Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 95, pp. 1–23.

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

WELL-POSEDNESS AND ENERGY DECAY OF A

TRANSMISSION PROBLEM OF KIRCHHOFF TYPE WAVE

EQUATIONS WITH DAMPING AND DELAY TERMS

ZHIQING LIU, CUNCHEN GAO, ZHONG BO FANG

Abstract. We consider a transmission problem of Kirchhoff type wave equa-
tions with delay and damping terms, subject to a memory condition on one

part of the boundary. Under appropriate hypotheses on the relaxation func-

tion and the relationship between weights of damping and delay terms, we
establish well-posedness of the problem. Using the Faedo-Galerkin approxi-

mation technique, and introducing suitable energy and Lyapunov functionals,

we obtain estimates for exponential, polynomial, and logarithmic decay.

1. Introduction

We consider a transmission problem of Kirchhoff type wave equations with damp-
ing and delay terms,

utt − (1 + ‖∇u‖2Ω1
)∆u+ µ1ut + µ2ut(t− τ) = 0, (x, t) ∈ S1, (1.1)

vtt − (1 + ‖∇v‖2Ω2
)∆v = 0, (x, t) ∈ S2, (1.2)

subject to boundary and transmission conditions

v = 0, (x, t) ∈ ∂S0, (1.3)

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
= (1 + ‖∇v‖2Ω2

)
∂v

∂ν
, u = v, (x, t) ∈ ∂S1, (1.4)

u+

∫ t

0

g(t− s)(1 + ‖∇u(s)‖2Ω1
)
∂u(s)

∂ν
ds = 0, (x, t) ∈ ∂S2, (1.5)

and initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω1, (1.6)

ut(x, t− τ) = f0(x, t− τ), (x, t) ∈ Ω1 × (0, τ), (1.7)

v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ Ω2. (1.8)

Here, Si := Ωi × (0,+∞) and ∂Sj := Γj × (0,+∞) with i = 1, 2 and j = 0, 1, 2,
where Ω ⊂ RN (N ≥ 2) is a bounded domain with smooth boundary ∂Ω = Γ0∪Γ2,
Γ0 ∩ Γ2 = ∅. Γ0 is the boundary of small ball B(x0) containing x0 in Ω, Ω2 ⊂ Ω
is a subdomain with smooth boundary Γ0 ∪ Γ1 in the outside of B(x0), and Ω1 =

2010 Mathematics Subject Classification. 35B40, 35L51.
Key words and phrases. Transmission problem; Kirchhoff type wave equations; damping;

delay term; well-posed; decay estimate.
©2021. This work is licensed under a CC BY 4.0 license.

Submitted April 5, 2021. Published December 2, 2021.

1



2 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Ω \ (Ω2 ∪ B(x0)) is a subdomain with smooth boundary Γ1 ∪ Γ2. ν denotes the
unit outer normal vector pointing towards the exterior of Ω1 and there exists δ > 0,
such that m · ν ≥ δ > 0 on Γ2, where m := m(x) = x − x0 (see Figure 1 for an
example). Moreover, µ1 and µ2 are positive constants, τ > 0 is the delay, g is a
positive function, and f0 is the given history belonging to suitable spaces.

Figure 1. Domain Ω.

Our transmission model (1.1)-(1.8) arises in several applications in physics and
biology, such as models of the transverse vibrations of a membrane composed by
two different materials in Ω1 and Ω2.

In the past decades, there many authors investigated wave equations and sys-
tems with damping terms and showed that the dissipation produced by internal or
boundary damping can lead to the decay of solutions, see [8, 9, 16, 17, 18, 19, 24, 33]
and the references therein. For examples, Cavalcanti et al. [16] studied the mixed
initial boundary value problem of linear degenerate wave equations with nonlinear
boundary damping and boundary memory sources

ρ1(x, t)utt + ρ2(x, t)ut −∆u = 0, (x, t) ∈ Ω× (0,+∞),

∂u

∂ν
+ u+ ut + g(t)|ut|ρut = g ∗ |u|γu, (x, t) ∈ ∂S0,

u = 0, (x, t) ∈ ∂S1,

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

where Ω ⊂ RN is a bounded domain with boundary ∂Ω of C2, ∂Ω = Γ0 ∪ Γ1,
Γ0 ∩ Γ1 = ∅. Meantime, Γ0 and Γ1 possess positive measures with

Γ0 := {x ∈ ∂Ω : ν · (x− x0) ≤ 0} ,
Γ1 := {x ∈ ∂Ω : ν · (x− x0) > 0} .

They established the existence and exponential decay estimates of the global solu-
tions. Later, Park and Bae [8] considered a Kirchhoff type wave equation

utt −M(‖∇u‖22)∆u−∆ut = 0, (x, t) ∈ Ω× (0,+∞),

and obtained the same conclusion with [16] under similar conditions. Santos et al.
[17] investigated the Kirchhoff type wave equation

utt −M(‖∇u‖22)∆u−∆ut + f(u) = 0, (x, t) ∈ Ω× (0,+∞),



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 3

with boundary conditions

u = 0, (x, t) ∈ ∂S0,

u+

∫ t

0

g(t− s)(M(‖∇u‖22)
∂u

∂ν
(s) +

∂us
∂ν

(s))ds = 0, (x, t) ∈ ∂S1.

They proved that the energy decays with the same rate to the relaxation function,
that is, the energy decays exponentially or polynomially provided the relaxation
function decays exponentially or polynomially, respectively. Bae [9] considered the
coupled wave equation of Kirchhoff type

utt − (1 + ‖∇u‖22 + ‖∇v‖22)∆u+ |u|αu = 0, (x, t) ∈ Ω× (0,+∞),

vtt − (1 + ‖∇u‖22 + ‖∇v‖22)∆v + |v|βv = 0, (x, t) ∈ Ω× (0,+∞),

subject to mixed boundary conditions, and obtained the similar conclusion with
[17]. We refer to [18, 19, 24] on the decay estimates of degenerate wave equations
with localized damping and viscoelastic damping and linear systems with boundary
memory dissipation.

Most recently, for the research advances on the ground state solutions for quasi-
linear equations of Kirchhoff type and multiple positive solutions to the fractional
Kirchhoff problem, one can see [13, 14].

