Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 21, pp. 1–24.

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

NONSTATIONARY LAMÉ SYSTEM WITHOUT

DEFINITE SIGN ENERGY

MANUEL MILLA MIRANDA, ALDO T. LOUREDO,
MARCONDES R. CLARK, GIOVANA SIRACUSA

Abstract. This article concerns the existence and decay of solutions of a

nonstationary Lamé system. This system has a nonlinear perturbation that
produces an energy without definite sign. We consider displacement and trac-

tion conditions at the boundary and a general nonlinear boundary damping.
We also obtain exponential decay of the energy.

1. Introduction

We consider an isotropic homogeneous elastic body that in its equilibrium posi-
tion occupies a bounded set Ω of R3. Suppose that at the instant t = 0 an external
force acts on the body and then stops. As a consequence of this, the particles of
the body begin to oscillate. The motion of these small oscillations of the particles
can be described by the nonstationary Lamé system

u′′(x, t)− µ∆u(x, t)− (λ+ µ)∇ div u(x, t) = 0, x ∈ Ω, t > 0, (1.1)

where u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) denotes the displacement of the particle
x of the body at the instant t, u′(x, t) = ∂

∂tu(x, t), ∆u(x, t) = (∆u1(x, t),∆u2(x, t),

∆u3(x, t)), ∇ =
(
∂
∂x1

, ∂
∂x2

, ∂
∂x3

)
, div u(x, t) =

∑3
i=1

∂ui
∂xi

(x, t) and λ, µ are the Lamé

coefficients of the material of the body with µ > 0 and λ+µ > 0 (see, for example,
Ciarlet [4], Duvaut and Lions [6] and Landau and Lifshitz [11]).

Existence of solutions of the mixed value problem for the system (1.1) can be
found, for example, in Marsden and Hughes [17]. The decay of solutions for (1.1)
with linear boundary conditions is analyzed in Caldas [3] and in Komornik [9]. The
decay of the energy for some variations of (1.1) is investigated in Bociu, Derochers
and Toundykov [2] and in Cordeiro, Santos and Raposo [5]. The inverse problem
and observability for system (1.1) are studied in Belishev and Lasiecka [1] and in
Imanuvilov and Yamamoto [7].

We introduce a nonlinear perturbation in (1.1) and consider a given nonlinear
boundary damping acting on a part of the boundary of Ω. This in the n−dimensional
frame work.

The objective of this paper is to investigate existence and decay of global solu-
tions of the above mixed problem. Thus we consider an open bounded set Ω of Rn

2020 Mathematics Subject Classification. 35L15, 35L20, 35K55, 35L60, 35L70.
Key words and phrases. Nonlinear elasticity; mixed boundary conditions;

existence of solutions.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted March 29, 2021. Published March 18, 2022.

1



2 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

whose boundary Γ, of class C2, is constituted of two parts Γ0 and Γ1, both with
positive Lebesgue measures, and Γ0 ∩ Γ1 = ∅. The unit exterior normal vector at
x ∈ Γ is denoted by ν(x). In these conditions we have the problem

u′′(x, t)− µ∆u(x, t)− (λ+ µ)∇ div u(x, t) + |u(x, t)|ρ = 0, in Ω× (0,∞); (1.2)

u(x, t) = 0, on Γ0 × (0,∞); (1.3)

µ
∂u

∂ν
(x, t) + (λ+ µ) div u(x, t)ν(x) + h(x, u′(x, t)) = 0, on Γ1 × (0,∞); (1.4)

u(x, 0) = u0(x), u1(x, 0) = u1(x), x ∈ Ω, (1.5)

where u(x, t) = (u1(x, t), . . . , un(x, t)), ∆u = (∆u1, . . . ,∆un), ∇ =
(
∂
∂x1

, . . . , ∂
∂xn

)
,

div u =
∑n
i=1

∂ui
∂xi

, ∂u
∂ν =

(
∂u1

∂ν , . . . ,
∂un
∂ν

)
, |u|ρ = (|u1|ρ, . . . , |un|ρ) (ρ is a positive

real number) and h(x, u) = (h1(x, u1), . . . , hn(x, un)), with hi(x, s) is measurable
in Γ1 and continuous in R, i = 1, . . . , n.

To obtain the existence of global solutions of the above problem we must over-
come two serious difficulties. First, note that∫

Ω

|u(x, t)|ρ.u′(x, t)dx =

n∑
i=1

∫
Ω

|ui(x, t)|ρu′i(x, t)dx

=

n∑
i=1

1

ρ+ 1

d

dt

∫
Ω

|ui(x, t)|ρui(x, t)dx.

Then ∫ t

0

∫
Ω

|u(x, t)|ρ.u′(x, t)dxds =

n∑
i=1

1

ρ+ 1

∫
Ω

|ui(x, t)|ρui(x, t)dx

−
n∑
i=1

1

ρ+ 1

∫
Ω

|ui(x, 0)|ρui(x, 0)dx.

Note that each term
∫

Ω
|ui(x, t)|ρui(x, t)dx does not have definite sign. Thus the

energy method does not work in the present problem to obtain global solutions.
To overcome this difficulty, we introduce a significative generalization of an idea of
Tartar [24] (cf. [13, 19, 20, 21] for a direct application in [24]). This method simplify
the potential well method due to Sattinger [22]. Of course, the norm of initial data
u0 and u1 are related to ρ and this ρ depends on the embedding of Sobolev spaces.

The second difficulty is caused by the generality of the functions hi(x, s). As-
suming that each hi(x, s) is strongly monotone in R, hi(x, 0) = 0 a.e. in x ∈ Γ1 and
using an approximation of continuous function by Lipschitz continuous function (cf.
[16, 23]), we overcome this difficulty. Also in this part we introduce a trace result
given by a theorem for non-smooth functions.

In our approach on the existence of solutions we use the Galerkin method with
an special basis in order to obtain a second a priori estimate of the approximate
solutions. This choose is related to the boundary condition (1.4) at t = 0. In the
passage to the limit on the nonlinear terms of the approximate problem, we use
compactness arguments (see Lions [13]) and a result by Strauss [23].

The decay of solutions is derived by the multiplier method (see Komornik and
Zuazua [10]). In our approach considered a boundary damping given by a strongly
monotone Lipschitz continuous function. We note that Lasiecka and Tataru [12] and



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 3

in Komornik and Zuazua [8] considered a wave equation with boundary damping
given by a function h(s) with |h(s)| ≤ L|s| for |s| large enough.

2. Notation and main results

The notation introduced in the previous section for the n-dimensional case will
now be complemented for establishing our results.

Let L2(Ω) be the usual Hilbert space equipped with the following inner product
and norm:

(f, g) =

∫
Ω

f(x)g(x)dx and |f |2 = (f, f).

We use the Hilbert space

H1
Γ0

(Ω) := {w ∈ H1(Ω) : w = 0 on Γ0}
equipped with the following inner product and norm:

((u, z)) =

n∑
i=1

∫
Ω

∂w

∂xi
(x)

∂z

∂xi
(x)dx and ‖w‖2 = ((w,w)).

We use capital letters with double trace to represent the n−product of the same
space. Thus L2(Ω) = (L2(Ω))n, H1

0(Ω) = (H1
0 (Ω))n, L2(Γ1) = (L2(Γ1))n,

H1/2(Γ1) = (H1/2(Γ1))n. Each of such space is endowed with its product topology.
The dual of H1/2(Γ1) is denoted by H−1/2(Γ1).

Remark 2.1. We consider H1
Γ0

(Ω) with its usual product topology and V =

H1
Γ0

(Ω) with the inner product

((u, v))V = µ((u, v)) + (λ+ µ)(div u,div v)L2(Ω).

We have

µ1/2‖u‖H1
Γ0

(Ω) ≤ ‖u‖V ≤ [µ+ n2(λ+ µ)]1/2‖u‖H1
Γ0

(Ω), ∀u ∈ H1
Γ0

(Ω).

