3) hold. If νλ1 > 2Lg then there exist
at least one weak stationary solution u∗ to (4.1) satisfying (4.2). In addition, if

ν ≥ Cfλ−1
1 + 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
Lgλ

−1
1√

1− ρ∗
then the stationary solution u∗ is unique, stable, and satisfies∫ ∞

0

(|u(s)− u∗|2 + α2‖u(s)− u∗‖2)ds

≤ |φ(0)− u∗|2 + α2‖φ(0)− u∗‖2 +
Lg√

1− ρ∗
‖φ− u∗‖2L2(−ρ(0),0;H).

(4.9)

for any solution u to (4.6) with φ ∈ BCL−∞(H). Furthermore, if

ν > Cfλ
−1
1 + 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
Lgλ

−1
1√

1− ρ∗
then u∗ is asymptotically stable.

Proof. Since all the assumptions of Theorem 4.2 are satisfied, there exists a unique
stationary solution u∗ to (4.1) satisfying (4.2). Let us set w(t) = u(t) − u∗. Then
it satisfies

d

dt
(w(t) + α2Aw(t)) = −νAw(t)−B(u(t), u(t)) +B(u∗, u∗)

− f(u) + f(u∗) + P (G(u(t− ρ(t)))−G(u∗)),
(4.10)

with initial condition w(θ) = φ(θ) − u∗, θ ∈ (−∞, 0]. For any φ ∈ BCL−∞(H),
and any t > 0 we define

U(t, φ) = |φ(0)− u∗|2 + α2‖φ(0)− u∗‖2 +
c

1− ρ∗

∫ t

t−ρ(t)
|u(s)− u∗|2ds,



14 L. T. THUY EJDE-2022/07

where the constant c > 0 is to be chosen later. Then for any u(.;φ) of (4.6) with
initial data φ ∈ BCL−∞(H), we have

U(t, ut) = |u(t)− u∗|2 + α2‖u(t)− u∗‖2 +
c

1− ρ∗

∫ t

t−ρ(t)
|u(s)− u∗|2ds. (4.11)

From (4.10) and using an estimate similar to (2.4) we obtain

d

dt
U(t, wt)

= 2
〈 d
dt

(w(t) + α2Aw(t)), w(t)
〉

+
c

1− ρ∗
|w(t)|2 − c(1− ρ′(t))

1− ρ∗
|w(t− ρ(t))|2

≤ −2ν‖w(t)‖2 + 4c0λ
−1/4
1 ‖u∗‖‖w(t)‖2 + 2Cf |w(t)|2

+ 2Lg|w(t− ρ(t))||w(t)|+ c

1− ρ∗
|w(t)|2 − c(1− ρ′(t))

1− ρ∗
|w(t− ρ(t))|2.

By Cauchy’s inequality, Poincaré’s inequality (2.1) and (4.2), we obtain

d

dt
U(t, wt)

≤ −2ν‖w(t)‖2 + 4c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

‖w(t)‖2 + 2Cf |w(t)|2

+ 2Lg|w(t− ρ(t))||w(t)|+ c

1− ρ∗
|w(t)|2 − c|w(t− ρ(t))|2

≤ −2ν‖w(t)‖2 + 4c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

‖w(t)‖2 + 2Cf |w(t)|2

+ 2
( c

2
|w(t− ρ(t))|2 +

L2
g

2c
|w(t)|2

)
+

c

1− ρ∗
|w(t)|2 − c|w(t− ρ(t))|2

≤ −2
(
ν − 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

−
L2
gλ
−1
1

2c
− cλ−1

1

2(1− ρ∗)
− Cfλ−1

1

)
× ‖w(t)‖2.

If we choose c = Lg
√

1− ρ∗, then the coefficient in the right-hand side takes it
minimum value. We conclude that

d

dt
U(t, wt)

≤ −2
(
ν − 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

− Lgλ
−1
1√

1− ρ∗
− Cfλ−1

1

)
‖w(t)‖2.

(4.12)

Integrating (4.12) from 0 to t, we obtain

U(t, wt) + 2
(
ν − 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

− Lgλ
−1
1√

1− ρ∗
− Cfλ−1

1

)∫ t

0

‖w(s)‖2ds

≤ U(0, u0).

(4.13)

From (4.11), we have

U(t, wt) ≥ |u(t)− u∗|2 + α2‖u(t)− u∗‖2



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 15

and

U(0, u0) = |φ(0)− u∗|2 + α2‖φ(0)− u∗‖2 +
Lg√

1− ρ∗
‖φ− u∗‖2L2(−ρ(0),0;H).

Then using the Poincaré inequality (2.1), inequality (4.13) becomes

|u(t)− u∗|2 + α2‖u(t)− u∗‖2

+ 2λ1

(
ν − 2c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

− Lgλ
−1
1√

1− ρ∗
− Cfλ−1

1

)
×
∫ t

0

|u(s)− u∗|2ds

≤ |φ(0)− u∗|2 + α2‖φ(0)− u∗‖2 +
Lg√

1− ρ∗
‖φ− u∗‖2L2(−ρ(0),0;H).

(4.14)

Therefore, if

ν ≥ 2c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
Lgλ

−1
1√

1− ρ∗
+ Cfλ

−1
1 ,

then the stationary solution u∗ is stable and satisfies (4.9). If

ν > 2c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
Lgλ

−1
1√

1− ρ∗
+ Cfλ

−1
1 ,

then from (4.14) we obtain∫ ∞
0

(|u(s)− u∗|2 + α2‖u(s)− u∗‖2)ds

≤ |φ(0)− u∗|2 + α2‖φ(0)− u∗‖2 +
Lg√

1− ρ∗
‖φ− u∗‖2L2(−ρ(0),0;H).