It is well known that delay effects, which arise in many practical problems, may
be the sources of instability. Hence, the control of PDEs with delay effects has
become an active area of research in recent years. For examples, it was proved in
[6, 21, 22, 25, 26] that an arbitrarily small delay may destabilize a system which is
uniformly asymptotically stable in the absence of delay, unless additional conditions
or control terms were imposed. A boundary stabilization problem for the wave
equation with interior delay was studied in [15]. The authors proved an exponential
stability result under some Lions geometric conditions. Kirane and Said-Houari [20]
considered the viscoelastic wave equation with delay

utt −∆u+

∫ t

0

g(t− s)∆u(s)ds+ µ1ut + µ2ut(t− τ) = 0, (x, t) ∈ Ω× (0,+∞),

where µ1 and µ2 are positive constants. Under the hypothesis of 0 ≤ µ1 ≤ µ2,
they established general decay estimate of the energy. Later, Liu [31] improved this
result by considering the equation with a time-varying delay term, with coefficient
µ2 not necessarily positive.

For the transmission problems, we can see [1, 2, 4, 10, 11, 23, 32] for the stud-
ies of existence, regularity, controllability and decay estimates of solutions for the
transmission problems with Laplacian operators. For example, Marzocchi [1] proved
that the solution for a semilinear transmission problem between an elastic and ther-
moelastic material in one-dimensional space decays exponentially. This result was
extended to the case of N -dimensional space by Marzocchi and Naso [2]. Bastos
and Raposo [32] investigated the transmission problem with frictional damping and
showed the well-posedness and exponential stability of the total energy. Recently,
There are many new results on transmission problems with operators of Kirchhoff
type, see [3, 5, 7, 12, 27, 28, 29, 30]. Bae [12] concerned the transmission problem
for the wave equations given by

utt − ‖∇u‖2Ω1
∆u+ |u|αu = 0, (x, t) ∈ S1,

vtt − ‖∇v‖2Ω2
∆v + |v|βv = 0, (x, t) ∈ S2,



4 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

subject to boundary and transmission conditions

v = 0, (x, t) ∈ ∂S0 × (0,+∞),

u = v, ‖∇u‖2Ω1

∂u

∂ν
= ‖∇v‖2Ω2

∂v

∂ν
, (x, t) ∈ ∂S1,

u+

∫ t

0

g(t− s)‖∇u(s)‖2Ω1

∂u(s)

∂ν
ds = 0, (x, t) ∈ ∂S2,

and initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω1,

v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ Ω2.

He studied the global existence of solutions and showed that if the relaxation func-
tion decays exponentially or polynomially, the solutions decays with the same rates.
Later, Park [27, 28] considered the transmission problem of the Kirchhoff type wave
equations

utt − (1 + ‖∇u‖2Ω1
)∆u = 0, (x, t) ∈ S1,

vtt − (1 + ‖∇v‖2Ω2
)∆v = 0, (x, t) ∈ S2,

subject to the same boundary and transmission conditions with [12]. He established
general decay results depending on the behavior of the relaxation function.

On the other hand, for the transmission problems with delay terms, Benseghir [3]
investigated the linear transmission problem with a delay term in one-dimensional
space

utt − auxx + µ1ut + µ2ut(t− τ) = 0, (x, t) ∈ Ω× (0,+∞),

vtt − bvxx = 0, (x, t) ∈ (L1, L2)× (0,+∞),

subject to boundary and transmission conditions

u(0, t) = v(L3, t) = 0,

u(Li, t) = v(Li, t), aux(Li, t) = bvx(Li, t), i = 1, 2,

and initial conditions

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

ut(x, t− τ) = f0(x, t− τ), (x, t) ∈ Ω× (0, τ),

v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ (L1, L2),

where 0 < L1 < L2 < L3, Ω = (0, L1) ∪ (L2, L3), a and b are positive constants.
Under the assumption µ2 < µ1, he showed the exponential stability of the solution
by introducing a suitable Lyaponov functional. Li et al. [7] studied the linear
transmission system with long time memory and delay terms

utt − auxx +

∫ +∞

0

g(t− s)uxx(s)ds+ µ1ut + µ2ut(t− τ) = 0, (x, t) ∈ Ω× (0,+∞),

vtt − bvxx = 0, (x, t) ∈ (L1, L2)× (0,+∞),

with the same boundary, transmission and initial conditions with [3]. Under the
assumption µ2 ≤ µ1, they proved the well-posedness result by means of semigroup
theory and Hille-Yosida theorem. Furthermore, they established a general decay



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 5

result, of which the exponential and polynomial decays are only special cases. More-
over, we refer to [5] for the similar transmission problem with short time memory
term.

In view of the works mentioned above, on can see that the studies on transmis-
sion problem for a Kirchhoff type wave system (1.1)-(1.8) with damping and delay
terms has not been started. The main difficulty encountered arises from the simul-
taneous appearance of the Kirchhoff type operators, delay and damping terms and
memory damping on one part of the boundary. Our first goal is to establish the
well-posedness of problem (1.1)-(1.8) by means of Faedo-Galerkin approximation
together with priori energy estimates. As for the asymptotic behavior, we establish
a general decay result under a wider class of relaxation functions and some condi-
tions on the boundary, by introducing suitable energy and Lyapunov functionals.

The remaining of this paper is organized as follows: In Sect.2, we present some
preliminaries and state the main results. In Sect.3, we establish well-possedness of
problem (1.1)-(1.8) and the general decay estimate of energy is derived in Sect.4.

2. Preliminaries and main results

In this section, we present some materials needed in the proof and state the main
results. Throughout this paper, we define

H1
Γ(Ω2) := {v ∈ H1(Ω2) : v = 0 on Γ0},

V := {(u, v) ∈ H1(Ω1)×H1
Γ(Ω2) : u = v, (1 + ‖∇u‖2Ω1

)
∂u

∂ν
= (1 + ‖∇v‖2Ω2

)
∂v

∂ν
},

(u, v)Ωi
:=

∫
Ωi

u(x)v(x)dx, i = 1, 2, (u, v)Γj
:=

∫
Γj

u(x)v(x)dx, j = 1, 2.

For a Banach space X, ‖ · ‖X denotes the norm of X. For simplicity, we denote
‖ · ‖L2(Ωi) and ‖ · ‖L2(Γj) by ‖ · ‖Ωi

and ‖ · ‖Γj
, respectively.

We use al the notation

(h ∗ u)(t) :=

∫ t

0

h(t− s)u(s)ds,

(h ◦ u)(t) :=

∫ t

0

h(t− s)[u(t)− u(s)]ds,

(h � u)(t) :=

∫ t

0

h(t− s)|u(t)− u(s)|2ds.