We will use the notation H = L2(Ω) equipped with the inner product (u, v)H =∑n
i=1(ui, vi). Let B be the positive self-adjoint operator of H defined by triplet

{V,H, ((u, v))V } (see Lions [14]). Then

(Bu, v) = ((u, v))V , ∀u ∈ D(B), ∀v ∈ V,
and

B = −µ∆− (λ+ µ)∇div, D(B) = {v ∈ V : Bu ∈ H, γ1u = 0 on Γ1}, (2.1)

where

γ1u = µ
∂u

∂ν
+ (λ+ µ)(div u)ν. (2.2)

Note that γ1 is well defined (cf. Theorem 3.3).
We make the following restrictions on ρ:

ρ > 1 if n = 1, 2, (2.3)

n+ 1

n
≤ ρ ≤ n

n− 2
if n ≥ 3. (2.4)

The notation X ↪→ Y indicates that the space X is continuously embedded in Y .
From restrictions (2.4), we have

H1
Γ0

(Ω) ↪→ Lq
∗
(Ω) ↪→ L2ρ(Ω) ↪→ Lρ+1(Ω) ↪→ Lρ(Ω),



4 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

Lq
∗
(Ω) ↪→ Ln(ρ−1)(Ω),

where q∗ = 2n
n−1 if n ≥ 3. Then there exist positive constants k0, . . . , k5 such that

‖w‖Lρ+1(Ω) ≤ k0‖w‖, ‖w‖Lρ(Ω) ≤ k1‖w‖, (2.5)

‖w‖L2ρ(Ω) ≤ k2‖w‖, ‖w‖Ln(ρ)−1(Ω) ≤ k3‖w‖, (2.6)

‖w‖Lq∗ (Ω) ≤ k4‖w‖, |w| ≤ k5‖w‖, (2.7)

for all w ∈ H1
Γ0

(Ω). Note that H
1/2
Γ0

(Γ1) ↪→ Lq
∗
1 (Γ1) where q∗1 = 2(n−1)

n−1 , n ≥ 3, and
q∗1 ≥ ρ+ 1. Thus

H1
Γ0

(Ω) ↪→ H1/2(Γ1) ↪→ Lq
∗
1 (Γ1) ↪→ Lρ+1(Γ1) ↪→ L2(Γ1).

Then there exist positive constants k6 and k7 such that

‖w‖Lρ+1(Γ1) ≤ k6‖w‖, ‖w‖L2(Γ1) ≤ k7‖w‖, for all w ∈ H1
Γ0

(Γ1). (2.8)

We assume that the function h = (h1, h2, . . . , hn) satisfies

h ∈ C0(R,L∞(Γ1)); (2.9)

hi(x, 0) = 0 a.e. for x in Γ1, i = 1, 2, . . . , n; (2.10)

[hi(x, s)− hi(x, r)](s− r) ≥ d0(s− r)2, ∀s, r ∈ R, a.e. x in Γ1, (2.11)

where i = 1, 2, . . . , n and d0 is a positive constant.

Remark 2.2. An example of functions hi(x, s), i = 1, 2, . . . , n, satisfying (2.9)–
(2.11) is given by

hi(x, s) = δ(x)(s+ |s|αs), x ∈ Γ1, s ∈ R

δ ∈ L∞(Γ1), δ(x) ≥ δ0 > 0 and α > 1, α constant.

Consider

λ∗ =
( 1

4N

) 1
ρ−1

and N =
nkρ+1

0

(ρ+ 1)µ
ρ+1

2

. (2.12)

We make the following assumptions on the initial data u0 and u1:

u0 ∈ D(B), u1 ∈ H1
0(Ω), (2.13)

‖u0‖V < λ∗, (2.14)

1

2
‖u1‖2H +

1

2
‖u0‖2V +

nkρ+1
0

(ρ+ 1)µ
ρ+1

2

‖u0‖ρ+1
V <

1

4
(λ∗)2. (2.15)

Let A be the operator

A = −µ∆− (λ+ µ)∇ div, (2.16)

and the Hilbert space

W = {u ∈ V : Au ∈ H} (2.17)

be provided with the inner product

(u, v)W = ((u, v))V + (Au,Av)H . (2.18)

Note that the operators A and B, introduced in (2.1) and (2.16), respectively, have
the same form but D(B) is contained in W .



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 5

Theorem 2.3. Suppose that (2.3), (2.4), (2.9)-(2.11) and (2.13)-(2.15) hold. Then
there exist a function u such that

u ∈ L∞(0,∞;V ) ∩ L2
loc(0,∞;W ), (2.19)

u′ ∈ L∞(0,∞;H) ∩ L∞loc(0,∞;V ), (2.20)

u′′ ∈ L∞loc(0,∞;H), div u ∈ L∞(0,∞;L2(Ω)), (2.21)

satisfying

u′′ − µ∆u− (λ+ µ)∇div u+ |u|ρ = 0 in L2
loc(0,∞;H), (2.22)

γ1u+ h(·, u′) = 0 inL1
loc(0,∞;H−1/2(Γ1) + L1(Γ1)), (2.23)

u(0) = u0, u′(0) = u1. (2.24)

It is worth noting that

(1) The energy E of system (2.22)–(2.24), which is defined in (2.42), does not
has definite sign.

(2) The uniqueness of solution of Problem (2.22)–(2.24) is an open problem.
The difficulty is due to the general assumption made about the function
h(·, u′).

To obtain the decay of the energy of the Problem (1.2)–(1.5), we make some
restrictions on Γ and hi(x, s). This will lead us to a new theorem on the existence
of solutions. The justification of this new theorem will be find in the proof of the
decay of the energy, in Section 5. We assume that there is x0 ∈ Rn such that

Γ0 = {x ∈ Γ : m(x) · ν(x) ≤ 0} and Γ1 = {x ∈ Γ : m(x) · ν(x) > 0}, (2.25)

where m(x) = x− x0, x ∈ Γ, and x · y is the inner product of Rn. Let

R = max{||m(x)|| : x ∈ Γ1}, (2.26)

0 < b0 = min{m(x)ν(x);x ∈ Γ1}. (2.27)

Assume also that each hi(x, s) has the form

hi(x, s) = [m(x) · ν(x)]pi(s),

where pi(s) is strongly monotone and Lipschitiz continuous, namely,

[pi(r)− pi(s)](r − s) ≥ d∗0(r − s)2, ∀r, s ∈ R, pi(0) = 0, i = 1, 2, . . . , n, (2.28)

|pi(s)| ≤ L|s|, ∀s ∈ R, i = 1, 2, . . . , n, (2.29)

where d∗0 and L are positive constants.
Let

N1 =
1

µ
ρ+1

2

{[ 1

ρ+ 1
+
∣∣ 2n

ρ+ 1
− (n− 1)

∣∣]kρ+1
0 n+

2R

ρ+ 1
kρ+1

6 n
}
, (2.30)

where k0 and k6 were defined in (2.5) and (2.8), respectively. We consider the real
number

λ∗1 =
( 1

4N1

) 1
ρ−1

. (2.31)

and introduce the hypotheses

u0 ∈ D(B), u1 ∈ H1
0(Ω), (2.32)

‖u0‖V < λ∗1, (2.33)



6 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

1

2
‖u1‖2H +

1

2
‖u0‖2V +

nkρ+1
0

(ρ+ 1)µ
ρ+1

2

‖u0‖ρ+1
V <

1

4
(λ∗1)2. (2.34)

Note that λ∗1 < λ∗, where λ∗ was given in (2.12).

Theorem 2.4. Assume (2.3), (2.4), (2.28), (2.29), (2.32)–(2.34) hold. Then there
exist a unique function u in the class (2.19)–(2.21) such that u satisfies

u′′ − µ∆u− (λ+ µ)∇div u+ |u|ρ = 0 in L2
loc(0,∞;H), (2.35)

γ1u+ [m(·)ν(·)]p(u′) = 0 in L2
loc(0,∞;H1/2(Γ1)), (2.36)

u(0) = u0, u′(0) = u1. (2.37)

We assume that the solution u given by Theorem 2.4 has the regularity

u ∈ L2
loc(0,∞;H2(Ω)) (2.38)

Note that the u given by Theorem 2.4 is solution of an equation

Au = f in Ω× (0,∞) (f ∈ L2
loc(0,∞;H)),

u = 0 on Γ0 × (0,∞),

γ1u = g on Γ1 × (0,∞), (g ∈ L2
loc(0,∞;H1/2(Γ1)).

If Ω is a bounded domain of R3 with boundary Γ ∈ C2, then u has regularity (2.38)
(see Ciarlet [4, Theorem 63-6, p. 296]).