By the continuity in time of u in H, we deduce that limt→∞ |u(t) − u∗|2 = 0, i.e.
the stationary solution u∗ is asymptotically stable. �

Since
(2−ρ∗)Lgλ−1

1

2(1−ρ∗) >
Lgλ

−1
1√

1−ρ∗
for ρ∗ ∈ (0, 1), Theorem 4.5 is an improvement of

Theorem 4.4.

4.2.3. Polynomial stability: the proportional delay case. We now consider the 3D
Kelvin-Voigt-Brinkman-Forchheimer equations with proportional delay, which is a
particular case of unbounded variable delay. More precisely, we assume ρ(t) =
(1 − q)t with q ∈ (0, 1). We will show the polynomial stability of the stationary
solution.

First, we consider the pantograph equation

x′(t) = ax(t) + bx(qt), t > 0, x(0) = x0, q ∈ (0, 1). (4.15)

The following lemmas are key tools in the proof of polynomial stability results.

Lemma 4.6 ([4, Lemma 3.4]). Let a ∈ R, b > 0 and q ∈ (0, 1). Assume x is the
solution to (4.15) with x(0) > 0. Suppose p ∈ C(R+,R+) satisfies

D+p(t) ≤ ap(t) + bp(qt), t ≥ 0,

with 0 < p(0) < x(0) and where D+ denotes the Dini derivative. Then p(t) ≤ x(t)
for all t ≥ 0.



16 L. T. THUY EJDE-2022/07

Lemma 4.7 ([4, Lemma 3.5]). Let x be a solution to (4.15). If a < 0, b ∈ R, then
there exists C = C(a, b, q) > 0 such that

lim sup
t→∞

|x(t)|
tµ

= C|x(0)|,

where µ ∈ R satisfies 0 = a+ |b|qµ. Then, for some (possibly new) C = C(a, b, q) >
0, we have

|x(t)| ≤ C|x(0)|(1 + t)µ, t ≥ 0.

We are ready to state the main result in this subsection.

Theorem 4.8. Assume that f ∈ V ′ and consider (4.6) with ρ(t) = (1 − q)t, for
q ∈ (0, 1). If ν > Lgλ

−1
1 then there exists at least one weak stationary solution u∗

to (4.1) satisfying (4.2). Furthermore, if

ν > 2c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+
1

2
Lgλ

−1
1 + Cfλ

−1
1 , (4.16)

then u∗ is asymptotically stable with polynomial rate, that is, there exists C =
C(ν, Lg, q) > 0 such that

|u(t)− u∗|2 + α2‖u(t)− u∗‖2 ≤ C(|φ(0)− u∗|2 + α2‖φ(0)− u∗‖2)(1 + t)µ,

for all t ≥ 0, where

µ = logq

(2ν − 4c0λ
1/4
1

( (2νK|Ω|+‖h‖2∗)
ν(λ1ν−2Lg)

)1/2 − Lgλ−1
1 − 2Cfλ

−1
1

Lgλ
−1
1

)
< 0. (4.17)

Proof. The existence and uniqueness of a stationary solution u∗ to (4.1) follows
from Theorem 4.2. Let v(t) = u(t)− u∗, then v(t) satisfies

d

dt
v(t) + νAv(t)−B(u(t), u(t)) +B(u∗, u∗) + f(u)− f(u∗)

= P (G(u(qt))−G(u∗)),

with initial condition v(θ) = φ(θ)−u∗, θ ∈ (−∞, 0]. Then, using the same argument
as in estimate (4.8), taking ρ(t) = (1− q)t, and using the bound (4.2), we have

d

dt
(|v(t)|2 + α2‖v(t)‖2)

≤
(

4c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+ Lgλ
−1
1 + 2Cfλ

−1
1 − 2ν

)
× (‖v(t)‖2 + α2‖v(t)‖2) + Lg(|v(qt)|2 + α2‖v(qt)‖2).

(4.18)

Denoting x(t) = |v(t)|2 + α2‖v(t)‖2, noting that

4c0λ
1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+ Lgλ
−1
1 + 2Cfλ

−1
1 − 2ν < 0,

and using the Poincaré inequality (2.1), we obtain from (4.18) that

x′(t) ≤ λ1

(
4c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+Lgλ
−1
1 + 2Cfλ

−1
1 − 2ν

)
x(t) +Lgx(qt).

Applying Lemmas 4.6 and 4.7, there exists C = C(ν, Lg, q) > 0 such that

x(t) ≤ Cx(0)(1 + t)µ, ∀t ≥ 0,



EJDE-2022/07 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 17

where µ satisfies

λ1

(
4c0λ

1/4
1

( (2νK|Ω|+ ‖h‖2∗)
ν(λ1ν − 2Lg)

)1/2

+ Lgλ
−1
1 + 2Cfλ

−1
1 − 2ν

)
+ Lgq

µ = 0.

that is, µ is given by (4.17). Note that if µ < 0, we obtain the polynomial stability
of the stationary solution u∗. The proof is complete. �

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Le Thi Thuy

Department of Mathematics, Electric Power University, 235, Hoang Quoc Viet, Bac

Tu Liem, Hanoi, Vietnam
Email address: thuylt@epu.edu.vn, thuylephuong@gmail.com


	1. Introduction
	2. Preliminaries
	3. Existence and uniqueness of weak solutions
	4. Existence and stability of stationary solutions
	4.1. Existence of weak stationary solutions
	4.2. Stability results

	References