A direct calculation shows that

(h ∗ u, ut)Γ2 =− 1

2

d

dt

[ ∫
Γ2

(h � u)(t)dΓ−
(∫ t

0

h(s)ds
)
‖u‖2Γ2

]
− 1

2
h(t)‖u‖2Γ2

+
1

2

∫
Γ2

(h′ � u)(t)dΓ,

(2.1)

and

‖(h ◦ u)(t)‖2Γ2
≤
(∫ t

0

|h(s)|ds
)∫

Γ2

(|h| � u)(t)dΓ. (2.2)

Differentiating (1.3), we arrive at the following Volterra equation

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
+

1

g(0)
g′ ∗ (1 + ‖∇u‖2Ω1

)
∂u

∂ν
= − 1

g(0)
ut.



6 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Applying the Volterra’s inverse operator, we obtain

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
= − 1

g(0)
(ut + k ∗ ut),

where the resolvent kernel satisfies k(t) + 1
g(0) (g′ ∗ k)(t) = − 1

g(0)g
′(t). Denoting

r = 1/g(0), then the aforementioned equality can be written as

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
= −r[ut + k(0)u− k(t)u0 + (k′ ∗ u)(t)]. (2.3)

which (2.3) implies (1.3).
For the resolvent kernel function k, as in [27, 28], we assume that

(H1) k : R+ → R+ is a function of C2 such that

k(0) > 0, lim
t→∞

k(t) = 0, k′(t) ≤ 0,

and there exists a non-increasing continuous function ξ : R+ → R+ satisfy-
ing

k′′(t) ≥ −ξ(t)k′(t), ∀t ≥ 0, and

∫ +∞

0

ξ(s)ds = +∞.

As in [26], we introduce the variable

z(x, ρ, t) = ut(x, t− τρ), (x, ρ, t) ∈ S1 × (0, 1).

Then z satisfies

τzt(x, ρ, t) + zρ(x, ρ, t) = 0, (x, ρ, t) ∈ S1 × (0, 1).

Therefore, problem (1.1)-(1.8) can be rewritten as

utt − (1 + ‖∇u‖2Ω1
)∆u+ µ1ut + µ2z(x, 1, t) = 0, (x, t) ∈ S1, (2.4)

τzt(x, ρ, t) + zρ(x, ρ, t) = 0, (x, ρ, t) ∈ S1 × (0, 1), (2.5)

vtt − (1 + ‖∇v‖2Ω2
)∆v = 0, (x, t) ∈ S2, (2.6)

v = 0, (x, t) ∈ ∂S0, (2.7)

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
= (1 + ‖∇v‖2Ω2

)
∂v

∂ν
, u = v, (x, t) ∈ ∂S1, (2.8)

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
= −r[ut + k(0)u− k(t)u0 + (k′ ∗ u)(t)], (x, t) ∈ ∂S2, (2.9)

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω1, (2.10)

z(x, 1, t) = f0(x, t− τ), (x, t) ∈ Ω1 × (0, τ), (2.11)

z(x, 0, t) = ut, (x, t) ∈ S1, (2.12)

v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ Ω2. (2.13)

Therefore, it is sufficient to consider problem (2.4)-(2.13), which is equivalent to
(1.1)-(1.8).

Firstly, we present the definition of weak solution of (2.4)-(2.13).

Definition 2.1. Let the initial data (u0, v0) ∈ H2
0 (Ω1)×H2

0 (Ω2), (u1, v1) ∈ V , and
f0 ∈ L2(Ω1 × (−τ, 0)) be given. Functions (u, v, z) ∈ C(0, T ;V × L2(Ω1 × (0, 1)))



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 7

are called the weak solution of problem (2.4)-(2.13), if (u, v, z) satisfies the initial
conditions (u(0), v(0)) = (u0, v0), z(x, 1, t) = f0(x, t− τ), for all t ∈ (0, τ), and∫

Ω1

uttφdx+ (1 + ‖∇u‖2Ω1
)

∫
Ω1

∇u · ∇φdx+ µ1

∫
Ω1

utφdx+ µ2

∫
Ω1

z(x, 1, t)φdx

+

∫
Ω2

vttψdx+ (1 + ‖∇v‖2Ω2
)

∫
Ω2

∇v · ∇ψdx

= −r
∫

Γ2

[ut + k(0)u+ (k′ ∗ u)(t)− k(t)u(0)]φdx,∫
Ω1

τztϕdx+

∫
Ω1

zρϕdx = 0,

for all (φ, ψ) ∈ V , all ϕ(x, ρ) ∈ L2(Ω1 × (0, 1)), and all t ∈ [0, τ ].

As for the well-posedness of solution to problem (2.4)-(2.13), by the Feado-
Galerkin approximation technique, we obtain the following result.

Theorem 2.2. Suppose that µ2 ≤ µ1 and (H1) holds. Then for (u0, v0) ∈ H2
0 (Ω1)×

H2
0 (Ω2), (u1, v1) ∈ V , f0 ∈ L2(Ω1 × (−τ, 0)) satisfying the compatibility conditions

(1 + ‖∇u0‖2Ω1
)
∂u0

∂ν
+ ru1 = 0, on Γ2, v0 = 0, on Γ0,

u0 = v0, (1 + ‖∇u0‖2Ω1
)
∂u0

∂ν
= (1 + ‖∇v0‖2Ω2

)
∂v0

∂ν
, on Γ1,

there exists a unique weak solution (u, v, z) of problem (2.4)-(2.13) such that

(u, v) ∈ C((0,+∞);V ) ∩ C1((0,+∞);L2(Ω1)× L2(Ω2)),

z ∈ C((0,+∞);L2((0, 1)× Ω1)).

To state the result of uniform decay rate for energy, we define the energy func-
tional

E(t) :=
1

2
(‖ut‖2Ω1

+ ‖vt‖2Ω2
+ ‖∇u‖2Ω1

+ ‖∇v‖2Ω2
) +

1

4
(‖∇u‖4Ω1

+ ‖∇v‖4Ω2
)

+
r

2
k(t)‖u‖2Γ2

− r

2

∫
Γ2

(k′ � u)(t)dΓ +
ζ

2

∫
Ω1

∫ 1

0

|z(x, ρ, t)|2dρdx,
(2.14)

where ζ is a positive constant such that

τµ2 < ζ < τ(2µ1 − µ2). (2.15)

Next, we establish a general decay estimate result.