We consider the constants

M =
1

µ1/2
[4R+ 2(n− 1)k5] , (2.39)

P =
1

µ
(n− 1)2R2L2k2

7 +
1

µ
R2L2 +R, (2.40)

σ = min
{ 1

2M
,
b0d
∗
0

P

}
, (2.41)

where k5 and k7 were defined in (2.7) and (2.8), respectively. We introduce the
energy

E(t) =
1

2
‖u′(t)‖H +

1

2
‖u(t)‖V +

1

ρ+ 1
(|u(t)|ρ, u(t))H , t ≥ 0. (2.42)

Theorem 2.5. Let u be the solution given by Theorem 2.4, and assume (2.38)
holds. Then

E(t) ≤ 3E(0)e−2σt/3, ∀t ≥ 0.

Before proving Theorem 2.3, we will show some previous results concerning to
the trace γ1u for a function u ∈W and on the approximation of the function h by
a Lipchitz continuous function hl.

3. Preliminary results

Let O be a star-shaped subset of Rn. Consider the linear homotetic transforma-
tion ση(x) = ηx, η > 0. Note that for η > 1,

O ⊂ O ⊂ ση(O). (3.1)

We consider a vectorial function v defined in O. For η > 0 introduce the function

ση ◦ v : ση(O)→ Rn, (ση ◦ v)(y) = v(σ1/n(y)).



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 7

Note that when η > 1, the domain of the function ση ◦ v contains the domain of w
(see (3.1)).

Proposition 3.1. Let S ∈ D′(O). Then

(1) ση ◦ S defined by

〈ση ◦ S, θ〉 =
1

ηn
〈S, σ 1

η
◦ θ〉, θ ∈ D(ση(O))

belongs to D′(ση(O)) (η > 0).

(2) ∂
∂yi

(ση ◦ S) = ηση ◦
(
∂
∂yi

S
)

(η > 0).

(3) If η > 1, η → 1, the restriction to O of ση ◦ S converges in the distribution
sense to S.

(4) If v ∈ Lp(O) (1 ≤ p <∞), ση ◦ v ∈ Lp(ση(O)) (η > 0). For η > 1, η → 1,
the restriction to O of ση ◦ v converges to v in Lp(O).

The proof of the above proposition can be found in Temam [25].

Theorem 3.2. The space (D(Ω))n is dense in W .

Proof. Let U be an open set of Rn with boundary ∂U of class C2. We introduce
the Hilbert space

X(U) = {u ∈ H1(Ω) : Au ∈ L2(Ω)}
equipped with the scalar product

(u, v)X(U) = (u, v)H1(U) + (Au,Au)L2(U).

The proof will be divided into four steps:

Step 1. By truncation and regularization (see Lions [14]), we prove that (D(Rn))n

is dense in X(Rn).

Step 2. Let (Ul)1≤l≤m be an open cover of Γ0 and Γ1 with U+
l = Ω∩Ul star-shaped

with respect to one of its points, l = 1, 2, . . . ,m. Let (ϕl)0≤l≤m be a C∞ partition

of unity subordinate to the open convering Ω, (Ul)1≤l≤n of Ω. Thus

ϕ0(x) +

n∑
l=1

ϕl(x) = 1, ∀x ∈ Ω, ϕ0 ∈ D(Ω), ϕl ∈ D(Ul), l = 1, 2, . . . ,m.

Considering u ∈W ,

u = ϕ0u+

m∑
l=1

ϕlu. (3.2)

We use the notation vl = ϕlu for l = 0, 1, . . . ,m.
Analysis of v0 = (v01, . . . , v0n). Represent by U0 an open of Rn such that

suppϕ0 ⊂ U0 ⊂ Ω is contained in U0 and U0 is star-shaped with respect to one of
its points. After translation, we consider U0 as being star-shaped with respect to
0 ∈ Rn. We define

ση ◦ v0 = (ση ◦ v01, . . . , ση ◦ v0n), η > 1.

Then by (3.1) and Proposition 3.1 part 1, we have that ση ◦ v0 is defined in ση(U0).
Consider ψ ∈ D(ση(U0)) such that ψ = 1 on U0, and

w0η = ψ[ση ◦ v0], η > 1. (3.3)



8 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

Then suppw0η is contained in ση(U0). By Proposition 3.1 part 2, we obtain

∂

∂xi
w0η =

∂ψ

∂xi
[ση ◦ v0] + ηψ

(
ση ◦

∂v0

∂xi

)
, (3.4)

Aw0η = f(η)ψ[ση ◦Av0] +
∑

1≤|α|≤2
|β|≤1

gβ(η)Pα(ψ)[ση ◦Qβ(v0)], (3.5)

where f(η) and gβ(η) are real functions of η with f(η)→ 1 as η → 1, α and β, are
multi-indices α = (α1, . . . , αn), β = (β1, . . . , βn) and Pα, Qβ are partial differential
operators.

By the two preceding equalities, we obtain that w0η ∈ X(ση(U0)). Consider w̃0η

the extension of w0η by zero outside of ση(U0). Then w̃0η ∈ X(Rn). By the first
part, we have that w̃0η can be approximated in X(Rn) by functions of (D(Rn))n.
Consequently

w0η can be approximate in X(ση(U0)) by functions of (D(ση(U0)))n. (3.6)

By (3.3)–(3.5) we have

w0η

∣∣
U0

= ση ◦ v0

∣∣
U0
,

∂w0η

∂xi

∣∣
U0

= ηση ◦
v0

∂xi

∣∣
U0
,

Aw0η

∣∣
U0

= f(η)[ση ◦Av0]
∣∣
U0
.

Then by Proposition 3.1 part 3, we obtain

w0η

∣∣
U0
→ v0 in L2(Ω) as η → 1,

∂w0η

∂xi

∣∣
U0
→ ∂v0

∂xi
in L2(Ω) as η → 1,

Aw0η

∣∣
U0
→ Av0 in L2(Ω) as η → 1.

By (3.6) and the last three convergences we conclude that v0 can be approximated
in X(U0) by functions of D(U0).

Step 3. Analysis of vl, l = 1, 2, . . . ,m. In this case we apply similar arguments to
those used previously for v0. Thus we take U+

l instead of U0. We can assume the

U+
l is star-shaped with respect to 0 ∈ Rn. Consider ση(U+

l ) instead of ση(U0). We
introduce

ψ ∈ D(ση(U+
l ) with ψ ≡ 1 on U+

l .

Consider wlη = ψ[ση ◦ vl], η > 1. Then

wlη ∈ X(ση(U+
l )),

suppwlη is contained in ση(U+
l ),

w̃lη belongs to X(Rn),

wlη
∣∣
U+
l

→ vl in X(U+
l ) as η → 1.

Thus vl can be approximated in X(U+
l ) by a function of (D(U+

l ))n. By (3.2)
and the above results we conclude that v ∈ W can be approximated in X(Ω) by
functions of (D(Ω))n.

Step 4. Theorem follows because X(Ω) and W have equivalent norms in W . �



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 9

Theorem 3.3. There exists a linear continuous map γ1 : W → H−1/2(Γ1), u 7→
γ1u such that

γ1u = µ
∂u

∂ν
+ (λ+ µ)(div u)ν, ∀u ∈ (D(Ω))n. (3.7)

Furthermore

(Au, v)H = ((u, v))V − 〈γ1u, v〉H−1/2(Γ1)×H1/2(Γ1), ∀u ∈W, v ∈ V. (3.8)

Proof. Let u ∈ (D(Ω))n and v ∈ V . By Gauss’s Divergence Theorem, we obtain

(Au, v)H = ((u, v))V − (γ1u, v)L2(Γ1). (3.9)

Then

|(γ1u, v)L2(Γ1)| ≤ ‖u‖V ‖v‖V + C‖Au‖H‖v‖V .
Therefore,

|〈γ1u, v〉H−1/2(Γ1)
×H1/2(Γ1)

| ≤ C‖u‖W ‖v‖V . (3.10)

Let ξ ∈ H1/2(Γ1), then by the Trace Theorem there is v ∈ V such that γ0v = ξ and
the map

H1/2(Γ1)→ V, ξ 7→ v

is continuous. By the above map and (3.10), we obtain∣∣〈γ1u, ξ〉H−1/2(Γ1)
×H1/2(Γ1)

∣∣ ≤ C‖u‖W ‖ξ‖H1/2(Γ1), ∀u ∈ (D(Ω))n, ξ ∈ H1/2(Γ1).