Theorem 2.3. Let (u, v, z) be the solution of (2.4)-(2.13), assuming µ2 < µ1 and
(H1) holds. Then for t0 > 0 large enough, there exist constants C0 > 0 and $ > 0
such that

(i) E(t) ≤ C0E(0)e−$
∫ t
0
ξ(s)ds for all t ≥ t0, if u0 = 0 on Γ2,

(ii) otherwise, E(t) ≤ C0[E(0) + ‖u0‖2Γ2

∫ t
0
k2(s)e$

∫ s
0
ξ(r)drds]e−$

∫ t
0
ξ(s)ds for

all t ≥ t0.

Remark 2.4. The exponential decay and polynomial decay in previous literatures
are special cases of the result in Theorem 2.3. In fact, if we take

k(t) = e−σt, σ > 0, ξ(t) = σ; k(t) =
1

(1 + t)σ
, σ > 0, ξ(t) =

1 + σ

1 + t
;



8 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

k(t) =
1

ln(ln(3 + t))
, ξ(t) =

1

ln(3 + t)(3 + t)
,

then by the result of Theorem 2.3, the energy may decay exponentially, polynomi-
ally, and logarithmically, respectively.

3. Well-posedness

In this section, by using Feado-Galerkin approximation technique and some prior
estimates, we establish the well-posedness of problem (2.4)-(2.13).

Proof of Theorem 2.2. We divide the proof into four steps.

Step 1. Feado-Galerkin approximation. Let {(φj , ψj)}j∈N+ be a basis for
V , which is orthogonal in L2(Ω1) × L2(Ω2). For all n ≥ 1, denoting Vn :=
span{(φ1, ψ1), (φ2, ψ2), . . . , (φn, ψn)} and defining the sequence {ϕj(x, ρ)}1≤j≤n as
follows:

ϕj(x, 0) = φj(x).

Then we may extend ϕj(x, 0) by ϕj(x, ρ) over L2(Ω1 × (0, 1)) and denote Wn =
span = {ϕ1, ϕ2 . . . , ϕn}.

We define the approximations:

(u(n)(x, t), v(n)(x, t)) :=

n∑
j=1

bjn(t)(φj(x), ψj(x)),

z(n)(x, ρ, t) :=

n∑
j=1

cjn(t)ϕj(x, ρ),

where (u(n), v(n), z(n)) are solutions to the following finite dimensional Cauchy prob-
lem: ∫

Ω1

u
(n)
tt φjdx+ (1 + ‖∇u(n)‖2Ω1

)

∫
Ω1

∇u(n) · ∇φjdx

+ µ1

∫
Ω1

u
(n)
t φjdx+ µ2

∫
Ω1

z(n)(x, 1, t)φjdx

+

∫
Ω2

v
(n)
tt ψjdx+ (1 + ‖∇v(n)‖2Ω2

)

∫
Ω2

∇v(n) · ∇ψjdx

= −r
∫

Γ2

[u
(n)
t + k(0)u(n) + (k′ ∗ u(n))(t)− k(t)u0n]φjdx,

(3.1)

∫
Ω1

τz
(n)
t ϕjdx+

∫
Ω1

z(n)
ρ ϕjdx = 0, (3.2)

z(n)(x, 0, t) = u
(n)
t (x, t), (3.3)

and

(u0n, v0n) = (u(n)(0), v(n)(0))→ (u0, v0), in H2
0 (Ω1)×H2

0 (Ω2), (3.4)

(u1n, v1n) = (u
(n)
t (0), v

(n)
t (0))→ (u1, v1), in V, (3.5)

z0n = z(n)(x, 1, t)→ f0(x, t− τ), in L2(Ω1 × (−τ, 0)). (3.6)

According to the standard theory of ordinary differential equations, the finite di-
mensional problem (3.1)-(3.3) possesses a unique solution (bjn(t), cjn(t))j=1,...,n on



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 9

[0, Tn), Tn > 0. The extension of these solutions to the whole interval [0, T ], for all
T > 0, is a consequence of the first estimate which we are going to prove below.

Step 2. Energy estimates. A prior estimate I: Multiplying (3.1) by b′jn(t)
and summing on j, then using (2.1) we have

d

dt

{1

2
(‖u(n)

t ‖2Ω1
+ ‖v(n)

t ‖2Ω2
+ ‖∇u(n)‖2Ω1

+ ‖∇v(n)‖2Ω2
)

+
1

4
(‖∇u(n)‖4Ω1

+ ‖∇v(n)‖4Ω2
)
}

= −µ1‖u(n)
t ‖2Ω1

− µ2

∫
Ω1

u
(n)
t z(n)(x, 1, t)dx− r‖u(n)

t ‖2Γ2

+ rk(t)

∫
Γ2

u0nu
(n)
t dx− r

2

∫
Γ2

(k′′ � u(n))(t)dΓ +
r

2
k′(t)‖u(n)‖2Γ2

+
d

dt

[r
2

∫
Γ2

(k′ � u(n))(t)dΓ− r

2
k(t)‖u(n)‖2Γ2

]
.

(3.7)

Multiplying (3.2) by ζ
τ c
′
jn(t) and integrating over (0, 1) on ρ and then summing on

j, we obtain

ζ

2

d

dt

∫
Ω1

∫ 1

0

|z(n)(x, ρ, t)|2dρdx = − ζ

2τ
[‖z(n)(x, 1, t)‖2Ω1

− ‖u(n)
t ‖2Ω1

], (3.8)

where ζ is a positive constant such that

τµ2 ≤ ζ ≤ τ(2µ1 − µ2).

Combining (3.7) and (3.8), we can derive

d

dt
E(n)(t) + r‖u(n)

t ‖2Γ2
+
r

2

∫
Γ2

(k′′ � u(n))(t)dΓ− r

2
k′(t)‖u(n)‖2Γ2

= −(µ1 −
ζ

2τ
)‖u(n)

t ‖2Ω1
− ζ

2τ
‖z(n)(x, 1, t)‖2Ω1

− µ2

∫
Ω1

u
(n)
t z(n)(x, 1, t)dx+ rk(t)

∫
Γ2

u0nu
(n)
t dΓ,

(3.9)

where

E(n)(t) =
1

2
‖u(n)

t ‖2Ω1
+

1

2
‖v(n)
t ‖2Ω2

+
1

2
‖∇u(n)‖2Ω1

+
1

2
‖∇v(n)‖2Ω2

+
1

4
‖∇u(n)‖4Ω1

+
1

4
‖∇v(n)‖4Ω2

− r

2

∫
Γ2

(k′ � u(n))(t)dΓ

+
r

2
k(t)‖u(n)‖2Γ2

+
ζ

2

∫
Ω1

∫ 1

0

|z(n)(x, ρ, t)|2dρdx.

(3.10)

It follows from Young’s inequality that

| − µ2

∫
Ω1

u
(n)
t z(n)(x, 1, t)dx| ≤ µ2

2
‖u(n)

t ‖2Ω1
+
µ2

2
‖z(n)(x, 1, t)‖2Ω1

, (3.11)

rk(t)

∫
Γ2

u0nu
(n)
t dΓ ≤ r

2
‖u(n)

t ‖2Γ2
+
rk2(t)

2
‖u0n‖2Γ2

. (3.12)



10 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Substituting (3.11) and (3.12) into (3.10) we obtain

d

dt
E(n)(t) ≤− r

2

[
‖u(n)

t ‖2Γ2
+

∫
Γ2

(k′′ � u(n))(t)dΓ− k′(t)‖u(n)‖2Γ2

]
− (µ1 −

µ2

2
− ζ

2τ
)‖u(n)

t ‖2Ω1

−
( ζ

2τ
− µ2

2

)
‖z(n)(x, 1, t)‖2Ω1

+
rk2(t)

2
‖u0n‖2Γ2

.

(3.13)

Integrating (3.13) over (0, t), 0 < t ≤ T , and then using Gronwall’s lemma and
(3.4)-(3.6), we obtain the first estimate

‖u(n)
t ‖2Ω1

+ ‖v(n)
t ‖2Ω2

+ ‖∇u(n)‖2Ω1
+ ‖∇v(n)‖2Ω2

+ ‖∇u(n)‖4Ω1
+ ‖∇v(n)‖4Ω2

+ ‖z(n)(x, ρ, t)‖2L2(Ω1×(0,1)) +

∫ t

0

‖u(n)
t (s)‖2Γ2

ds ≤ L1,
(3.14)

where L1 > 0 is a constant independent of n.

A prior estimate II: First of all, it can be deduced easily from the assumptions

on initial data in Theorem 2.2 that ‖u(n)
tt (0)‖2Ω1

+ ‖v(n)
tt (0)‖2Ω2

≤ C, where C > 0
is independent of n. Differentiating (3.1) with respect to t and multiplying it by
b′′jn(t), and summing on j, we have

d

dt

{1

2
‖u(n)

tt ‖2Ω1
+

1

2
‖v(n)
tt ‖2Ω2

+
1

2
‖∇u(n)

t ‖2Ω1
+

1

2
‖∇v(n)

t ‖2Ω2

+
1

2
‖∇u(n)‖2Ω1

‖∇u(n)
t ‖2Ω1

+
1

2
‖∇v(n)‖2Ω2

‖∇v(n)
t ‖2Ω2

+
rk(0)

2
‖u(n)

t ‖2Γ2

+
(∫

Ω1

∇u(n) · ∇u(n)
t dx

)2

+
( ∫

Ω2

∇v(n) · ∇v(n)
t dx

)2}
= 3‖∇u(n)

t ‖2Ω1

∫
Ω1

∇u(n) · ∇u(n)
t dx+ 3‖∇v(n)

t ‖2Ω2

∫
Ω2

∇v(n) · ∇v(n)
t dx

− µ1‖u(n)
tt ‖2Ω1

− µ2

∫
Ω1

u
(n)
tt z

(n)
t (x, 1, t)dx+ rk′(t)

∫
Γ2

u0nu
(n)
tt dΓ

− r‖u(n)
tt ‖2Γ2

− r
∫

Γ2

(k′′ ∗ u(n))u
(n)
tt dΓ− rk′(0)

∫
Γ2

u(n)u
(n)
tt dΓ.

(3.15)

Differentiating (3.2) with respect to t and multiplying it by ζ
τ c
′′
jn(t), integrating

over (0, 1) on ρ and then summing on j, we obtain

ζ

2

d

dt

∫
Ω1

∫ 1

0

|z(n)
t (x, ρ, t)|2dρdx = − ζ

2τ

[
‖z(n)
t (x, 1, t)‖2Ω1

− ‖u(n)
tt ‖2Ω1

]
. (3.16)



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 11

Combining (3.15) and (3.16), we can derive

d

dt
E

(n)
1 (t) + r‖u(n)

tt ‖2Γ2

= 3‖∇u(n)
t ‖2Ω1

∫
Ω1

∇u(n) · ∇u(n)
t dx+ 3‖∇v(n)

t ‖2Ω2

∫
Ω2

∇v(n) · ∇v(n)
t dx

− (µ1 −
ζ

2τ
)‖u(n)

tt ‖2Ω1
− µ2

∫
Ω1

u
(n)
tt z

(n)
t (x, 1, t)dx

+ rk′(t)

∫
Γ2

u0nu
(n)
tt dΓ− r

∫
Γ2

(k′′ ∗ u(n))u
(n)
tt dΓ

− rk′(0)

∫
Γ2

u(n)u
(n)
tt dΓ− ζ

2τ
‖z(n)
t (x, 1, t)‖2Ω1

,

(3.17)

where

E
(n)
1 (t) =

1

2
‖u(n)

tt ‖2Ω1
+

1

2
‖v(n)
tt ‖2Ω2

+
1

2
‖∇u(n)

t ‖2Ω1
+

1

2
‖∇v(n)

t ‖2Ω2

+
1

2
‖∇u(n)‖2Ω1

‖∇u(n)
t ‖2Ω1

+
1

2
‖∇v(n)‖2Ω2

‖∇v(n)
t ‖2Ω2

+
(∫

Ω1

∇u(n) · ∇u(n)
t dx

)2

+
(∫

Ω2

∇v(n) · ∇v(n)
t dx

)2

+
rk(0)

2
‖u(n)

t ‖2Γ2
+
ζ

2

∫
Ω1

∫ 1

0

|z(n)
t (x, ρ, t)|2dρdx.