By this inequality and the density of (D(Ω))n in W given by Theorem 3.2, we
obtain (3.7). The equality (3.9) and Theorem 3.2 provide (3.8). �

Proposition 3.4. Let h be a function satisfying (2.9)–(2.11). Then for each i =
1, . . . , n there exists a sequence (hil) of functions of C0(R, L∞(Γ1)) such that

hil(x, 0) = 0 for a.e. x in Γ1; (3.11)

[hil(x, s)− hil(x, r)](s− r) ≥ d0(s− r)2, ∀s, r ∈ R for a.e. x ∈ Γ1; (3.12)

There exists a function cl ∈ L∞(Γ1) satisfying (3.13)

|hil(x, s)− hil(x, r)| ≤ cl|s− r|, ∀s, r ∈ R, for a.e. x in Γ1;

(hil) converges to hi uniformly on boundary sets for R for a.e. x ∈ Γ1. (3.14)

4. Proof of Theorem 2.3

In this section, we will prove the existence of solution of problem (2.22)–(2.24).

Proof of Theorem 2.3. We employ the Faedo-Galerkin’s method with a special basis
of V (see [16] or [18] for other special basis). Let (u1

l ) be a sequence of (D(Ω))n

such that

u1
l → u1 in H1

0(Ω). (4.1)

Fix l ∈ N. With u0 and u1
l we constructed a basis

{wl1, wl2, . . . } (4.2)

of V such that u0, u1
l belong to the subspace [wl1, w

l
2] generated by wl1 and wl2.

Consider the approximation (hil) of hi (i = 1, . . . , n) given by Proposition 3.4.
Here we will denote (hl) = (h1l, . . . , hnl).



10 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

Remark 4.1. Since u0 ∈ D(B) and u1
l ∈ H1

0(Ω), we have γ1u = 0 and hl(·, u1
l ) = 0

on Γ1. Thus

γ1(u0) + hl(·, u1
l ) = 0 in Γ1,∀l ∈ N.

Consider V lm = [wl1, . . . , w
l
m] the subspace of V generated by the first m vectors

of the basis of (4.2). Let us find the approximate solution ulm(t) ∈ V lm of Problem
(1.2)–(1.5), that is,

ulm(t) =

m∑
j=1

glmj(t)w
l
j ,

where ulm(t) ∈ V lm is the solution of the system

(u′′lm(t), v)H + ((ulm(t), v))V + (|ulm(t)|ρ, v)H

+ (hl(·, u′lm(t)), v)L2(Γ1) = 0, ∀v ∈ V lm
(4.3)

ulm(0) = u0, u′lm(0) = u1
l . (4.4)

System (4.3)–(4.4) has a solution on an interval [0, tlm) with tlm < T . This solution
can be extended to the interval [0, T ] as a consequence of the a priori estimates
that shall be proved.

First estimate. Taking v = u′lm(t) ∈ V lm in (4.3), we obtain

1

2

d

dt
‖u′lm(t)‖2H +

1

2

d

dt
‖ulm(t)‖2V + (|ulm(t)|ρ, u′lm(t))V

+ (hl(·, u′lm(t)), u′lm(t))L2(Γ1) = 0.
(4.5)

Then

(|ulm(t)|ρ, u′lm(t))H =
1

ρ+ 1

d

dt
(|ulm(t)|ρ, ulm(t))H

and by (3.12),

(hl(·, u′lm(t)), u′lm(t))L2(Γ1) ≥ d0‖u′lm(t)‖2L2(Γ1).

Remark 4.2. As h(x, u) = (h1(x, u1), . . . , hn(x, un)) where hi(x, s) is measurable
in Γ1 and continuous in R, i = 1, . . . , n, by (3.12) we have

((hl(·, u′lm(t)), u′lm(t))L2(Γ1) =

n∑
i=1

(hil(·, u′kmi(t)), u′kmi(t))

≥
n∑
i=1

d0|u′kmi(t)|2 = d0‖u′km(t)‖2L2(Γ1).

(4.6)

Putting the above two expressions in (4.5) and then integrating on [0, t], 0 < t <
tlm, we obtain

1

2
‖u′lm(t)‖2H +

1

2
‖ulm(t)‖2V +

1

ρ+ 1
(|ulm(t)|ρ, ulm(t))H

+ d0

∫ t

0

‖u′lm(s)‖2L2(Γ1)ds

≤ 1

2
‖u1

l ‖2H +
1

2
‖u0‖2V +

1

ρ+ 1
(|u0|ρ, u0)H .

(4.7)



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 11

Next, our goal is to determine for which t ∈ (0, tlm) the first member of (4.7)
becomes non-negative. By (2.5) and Remark 2.1, we obtain∣∣ 1

ρ+ 1
(|ulm(t)|ρ, ulm(t))H

∣∣ ≤ 1

ρ+ 1
‖ulm(t)‖ρ+1

Lρ+1(Ω) ≤
nkρ+1

0

(ρ+ 1)µ
ρ+1

2

‖ulm(t)‖ρ+1
V .

Also ∣∣ 1

ρ+ 1
(|u0|ρ, u0)H

∣∣ ≤ nkρ+1
0

(ρ+ 1)µ
ρ+1

2

‖u0‖ρ+1
V .

Then by (2.15) we find a positive real number τ such that

1

2
‖u1

l ‖2H +
1

2
‖u0‖2V +

nkρ+1
0

(ρ+ 1)µ
ρ+1

2

‖u0‖ρ+1
V < τ <

1

4
(λ∗)2, ∀l ≥ l∗0. (4.8)

Taking into account the three inequalities in (4.7), we obtain

1

2
‖u′lm(t)‖2H +

1

2
‖ulm(t)‖2V −

nkρ+1
0

(ρ+ 1)µ
ρ+1

2

‖ulm(t)‖ρ+1
V

+ d0

∫ t

0

‖u′lm(s)‖2L2(Γ1)ds

≤ τ < 1

4
(λ∗)2, ∀l ≥ l∗0, ∀t ∈ [0, tlm).

(4.9)

We analyze for which t ∈ (0, tlm), we would have

1

4
‖ulm(t)‖2V −

nkρ+1
0

(ρ+ 1)µ
ρ+1

2

‖ulm(t)‖ρ+1
V ≥ 0.

Motivated by the above inequality, we consider the function

J(λ) =
1

4
λ2 − nkρ+1

0

(ρ+ 1)µ
ρ+1

2

λρ+1, λ ≥ 0.

We find that

J(λ) = λ2
[1
4
− nkρ+1

0

(ρ+ 1)µ
ρ+1

2

λρ−1
]

which implies

J(λ) ≥ 0 if 0 ≤ λ ≤
[ (ρ+ 1)µ

ρ+1
2

4nkρ+1
0

] 1
ρ−1 = λ∗. (4.10)

To continue the proof we need the following result.

Lemma 4.3. We have

‖ulm(t)‖V < λ∗, ∀t ∈ [0,∞), ∀l ≥ l0, ∀m ∈ N.

Proof. Fix m ∈ N. We argue by contradiction. Suppose that there exists t1 ∈
(0, tlm) such that ‖ulm(t1)‖V ≤ λ∗ and let θ(t) = ‖ulm(t)‖V . As θ is continuous
on [0, t1], by the Intermediate Value Theorem we have that there exists τ1 ∈ (0, t1]
such that θ(τ1) = λ∗. Let

t∗ = inf{τ ∈ (0, tlm) : θ(τ) = λ∗}.
We have

θ(t∗) = λ∗ because θ is continuous on [0, tlm); (4.11)

0 < t∗ < tlm because θ(0) = ‖u0‖V < λ∗; (4.12)



12 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

θ(t) < λ∗, ∀t ∈ [0, t∗). (4.13)

By (4.10) and (4.13), we obtain

J(‖ulm(t)‖V ‖) ≥ 0, ∀t ∈ [0, t∗).

Putting this inequality in (4.8), we find

1

4
‖ulm(t)‖2V ≤ τ <

1

4
(λ∗)2, ∀t ∈ [0, t∗).

Taking the limit in this inequality as t → t∗, t < t∗ and using (4.11), we arrive to
a contradiction. Thus the lemma is proved. �

By (4.8), Lemma 4.3 and (4.10), we obtain

1

2
‖u′lm(t)‖2H +

1

4
‖ulm(t)‖2V + d0

∫ t

0

‖u′lm(s)‖L2(Γ1)ds

<
1

4
(λ∗)2, ∀t ∈ [0,∞), ∀l ≥ l0, ∀m ∈ N.