(3.18)

By Young’s inequality, we obtain

| − µ2

∫
Ω1

u
(n)
tt z

(n)
t (x, 1, t)dx| ≤ µ2

2
‖u(n)

tt ‖2Ω1
+
µ2

2
‖z(n)
t (x, 1, t)‖2Ω1

, (3.19)

|rk′(t)
∫

Γ2

u0nu
(n)
tt dΓ| ≤ ηr‖u(n)

tt ‖2Γ2
+

r

4η
(k′(t))2‖u0n‖2Γ2

, (3.20)

∣∣− r ∫
Γ2

(k′′ ∗ u(n))(t)u
(n)
tt dΓ

∣∣
≤ ηr‖u(n)

tt ‖2Γ2
+
r(k′(t))2

4η
‖k′′(t)‖L1(0,+∞)

∫ t

0

k′′(t− s)‖u(n)(s)‖2Γ2
ds,

(3.21)

| − rk′(0)

∫
Γ2

u(n)u
(n)
tt dΓ| ≤ ηr‖u(n)

tt ‖2Γ2
+

r

4η
(k′(0))2‖u(n)‖2Γ2

, (3.22)

where 0 < η < 1/3 is a constant. Substituting (3.19)-(3.22) into (3.17), we can
derive

d

dt
E

(n)
1 (t) + r(1− 3η)‖u(n)

tt ‖2Γ2

≤ 3‖∇u(n)‖Ω1‖∇u
(n)
t ‖3Ω1

+ 3‖∇v(n)‖Ω2‖∇v
(n)
t ‖3Ω2

− (µ1 −
µ2

2
− ζ

2τ
)‖u(n)

tt ‖2Ω1
− (

ζ

2τ
− µ2

2
)‖z(n)

t (x, 1, t)‖2Ω1

+
r

4η
(k′(t))2‖k′′(t− s)‖L1(0,+∞)

∫ t

0

k′′(t− s)‖u(n)(s)‖2Γ2
ds

+
r

4η
(k′(t))2‖u0n‖2Γ2

+
r

4η
(k′(0))2‖u(n)‖2Γ2

.

(3.23)



12 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Integrating (3.23) over (0, t), 0 < t ≤ T and then using Gronwall’s lemma, we
obtain the second estimate

‖u(n)
tt ‖2Ω1

+ ‖v(n)
tt ‖2Ω2

+ ‖∇u(n)
t ‖2Ω1

+ ‖∇v(n)
t ‖2Ω2

+ ‖z(n)
t (x, ρ, t)‖2L2(Ω1×(0,1)) +

∫ t

0

‖u(n)
tt (s)‖2Γ2

ds ≤ L2,
(3.24)

where L2 > 0 is a constant independent of n.

Step 3. Pass to the limit. It follows from the first prior estimate (3.14) and
second prior estimate (3.24) that there exist subsequences of {u(n)}, {v(n)}, {z(n)}
(we still denote the subsequences by {u(n)}, {v(n)}, {z(n)} for convenience) such
that

(u(n), v(n))→ (u, v) strongly in C(0, T ;V ),

(u
(n)
t , v

(n)
t )→ (ut, vt) strongly in C(0, T ;L2(Ω1)× L2(Ω2)),

z(n) → z strongly in C(0, T ;L2(Ω1 × (0, 1))),

(u(n), v(n))→ (u, v) weak star in L∞(0, T ;V ),

(u
(n)
t , v

(n)
t )→ (ut, vt) weak star in L∞(0, T ;L2(Ω1)× L2(Ω2)),

(u
(n)
tt , v

(n)
tt )→ (utt, vtt) weak star in L∞(0, T ;L2(Ω1)× L2(Ω2)),

z(n) → z weak star in L∞(0, T ;L2(Ω1 × (0, 1))),

z
(n)
t → zt weak star in L∞(0, T ;L2(Ω1 × (0, 1))),

u
(n)
t → ut strongly in L2(0, T ;L2(Γ2)).

The above convergence results are sufficient to pass to the limit in the linear terms of
(3.1) and (3.2). From the first estimate and taking the continuity of trace operator

γ0 : H1(Ω1)→ H
1
2 (Γ2) into account, we have

{u(n)} is bounded in L2(0, T ;H
1
2 (Γ2)),

{u(n)
t } is bounded in L2(0, T ;H

1
2 (Γ2)),

{u(n)
tt } is bounded in L2(0, T ;L2(Γ2)).

The second estimate (3.24) implies

(1 + ‖∇u(n)‖2Ω1
)u(n) → (1 + ‖∇u‖2Ω1

)u strongly in C(0, T ;H1
0 (Ω1)),

(1 + ‖∇v(n)‖2Ω2
)v(n) → (1 + ‖∇v‖2Ω2

)v strongly in C(0, T ;H1
0 (Ω2)).

Thus we can pass to the limit in (3.1) and (3.2) to obtain

utt − (1 + ‖∇u‖2Ω1
)∆u+ µ1ut + µ2z(x, 1, t) = 0, in L2(0,∞;L2(Ω1)),

vtt − (1 + ‖∇v‖2Ω2
)∆v = 0, in L2(0,∞;L2(Ω2)),

τzt(x, ρ, t) + zρ(x, ρ, t) = 0, in L2(0,∞;L2(Ω1)),

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
= −r[ut + k(0)u− k(t)u0 + k′ ∗ u], in L2(0,∞;H

1
2 (Γ2)).



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 13

Step 4. Uniqueness. Let (u, v, z) and (ũ, ṽ, z̃) be two solutions of problem (2.4)-
(2.13). Then (û, v̂, ẑ) = (u− ũ, v − ṽ, z − z̃) satisfies

ûtt − [(1 + ‖∇u‖2Ω1
)∆u− (1 + ‖∇ũ‖2Ω1

)∆ũ]

+ µ1ût + µ2ẑ(x, 1, t) = 0, (x, t) ∈ S1,
(3.25)

τ ẑt(x, ρ, t) + ẑρ(x, ρ, t) = 0, (x, t) ∈ S1 × (0, 1), (3.26)

v̂tt − [(1 + ‖∇v‖2Ω2
)∆v − (1 + ‖∇ṽ‖2Ω2

)∆ṽ] = 0, (x, t) ∈ S2, (3.27)

v = 0, (x, t) ∈ ∂S0, (3.28)

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
− (1 + ‖∇ũ‖2Ω1

)
∂ũ

∂ν

= (1 + ‖∇v‖2Ω2
)
∂v

∂ν
− (1 + ‖∇ṽ‖2Ω2

)
∂ṽ

∂ν
,

(3.29)

û = v̂, (x, t) ∈ ∂S1, (3.30)

[((1 + ‖∇u‖2Ω1
))
∂u

∂ν
− ((1 + ‖∇ũ‖2Ω1

))
∂ũ

∂ν
]

= −r[ût + k(0)û+ (k′ ∗ û)(t)], (x, t) ∈ ∂S2,
(3.31)

û(x, 0) = 0, ût(x, 0) = 0, x ∈ Ω1, (3.32)

ẑ(x, 1, t) = 0, (x, t) ∈ Ω1 × (0, τ), (3.33)

ẑ(x, 0, t) = ût, (x, t) ∈ S1, (3.34)

v̂(x, 0) = 0, v̂t(x, 0) = 0, x ∈ Ω2. (3.35)

Multiplying (3.25) and (3.27) by ût and v̂t, and integrating over Ω1 and Ω2, respec-
tively, using (3.28)-(3.31) and (2.1), we obtain

1

2

d

dt

{
‖ût‖2Ω1

+ ‖v̂t‖2Ω2
+ ‖∇û‖2Ω1

+ ‖∇v̂‖2Ω2

+ ‖∇u‖2Ω1
‖∇û‖2Ω1

+ ‖∇v‖2Ω2
‖∇v̂‖2Ω2

}
=

1

2
‖∇û‖2Ω1

∫
Ω1

∇u · ∇utdx−
∫

Ω1

∇û · (∇u+∇ũ)dx

∫
Ω1

∇ũ · ∇ûtdx

+
1

2
‖∇v̂‖2Ω2

∫
Ω2

∇v · ∇vtdx−
∫

Ω2

∇v̂ · (∇v +∇ṽ)dx

∫
Ω2

∇ṽ · ∇v̂tdx

− µ1‖ût‖2Ω1
− µ2

∫
Ω1

ûtẑ(x, 1, t)dx− r‖ût‖2Γ2
− r

2

∫
Γ2

(k′′ � û)(t)dΓ

+
r

2
k′(t)‖û‖2Γ2

+
d

dt

[r
2

∫
Γ2

(k′ � û)(t)dΓ− r

2
k(t)‖û‖2Γ2

]
.

(3.36)

Multiplying (3.26) by ζ
τ ẑ(x, ρ, t) and integrating over Ω1 × (0, 1) on x and ρ, re-

spectively, we obtain

ζ

2

d

dt

∫
Ω1

∫ 1

0

|ẑ(x, ρ, t)|2dρdx = − ζ

2τ
[‖ẑ(x, 1, t)‖2Ω1

− ‖ût‖2Ω1
], (3.37)



14 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Combining (3.36), (3.37) and using Young’s inequality and estimates (3.14), (3.24),
we can derive

1

2

d

dt

{
‖ût‖2Ω1

+ ‖v̂t‖2Ω2
+ ‖∇û‖2Ω1

+ ‖∇v̂‖2Ω2
+ ‖∇u‖2Ω1

‖∇û‖2Ω1

+ ‖∇v‖2Ω2
‖∇v̂‖2Ω2

− r
∫

Γ2

(k′ � û)dΓ + rk(t)‖û‖2Γ2

+ ζ

∫
Ω1

∫ 1

0

|ẑ(x, ρ, t)|2dρdx
}

≤ C(‖∇û‖2Ω1
+ ‖∇v̂‖2Ω2

).

(3.38)

Integrating (3.38) over (0, t), 0 < t ≤ T and then using Gronwall’s lemma, we
obtain

‖ût‖2Ω1
+ ‖v̂t‖2Ω2

+ ‖∇û‖2Ω1
+ ‖∇v̂‖2Ω2

+ ζ

∫
Ω1

∫ 1

0

|ẑ(x, ρ, t)|2dρdx = 0.

Hence, uniqueness follows.
With the above 4 steps, we obtain the well-posedness of solution for problem

(1.1)-(1.8). �

4. Decay estimates

In this section, we consider the asymptotic behavior of problem (2.4)-(2.13). For
the proof of Theorem 2.3, we need the following lemmas.

Lemma 4.1. Let (u, v, z) be the solution of (2.4)-(2.13), then we have

d

dt
E(t) ≤− r

2
‖ut‖2Γ2

− (µ1 −
ζ

2τ
− µ2

2
)‖ut‖2Ω1

− (
ζ

2τ
− µ2

2
)‖z(x, 1, t)‖2Ω1

+
r

2
k2(t)‖u0‖2Γ2

− r

2

∫
Γ2

(k′′ � u)(t)dΓ +
r

2
k′(t)‖u‖2Γ2

.
(4.1)

Proof. Multiplying (2.4) and (2.6) by ut and vt, and integrating over Ω1 and Ω2,
respectively, with the aid of (2.7)-(2.9) and (2.1), we obtain

d

dt

{1

2
(‖ut‖2Ω1

+ ‖vt‖2Ω2
+ ‖∇u‖2Ω1

+ ‖∇v‖2Ω2
) +

1

4
(‖∇u‖4Ω1

+ ‖∇v‖4Ω2
)
}

= −µ1‖ut‖2Ω1
− µ2

∫
Ω1

utz(x, 1, t)dx− r‖ut‖2Γ2
+ rk(t)

∫
Γ2

u0utdΓ

− r

2

∫
Γ2

(k′′ � u)dΓ +
r

2
k′(t)‖u‖2Γ2

+
d

dt

[r
2

∫
Γ2

(k′ � u)dΓ− r

2
k(t)‖u‖2Γ2

]
.

(4.2)

Multiplying (2.5) by ζ
τ z(x, ρ, t) and integrating over Ω1 × [0, 1] with respect to x

and ρ, respectively, we have

ζ

2

d

dt

∫
Ω1

∫ 1

0

|z(x, ρ, t)|2dρdx = − ζ

2τ
[‖z(x, 1, t)‖2Ω1

− ‖ut‖2Ω1
]. (4.3)



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 15

Combining (4.2), (4.3) and (2.14), we can obtain

d

dt
E(t) =− (µ1 −

ζ

2τ
)‖ut‖2Ω1

− ζ

2τ
‖z(x, 1, t)‖2Ω1

− µ2

∫
Ω1

utz(x, 1, t)dx− r‖ut‖2Γ2
+ rk(t)

∫
Γ2

u0utdx

− r

2

∫
Γ2

(k′′ � u)(t)dΓ +
r

2
k′(t)‖u‖2Γ2

.