(4.14)

Thus

(ulm) is bounded in L∞(0,∞;V ) ∀l ≥ l0, ∀m ∈ N; (4.15)

(u′lm) is bounded in L∞(0,∞;H), ∀l ≥ l0, ∀m ∈ N; (4.16)

(u′lm) is bounded in L2(0,∞;L2(Γ1)), ∀l ≥ l0, ∀m ∈ N. (4.17)

Second estimate. We differentiate the approximate equation (4.3) and then we
take v = u′′lm(t). We obtain

1

2

d

dt
‖u′′lm(t)‖2H +

1

2

d

dt
‖u′lm(t)‖V

+
(
ρ|ulm(t)|ρ−2ulm(t)u′lm(t), u′′lm(t)

)
H

+ (h′l(·, (u′lm(t))u′′lm(t), u′′lm(t))L2(Γ1) = 0.

(4.18)

From Hölder inequality applied to 1
n + 1

q∗ + 1
2 = 1, (2.6), (2.7) and estimate (4.15),

we find that

|(ρ|ulm(t)|ρ−2ulm(t)u′lm(t), u′′lm(t))H | ≤ C‖u′lm(t)‖2V +
1

2
‖u′′lm(t)‖2H ,

where C > 0 is a constant independent of l ∈ N andm ∈ N. By (3.12) of Proposition
3.4, we obtain

(h′l(·, u′lm(t))u′′lm(t), u′′lm(t))L2(Γ1) ≥ d0‖u′′lm(t)‖2L2(Γ1).

Taking into account the last two inequalities in (4.18), we obtain

1

2

d

dt
‖u′′lm(t)‖2H +

1

2

d

dt
‖u′lm(t)‖V + d0‖u′′lm(t)‖2L2(Γ1)

≤ C‖u′lm(t)‖2V +
1

2
‖u′′lm(t)‖2H .

(4.19)

Third estimate. We make t = 0 in (4.3) and then take v = u′′lm(0). We obtain

‖u′′lm(0)‖2H + ((u0, u′′lm(0)))V + (|u0|ρ, u′′lm(0))H + (hl(·, u1
l ), u

′′
lm(0))L2(Γ1) = 0.

From Remark 4.1 it follows that

‖u′′lm(0)‖2H + (Bu0, u′′lm(0)H + (|u0|ρ, u′′lm(0))H = 0. (4.20)



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 13

In this part the choose of the special basis (4.2) is crucial.
By applying hypothesis (2.13) and inequality (2.6) in (4.20), we have

‖u′′lm(0)‖2H ≤ C, ∀l ≥ l0, ∀m ∈ N. (4.21)

Consider a real number T > 0. Integrating both sides of inequality (4.19) on
[0, t], 0 ≤ t ≤ T , and taking into account the estimate (4.21) and the convergence
(4.1), we obtain

1

2
‖u′′lm(t)‖2H +

1

2
‖u′lm(t)‖2V + d0

∫ t

0

‖u′′lm(s)‖2L2(Γ1)ds

≤ C +

∫ t

0

[C‖u′lm(s)‖2V +
1

2
‖u′′lm(s)‖2H ]ds, 0 ≤ t ≤ T,

(4.22)

where the constant C > 0 is independent of l ≥ l0, m ∈ N and T > 0.
By Gronwall Lemma and noting that T > 0 was arbitrary, by (4.22) we obtain

(u′lm) is bounded in L∞loc(0,∞;V ), ∀l ≥ l0, ∀m ∈ N; (4.23)

(u′′lm) is bounded in L∞loc(0,∞;H), ∀l ≥ l0, ∀m ∈ N; (4.24)

(u′′lm) is bounded in L2
loc(0,∞;L2(Γ1)), ;∀l ≥ l0, ∀m ∈ N. (4.25)

Pass to limit in m. In what follows of the paper it will be understood that various
subsequences of the principal sequence will be considered. Also the diagonal process
will be applied to obtain convergence in all (0,∞).

By (4.15)–(4.17) and (4.23)–(4.25), we find that there exists a subsequence of
(ulm), still denoted by (ulm), and a function ul such that

ulm → ul weak star in L∞(0,∞;V ); (4.26)

u′lm → u′l weak star in L∞(0,∞;H) ∩ L∞loc(0,∞;V ); (4.27)

u′′lm → u′′l weak star in L∞loc(0,∞;H); (4.28)

u′′lm → u′′l weak in L2
loc(0,L2(Γ1)). (4.29)

It follows from (4.27) that

u′lm → u′l weak star in L∞loc(0,∞;H1/2(Γ1)). (4.30)

As the embedding of V in H is compact, we obtain by (4.26) and (4.27) that

ulm → ul in L∞loc(0,∞;H).

Thus

ulm(x, t)→ ul(x, t) a.e. in Ω× (0, T ).

On the other hand, by (4.15) and (2.6), we find that

(|ulm|ρ)m∈N is bounded in L2(0, T ;H).

These two last results, Lions Lemma [13] and the diagonal process imply that

|ulm|ρ → |ul|ρ weak in L2
loc(0,∞;H). (4.31)

In a similar way, noting that the embedding of H1/2(Γ1) in L2(Γ1) is compact,
by (4.30) and (4.29), we obtain

hl(x, u
′
lm)→ hl(x, u

′
l) for a.e. in Γ1 × (0, T ) (4.32)



14 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

and by (3.13) and (4.23) we obtain that

(hl(·, u′lm)) is bounded in L2(0, T ;L2(Γ1)).

Therefore

hl(x, u
′
lm)→ hl(x, u

′
l) weak in L2

loc(0,∞;L2(Γ1)). (4.33)

Convergences (4.26)–(4.29), (4.31) and (4.32) permit us to pass to the limit as
m→∞ in approximate equation (4.3). Thus for θ ∈ D(Ω) and noting that (4.2) is
a base of V , we obtain∫ ∞

0

(u′′l (t), θ(t)v)Hdt+

∫ ∞
0

((ul(t), θ(t)v))V dt

+

∫ ∞
0

(|ul(t))|ρ, θ(t)v)V dt+

∫ ∞
0

(hl(·, u′l(t)), θ(t)v)L2(Γ1)dt = 0.

As the set {θv; θ ∈ D(0,∞), v ∈ V } is total in L2(0,∞;V ), the above inequality
implies∫ ∞

0

(u′′l (t), ϕ)Hdt+

∫ ∞
0

((ul(t), ϕ))V dt+

∫ ∞
0

(|ul(t))|ρ, ϕ)V θ(t)dt

+

∫ ∞
0

(hl(·, u′l(t)), ϕ)L2(Γ1)dt = 0, ∀ϕ ∈ L2(0,∞;V ),

(4.34)

and suppϕ is bounded in[0,∞).
Taking ϕ ∈ D((0,∞)× (Ω)n) in (4.33), we obtain

u′′l +Aul + |ul|ρ = 0 in D′((0,∞)× (Ω)n).

As u′′l and |ul|ρ belong to L2
loc(0,∞;H), we obtain

u′′l +Aul + |ul|ρ = 0 in L2
loc(0,∞;H). (4.35)

From now on , ϕ denotes a function satisfying conditions (4.33). We take the
inner product of H with ϕ in both of sides of (4.35). We deduce∫ ∞

0

(u′′l (s), ϕ)Hds+

∫ ∞
0

(Aul(s), ϕ)Hds+

∫ ∞
0

(|ul(s)|ρ, ϕ)Hds = 0. (4.36)

Note that

ul ∈ L∞(0,∞;V ) and Aul ∈ L2
loc(0,∞;H).

Then by Theorem 3.3, Part (3.8), we obtain γ1ul ∈ L2
loc(0,∞;H−1/2(Γ1)) and thus∫ ∞

0

(Aul(s), ϕ)Hds =

∫ ∞
0

((ul(s), ϕ))V ds−
∫ ∞

0

〈γ1ul(s), ϕ〉Y ′×Y ds,

where Y = H1/2(Γ1). Replacing this equality in (4.36), we deduce∫ ∞
0

(u′′l (s), ϕ)Hds+

∫ ∞
0

((ul(s), ϕ))V ds−
∫ ∞

0

〈γ1ul(s), ϕ〉Y ′×Y ds

+

∫ ∞
0

(|ul(s)|ρ, ϕ)Hds = 0.

(4.37)

Comparing (4.33) and (4.37) and taking into account the regularity of hl(·, u′l), we
find

γ1ul + hl(·, u′l) = 0 in L2
loc(0,∞,L2(Γ1). (4.38)



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 15

Pass to limit in l. Estimates (4.15)–(4.17) and (4.23)–(4.25) are independent of
l ≥ l0. Then as in (4.26)–(4.29), (4.30) and (4.31), we obtain that there exist a
subsequence of (ul), still denoted by (ul), and a function u such that

ul → u weak star in L∞(0, ∞;V ); (4.39)

u′l → u′ weak star in L∞(0,∞;H) ∩ L∞loc(0,∞;V ); (4.40)

u′′l → u′′ weak star in L∞loc(0,∞;H); (4.41)

|ul|ρ → |u|ρ weakly in L2
loc(0,∞;H); (4.42)

u′l(x, t)→ u′(x, t) a.e. (x, t) ∈ Γ1 × (0, T ). (4.43)

Convergence (4.41) follows from the compact embedding of H1/2(Γ1) in L2(Γ1).
Take the limit in (4.33). Then by convergences (4.39)–(4.41) and applying similar
arguments used to obtain (4.35), we deduce

u′′ +Au+ |u|ρ = 0 in L2
loc(0,∞;H). (4.44)

By estimate (4.15) and equation (4.35), we obtain that

(ul) is bounded in L∞(0,∞;V ),

(Aul) is bounded in L2
loc(0,∞;H).

Then by Theorem 3.3, we obtain

γ1ul → γ1u in L2
loc(0,∞;H−1/2(Γ1)). (4.45)

Fix (x, t) ∈ Γ1 × (0, T ). Then by convergences (4.43) and condition (3.14) of
Proposition 3.4, we deduce

hl(x, u
′
l(x, t))→ h(x, u′(x, t)) a.e. x in Γ1 × (0, T ). (4.46)

On the other hand, by estimates (4.15), (4.23), (4.16), and (4.24), we find that

(ul) is bounded in C0([0, T ];V ); ∀T > 0; (4.47)

(u′l) is bounded in C0([0, T ];H); ∀T > 0. (4.48)

By (4.33) and noting that each hi(x, s) is increasing in s, we obtain

0 ≤
∫ T

0

(hl(·, u′l), u′l)L2(Γ1)dt

= −1

2
‖u′l(T )‖2H +

1

2
‖u1

l ‖2H −
1

2
‖ul(T )‖2V +

1

2
‖u0‖2H −

∫ T

0

(|ul|ρ, u′l)Hdt.

Then by (4.47), (4.48) and (4.42), and (4.40), we have

0 ≤
∫ T

0

(hl(·, u′l), u′l)L2(Γ1)dt ≤ C(T ). (4.49)

It follows from (4.46), (4.49) and a results due to Strauss [23] that

hl(·, u′l)→ h(·, u′) in L1(0, T ;L1(Γ1)).

As T > 0 was arbitrary it follows that

hl(·, u′l)→ h(·, u′) in L1
loc(0,∞;L1(Γ1)). (4.50)

Taking the limit in (4.38) and using (4.45) and (4.50), we find that

γ1u+ h(·, u′) = 0 in L1
loc(0,∞;H−1/2(Γ1) + L1(Γ1)).



16 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

By equation (4.44), we deduce that u ∈ L2
loc(0,∞;W ). Convergences (4.39)-(4.41)

provide the initial conditions (2.24). Thus the proof is complete. �

Remark 4.4. The proof of the existence of solutions of Theorem 2.4 follows by
applying similar arguments used to obtain Theorem 2.3. In this case, we consider
J1(λ) = 1

4λ
2 −N1λ

ρ+1, λ ≥ 0 and

hil(x, s) = [m(x) · ν(x)]pi(s), ∀l ∈ N, i = 1, . . . , n.

Note that λ∗1 < λ∗. The uniqueness of solution is derived by the energy method.

5. Proof of Theorem 2.3

By (2.16) we have

(Au)i = −µ∆ui − (λ+ µ)
∂

∂xi
div u, i = 1, . . . , n. (5.1)

Proposition 5.1. Let u ∈ H2(Ω). Then
n∑
i=1

2(−(Au)i,m∇ui)

= µ(n− 2)

n∑
i=1

∫
Ω

|∇ui|2dx− µ
n∑
i=1

∫
Γ

|∇ui|2(m · ν)dΓ

+ 2µ

n∑
i=1

∫
Γ

∂ui
∂γ

(m · ∇ui)dΓ + (λ+ µ)(n− 2)

∫
Ω

(div u)2dx

− (λ+ µ)

∫
Γ

(div u)2(m · ν)dΓ + 2(λ+ µ)

n∑
i=1

∫
Γ

(div u)(m · ∇ui)νidΓ

=

6∑
i=1

Mi,

where,

• M1 = µ(n− 2)
∑n
i=1

∫
Ω
|∇ui|2dx;

• M2 = µ
∑n
i=1

∫
Γ
|∇ui|2(m · ν)dΓ;

• M3 = 2µ
∑n
i=1

∫
Γ
∂ui
∂γ (m · ∇ui)dΓ;

• M4 = (λ+ µ)(n− 2)
∫

Ω
(div u)2dx;

• M5 = −(λ+ µ)
∫

Γ
(div u)2(m · ν)dΓ;

• M6 = 2(λ+ µ)
∑n
i=1

∫
Γ
(div u)(m · ∇ui)νidΓ.

Proof. Expression (5.1) provides
n∑
i=1

2(−(Au)i,m · ∇ui)

=

n∑
i=1

2µ(∆ui,m · ∇ui) +

n∑
i=1

2(λ+ µ)
( ∂

∂xi
div u,m · ∇ui

)
.

(5.2)

By the Rellich identity (see Komornik-Zuazua [8]), we have
n∑
i=1

2µ(∆ui,m · ∇ui) = M1 +M2 +M3. (5.3)



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 17

On the other hand( ∂

∂xi
div u,m · ∇ui

)
= −

(
div u,

∂

∂xi
(m · ∇ui)

)
+

∫
Γ

(div u)(m · ∇ui)νidΓ. (5.4)

Also

∂

∂νi
(m · ∇ui) =

∂ui
∂xi

+

n∑
l=1

ml
∂

∂xl

(∂ui
∂xi

)
,

n∑
i=1

(
div u,

∂

∂xi
(m · ∇ui)

)
=

∫
Ω

(div u)2dx+

n∑
l=1

∫
Ω

(div u)ml
∂

∂xl
[div u]dx

=

∫
Ω

(div u)2dx− n

2

∫
Ω

(div u)2dx+
1

2

∫
Γ

(div u)2(m · ν)dΓ.

Plugging the last expression in (5.4), we obtain

n∑
i=1

2(λ+ µ)
( ∂

∂xi
div u,m · ∇ui

)
= M4 +M5 +M6. (5.5)

The proposition follows from (5.3) and (5.5). �

Proof of Theorem 2.5. We take the inner product of H in both sides of (2.35) with
u′. Then by (2.27) and (2.36) we find that

E′(l) ≤ −τ0‖u′(t)‖2L2(Γ1), (5.6)

where τ0 = b0d
∗
0, with d∗0 defined in (2.28). We introduce the perturbed energy

Eε(t) = E(t) + εα(t), t ≥ 0, ε > 0, (5.7)

where

α(t) =

n∑
i=1

2(u′i(t),m · ∇ui(t)) + (n− 1)

n∑
i=1

(u′i(t), ui(t)). (5.8)

I. Equivalence between Eε(t) and E(t). First of all, we note that

1

4
‖u(t)‖V +

1

ρ+ 1
(|u(t)|ρ, u(t))H ≥ 0, ∀t ≥ 0. (5.9)

In fact, since

|(|u(t)|ρ, u(t))H | ≤ nkρ+1
0 ‖u(t)‖ρ+1

H1
Γ0

(Ω)
≤ nk

ρ+1

0

µ
ρ+1

2

‖u(t)‖ρ+1
V ,

it follows that ∣∣ 1

ρ+ 1
(|u(t)|ρ+1, u(t))H

∣∣ ≤ 1

µ
ρ+1

2

nk
ρ+1

0

ρ+ 1
‖u(t)‖ρ+1

V . (5.10)

Since

− 1

µ
ρ+1

2

nkρ+1
0

ρ+ 1
> −N1,



18 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

with N1 defined in (2.30), we obtain

1

4
‖u(t)‖2V −

1

µ
ρ+1

2

nk
ρ+1

0

ρ+ 1
‖u(t)‖ρ+1

V ≥ 1

4
‖u(t)‖2V −N1‖u(t)‖ρ+1

V ≥ 0, ∀t ≥ 0, (5.11)

because

J1(λ) =
1

4
λ2 −N1λ

ρ+1 ≥ 0, ∀0 ≤ λ ≤ λ∗1, and 0 ≤ ‖u(t)‖V < λ∗1

(see Remark 4.4). Inequalities (5.10) and (5.11) provide (5.9). Then by (5.9) we
find that

E(t) ≥ 1

4
‖u′(t)‖2H +

1

4
‖u(t)‖2V . (5.12)

On the other hand, we have

|α(t)| ≤ 2R‖u′(t)‖H‖u(t)‖H1
Γ0

(Ω) + (n− 1)‖u′(t)‖H‖u(t)‖H .

Thus

|α(t)| ≤ R

µ1/2

(
‖u(t)‖2H + ‖u(t)‖2V

)
+

(n− 1)k5

µ1/2

(
(
1

2
‖u′(t)‖2H +

1

2
‖u(t)‖2V

)
,

which implies

|α(t)| ≤M
(1

4
‖u′(t)‖2H +

1

4
‖u(t)‖2V

)
. (5.13)

By (5.12) and (5.13), we obtain |α(t)| ≤ME(t), ∀t ≥ 0. Thus

|Eε(t)− E(t)| = ε|α(t)| ≤ εME(t), ∀t ≥ 0.

Choosing ε1 = 1
2M , we have

1

2
E(t) ≤ Eε(t) ≤

3

2
E(t), ∀t ≥ 0, ∀0 < ε ≤ ε1. (5.14)

II. Relation between E′ε(t) and E(t). By (5.8) we obtain

α′(t) =

n∑
i=1

2(u′′i (t),m · ∇ui(t)) +

n∑
i=1

2(u′i(t),m · ∇u′i(t))

+ (n− 1)

n∑
i=1

(u′′i (t), ui(t)) + (n− 1)

n∑
i=1

|u′i(t)|2

= D(t) + F (t) +G(t) + I(t).

(5.15)

We have

D(t) =

n∑
i=1

2(−(Au(t))i,m · ∇ui(t))−
n∑
i=1

2(|ui(t)|ρ,m · ∇u′i(t))

= D1(t) +D2(t).

(5.16)

Analysis of D2(t). We find

(|ui(t)|ρ,m · ∇ui(t)) =

n∑
j=1

∫
Ω

|ui(t)|ρmj
∂ui(t)

∂xj
dx

=

n∑
j=1

∫
Ω

mj
1

ρ+ 1
[
∂

∂xj
|ui(t)|ρui(t)]dx



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 19

= − n

ρ+ 1

∫
Ω

|ui(t)|ρui(t)dx+
1

ρ+ 1

∫
Γ

|ui(t)|ρui(t)(m · ν)dΓ.

Then

D2(t) =
2n

ρ+ 1

n∑
i=1

∫
Ω

|ui(t)|ρui(t)dx−
2

ρ+ 1

n∑
i=1

∫
Γ

|ui(t)|ρui(t)(m · ν)dΓ.

This result and (5.16) provide

D(t) = D1(t) +
2n

ρ+ 1

n∑
i=1

∫
Ω

|ui(t)|ρui(t)dx

− 2

ρ+ 1

n∑
i=1

∫
Γ

|ui(t)|ρui(t)(m · ν)dΓ.

(5.17)

Note that D1(t) is given by Proposition 5.1.
Analysis of F (t). We have

(u′i(t),m · ∇u′i(t)) =

n∑
j=1

∫
Ω

mj
1

2

∂

∂xj
(u′(t))2dx

= −n
2

∫
Γ

(u′i(t))
2dx+

1

2

∫
Γ

(u′i(t))
2(m · ν)dΓ.

Thus

F (t) = −n
n∑
i=1

|u′i(t)|2 +

∫
Γ

( n∑
i=1

(u′i(t))
2
)

(m · ν)dΓ. (5.18)

Analysis of G(t). We obtain

(u′′i (t), ui(t)) = µ(∆ui(t), ui(t)) + (λ+ µ)
( ∂

∂νi
div u(t), ui(t)

)
− (|ui(t)|ρ, ui(t))

= l1(t) + l2(t) + l3(t),

where

l1(t) = −µ
∫

Ω

|∇ui(t)|2dx+ µ

∫
Γ

∂ui(t)

∂ν
ui(t)dΓ,

l2(t) = −(λ+ µ)

∫
Ω

(div u(t))
∂u(t)

∂xi
dx+ (λ+ µ)

∫
Γ

(div u(t))ui(t)νidΓ.

Then

G(t) = −(n− 1)µ

n∑
i=1

∫
Ω

|∇ui(t)|2dx+ (n− 1)µ

n∑
i=1

∫
Γ

∂ui(t)

∂ν
ui(t)dΓ

− (n− 1)(λ+ µ)

∫
Ω

(div u(t))2dx

+ (n− 1)(λ+ µ)

∫
Γ

(div u(t))
( n∑
i=1

ui(t)νi

)
dΓ

− (n− 1)

n∑
i=1

(|ui(t)|ρ, ui(t)).

(5.19)

By (5.4), Proposition (5.1), (5.17)–(5.19), we find that

α′(t) = D(t) + F (t) +G(t) + I(t),



20 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

where

D(t)

= µ(n− 2)

n∑
i=1

∫
Ω

|∇ui(t)|2dx− µ
n∑
i=1

∫
Γ

|∇ui(t)|2(m · ν)dΓ

+ 2µ

n∑
i=1

∫
Γ

∂ui(t)

∂ν
(m · ∇ui(t))dΓ

+ (λ+ µ)(n− 2)

∫
Ω

(div u(t))2dx− (λ+ µ)

∫
Γ

(div u(t))2(m · ν)dΓ

+ 2(λ+ µ)

n∑
i=1

∫
Γ

(div u(t))(m · ∇ui(t))νidΓ +
2n

ρ+ 1

n∑
i=1

(|ui(t)|ρ, ui(t))

− 2

ρ+ 1

n∑
i=1

∫
Γ1

|ui(t)|ρui(t)(m · ν)dΓ1,

(5.20)

F (t) = −n
n∑
i=1

|ui(t)|2 +

∫
Γ

[
n∑
i=1

(u′i(t))
2

]
(m · ν)dΓ, (5.21)

G(t) = −(n− 1)µ

n∑
i=1

∫
Ω

|∇ui(t)|2dx+ (n− 1)µ

n∑
i=1

∫
Γ1

∂ui(t)

∂ν
ui(t)dΓ1

− (n− 1)(λ+ µ)

∫
Ω

(div u(t))2dx

+ (n− 1)(λ+ µ)

∫
Γ

(div u(t))
( n∑
i=1

ui(t)νi

)
dΓ

− (n− 1)

n∑
i=1

(|ui(t)|ρ, ui(t))

(5.22)

and

I(t) = (n− 1)
n∑
i=1

|u′i(t)|2. (5.23)

The goal is to transform (5.15)–(5.18) into an inequality of the form

α′(t) ≤ −E(t)−
(1

4
‖u(t)‖2V −N1‖u(t)‖ρ+1

V

)
+ P‖u′(t)‖2L2(Γ1)

and then to find conditions to have

1

4
‖u(t)‖2V −N1‖u(t)‖ρ+1

V ≥ 0, ∀t ≥ 0.

This last inequality motivates the introduction of Theorem 2.4.
By reducing similar terms in (5.20)–(5.23), we obtain

α′(t) = −‖u′(t)‖2H − µ‖u(t)‖2H1
Γ0

(Ω) − (λ+ µ)|div u(t)|2

+
2n

ρ+ 1

n∑
i=1

(|ui(t)|ρ, ui(t))− (n− 1)

n∑
i=1

(|ui(t)|ρ, ui(t)) +Q(t),
(5.24)



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 21

where

Q(t) = −µ
n∑
i=1

∫
Γ

|∇ui(t)|2(m · ν)dΓ + 2µ

n∑
i=1

∫
Γ

∂ui(t)

∂ν
(m · ∇ui(t))dΓ

− (λ+ µ)

∫
Γ

(div u(t))2(m · ν)dΓ

+ 2(λ+ µ)

n∑
i=1

∫
Γ

(div u(t))2νi(m · ∇ui(t))dΓ

− 2

ρ+ 1

n∑
i=1

∫
Γ

|ui(t)|ρui(t)(m · ν)dΓ +

n∑
i=1

∫
Γ

(u′i(t))
2(m · ν)dΓ

+ (n− 1)µ

n∑
i=1

∫
Γ

(∂ui
∂ν

)
ui(t)dΓ

+ (n− 1)(λ+ µ)

n∑
i=1

∫
Γ

[(div u(t))νi]ui(t)dΓ

=

8∑
j=1

qj(t).

(5.25)

By (5.24), we have

α′(t) = −2E(t) +
2

ρ+ 1
(|u(t)|ρ, u(t))H +

2n

ρ+ 1
(|u(t)|ρ, u(t))H

− (n− 1)(|u(t)|ρ, u(t))H +Q(t),

which implies

α′(t) ≤ −E(t)− 1

2
‖u(t)‖2V +

1

ρ+ 1
(|u(t)|ρ, u(t))H

+
2n

ρ+ 1
(|u(t)|ρ, u(t))H − (n− 1)(|u(t)|ρ, u(t))H +Q(t).

(5.26)

By (2.5), we obtain

|(|ui(t)|ρ, ui(t))| ≤ kρ+1
0 ‖u(t)‖ρ+1

H1
Γ0

(Ω)
.

Then∣∣∣ 1

ρ+ 1
(|u(t)|ρ, u(t))H +

2n

ρ+ 1
(|u(t)|ρ, u(t))H − (n− 1)(|u(t)|ρ, u(t))H

∣∣∣
≤ ω‖u(t)‖ρ+1

H1
Γ0

(Ω)
,

(5.27)

where

ω =
∣∣1 + 2n

ρ+ 1
− (n− 1)

∣∣nkρ+1
0 . (5.28)

From (5.26) and (5.27) we obtain

α′(t) ≤ −E(t)− 1

2
‖u(t)‖2V + ω‖u(t)‖ρ+1

H1
Γ0

(Ω)
+Q(t). (5.29)



22 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

Let us analyze (5.25). Note that ∂ui(t)
∂xj

= ∂ui(t)
∂ν νj in Γ0 (see Lions [15]). Since

q1(t) = −µ
n∑
i=1

∫
Γ0

(∂ui(t)
∂ν

)2

(m · ν)dΓ− µ
n∑
i=1

∫
Γ1

|∇ui(t)|2(m · ν)dΓ,

q2(t) = 2µ

n∑
i=1

∫
Γ0

(∂ui(t)
∂ν

)2

(m · ν)dΓ + 2µ

∫
Γ1

∂ui(t)

∂ν
(m · ∇ui(t))dΓ,

by noting that m · ν ≤ 0 in Γ0, we find that

q1(t) + q2(t) ≤ −µ
n∑
i=1

∫
Γ1

|∇ui(t)|2(m · ν)dΓ + 2µ

n∑
i=1

∫
Γ1

∂ui(t)

∂ν
(m · ∇ui(t))dΓ.

(5.30)
Therefore,

q3(t) ≤ −(λ+ µ)

∫
Γ0

( n∑
i=1

∂ui(t)

∂ν
νi

)2

(m · ν)dΓ,

and

q4(t) = 2(λ+ µ)

∫
Γ0

( n∑
j=1

∂uj(t)

∂ν
νj

)2

(m · ν)dΓ

+ 2(λ+ µ)

n∑
i=1

∫
Γ1

[(div u(t))νi](m · ∇ui(t))dΓ.

Then

q3(t) + q4(t) ≤ 2(λ+ µ)

n∑
i=1

∫
Γ1

[(div u)νi](m · ∇ui(t))dΓ. (5.31)

By observing that µ∂ui(t)∂ν +(λ+µ)(div u(t))νi+(m·ν)hi(u
′
i(t)) = 0 on Γ1, it follows

from (5.26) and (5.27) that

q1(t) + · · ·+ q4(t)

≤ −µ
n∑
i=1

∫
Γ1

|∇ui(t)|2(m · ν)dΓ + 2

n∑
i=1

∫
Γ1

[−(m · ν)hi(u
′
i(t))](m · ∇ui(t))dΓ.

However ∣∣∣2 ∫
Γ1

[−(m · ν)hi(u
′
i(t))](m · ∇ui(t))dΓ

∣∣∣
≤ 1

µ
R3L2

∫
Γ1

|ui(t)|2dΓ + µ

∫
Γ1

|∇ui(t)|2(m · ν)dΓ.

Then the last two inequalities provide

q1(t) + · · ·+ q4(t) ≤ 1

µ
R3L2‖u′(t)‖2L2(Γ1). (5.32)

By noting that ui = 0 in Γ0, we find that

q7(t) + q8(t) = (n− 1)

n∑
i=1

∫
Γ1

[−(m · ν)hi(u
′
i(t))]ui(t)dΓ

≤ 1

µ
(n− 1)2R2L2k2

7‖u′(t)‖2L2(Γ1) +
µ

4
‖u(t)‖2H1

Γ0
(Ω),



EJDE-2022/21 NONSTATIONARY LAMÉ SYSTEM WITHOUT DEFINITE SIGN ENERGY 23

where k7 was defined in (2.8). Thus

q7(t) + q8(t) ≤ 1

µ
(n− 1)2R2L2k2

7‖u′(t)‖2L2(Γ1) +
1

4
‖u(t)‖2V . (5.33)

We have

|q5(t)| ≤ 2

ρ+ 1
R

n∑
i=1

∫
Γ1

|ui(t)|ρ+1dΓ

≤ 2R

ρ+ 1
kρ+1

6

n∑
i=1

‖ui(t)‖ρ+1
H1

Γ0
(Ω)

≤ 2R

ρ+ 1
kρ+1

6 n‖u(t)‖ρ+1
H1

Γ0
(Ω)
.

(5.34)

Since u′i = 0 on Γ0, it follows that

|q6(t)| ≤ R
n∑
i=1

∫
Γ1

(u′i(t))
2dΓ = R‖u′(t)‖2L2(Γ1). (5.35)

Taking into account (5.32)–(5.35) in (5.29), we obtain

α′(t) ≤ −E(t)−
[1
4
‖u(t)‖2V −N1‖u(t)‖ρ+1

V

]
+ P‖u′(t)‖2L2(Γ1), (5.36)

where N1 and P were defined in (2.30) and (2.40)), respectively. By applying
Theorem 2.4 to (5.36), we find that

α′(t) ≤ −E(t) + P‖u′(t)‖2L2(Γ1). (5.37)

Now let us to return to the perturbed energy Eε(t) given in (5.7). By (5.6) and
(5.37), we obtain

E′ε(t) = E′(t) + εα′(t) ≤ −εE(t)− (τ0 − εP )‖u′‖2L2(Γ1).

Choosing 0 < ε2 ≤ τ0
P , we have

E′ε(t) ≤ −εE(t), ∀0 < ε ≤ ε2. (5.38)

Thus for σ = min{ 1
2M , τ0ρ } we have that (5.9) and (5.38) hold for all 0 < ε ≤ σ. By

(5.38) and (5.9), we obtain

E′ε(t) ≤ −
2

3
σEε(t),

which implies

Eε(t) ≤ Eε(0)e−
2
3σt.

This inequality and (5.9) provide Theorem 2.5. �

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24 M. MILLA MIRANDA, A. T. LOUREDO, M. R. CLARK, G. SIRACUSA EJDE-2022/21

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Manuel Millla Miranda
Departamento de Matemática, Universidade Estadual da Paráıba, DM, PB, Brazil

Email address: mmillamiranda@gmail.com

Aldo Trajano Louredo

Departamento de Matemática, Universidade Estadual da Paráıba, DM, PB, Brazil
Email address: aldolouredo@gmail.com

Marcondes Rodrigues Clark
Departamento de Matemática, Universidade Federal do Piaúı, DM, PI, Brazil

Email address: marcondesclark@gmail.com

Giovana Siracusa Gouveia

Departamento de Matemática, Universidade Federal de Sergipe, DM, SE, Brazil
Email address: gisiracusa@gmail.com


	1. Introduction
	2. Notation and main results
	3. Preliminary results
	4. Proof of Theorem ??
	5. Proof of Theorem 2.3
	References