(4.4)

By Young’s inequality, we obtain

| − µ2

∫
Ω1

utz(x, 1, t)dx| ≤
µ2

2
‖ut‖2Ω1

+
µ2

2
‖z(x, 1, t)‖2Ω1

, (4.5)

rk(t)

∫
Γ2

u0utdx ≤
r

2
‖ut‖2Γ2

+
r

2
k2(t)‖u0‖2Γ2

. (4.6)

Substituting (4.5), (4.6) into (4.4), we obtain

d

dt
E(t) ≤− r

2
‖ut‖2Γ2

− (µ1 −
ζ

2τ
− µ2

2
)‖ut‖2Ω1

− (
ζ

2τ
− µ2

2
)‖z(x, 1, t)‖2Ω1

+
r

2
k2(t)‖u0‖2Γ2

− r

2

∫
Γ2

(k′′ � u)(t)dΓ +
r

2
k′(t)‖u‖2Γ2

.
(4.7)

Then we can derive the result of Lemma 4.1. �

Remark 4.2. From the range of ζ, we can see that µ1− ζ
2τ−

µ2

2 > 0 and ζ
2τ−

µ2

2 > 0.
However, since

r

2
k2(t)‖u0‖2Γ2

≥ 0,

E(t) may be not nonincreasing.

Now we define the functional

Φ1(t) :=

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx+

∫
Ω2

[(m · ∇v) +
(N

2
− θ
)
v]vtdx,

where 0 < θ < 1 is a constant which will be determined later.

Lemma 4.3. Let (u, v, z) be a solution of problem (2.4)-(2.13), then for t0 > 0
large enough, there exist α1 > 0 such that

d

dt
Φ1(t) ≤− (1− θ)‖∇u‖4Ω1

− α1‖∇u‖2Ω1
+
( 1

4η1
− θ
)
‖ut‖2Ω1

+
1

4η2
‖z(x, 1, t)‖2Ω1

+
(R

2
+
r2

η3

)
‖ut‖2Γ2

+
r2

η3
k2(t)‖u0‖2Γ2

− θ‖vt‖2Ω2
− (1− θ)(1 + ‖∇v‖2Ω2

)‖∇v‖2Ω2
,

where ηi(i = 1, 2, 3) are sufficiently small positive constants.



16 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Proof. By (2.4) and integration by parts, we can derive

d

dt

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

=

∫
Ω1

(m · ∇ut)utdx− (1 + ‖∇u‖2Ω1
)

∫
Ω1

∇(m · ∇u) · ∇udx

+
(N

2
− θ
)
‖ut‖2Ω1

−
(N

2
− θ
)
(1 + ‖∇u‖2Ω1

)‖∇u‖2Ω1

+

∫
∂Ω1

[(m · ∇u) +
(N

2
− θ
)
u](1 + ‖∇u‖2Ω1

)
∂u

∂ν
dΓ

− µ1

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

− µ2

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]z(x, 1, t)dx.

(4.8)

Noting that ∫
Ω1

(m · ∇ut)utdx = −N
2
‖ut‖2Ω1

+
1

2

∫
∂Ω1

(m · ν)|ut|2dΓ, (4.9)

and

−
∫

Ω1

∇(m · ∇u) · ∇udx

= −
∫

Ω1

ΣNi,j=1[
∂

∂xi

(
mj

∂u

∂xj

) ∂u
∂xi

]dx

= −
∫

Ω1

ΣNi,j=1

∂u

∂xi

∂u

∂xj

∂mj

∂xi
dx− 1

2

∫
Ω1

ΣNi,j=1

∂

∂xj

( ∂u
∂xi

)2
mjdx

= (
N

2
− 1)‖∇u‖2Ω1

− 1

2

∫
∂Ω1

(m · ν)|∇u|2dΓ.

(4.10)

Substituting (4.9), (4.10) into (4.8) to deduce that

d

dt

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

= −θ‖ut‖2Ω1
− (1− θ)(1 + ‖∇u‖2Ω1

)‖∇u‖2Ω1
+

1

2

∫
∂Ω1

(m · ν)|ut|2dΓ

− 1

2
(1 + ‖∇u‖2Ω1

)

∫
∂Ω1

(m · ν)|∇u|2dΓ

+

∫
∂Ω1

[(m · ∇u) +
(N

2
− θ
)
u](1 + ‖∇u‖2Ω1

)
∂u

∂ν
dΓ

− µ1

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

− µ2

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]z(x, 1, t)dx.

(4.11)



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 17

Similarly, using (2.6) and integration by parts, we can derive

d

dt

∫
Ω2

[(m · ∇v) +
(N

2
− θ
)
v]vtdx

= −θ‖vt‖2Ω2
− (1− θ)(1 + ‖∇v‖2Ω2

)‖∇v‖2Ω2
+

1

2

∫
∂Ω2

(m · ν̃)|vt|2dΓ

− 1

2
(1 + ‖∇v‖2Ω2

)

∫
∂Ω2

(m · ν̃)|∇v|2dΓ

+

∫
∂Ω2

[(m · ∇v) +
(N

2
− θ
)
v](1 + ‖∇v‖2Ω2

)
∂v

∂ν̃
dΓ,

(4.12)

where ν̃ denotes the outer normal vector pointing towards the exterior of Ω2.
Adding (4.11) to (4.12) and using transmission conditions (2.8), we obtain

d

dt
Φ1(t) =− θ‖ut‖2Ω1

− (1− θ)(1 + ‖∇u‖2Ω1
)‖∇u‖2Ω1

+
1

2

∫
Γ2

(m · ν)|ut|2dΓ− 1

2
(1 + ‖∇u‖2Ω1

)

∫
Γ2

(m · ν)|∇u|2dΓ

+

∫
Γ2

[(m · ∇u) +
(N

2
− θ
)
u](1 + ‖∇u‖2Ω1

)
∂u

∂ν
dΓ

− µ1

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

− µ2

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]z(x, 1, t)dx

− θ‖vt‖2Ω2
− (1− θ)(1 + ‖∇v‖2Ω2

)‖∇v‖2Ω2

+
1

2

∫
Γ0

(m · ν̃)|vt|2dΓ− 1

2
(1 + ‖∇v‖2Ω2

)

∫
Γ0

(m · ν̃)|∇v|2dΓ

+

∫
Γ0

[(m · ∇v) +
(N

2
− θ
)
v](1 + ‖∇v‖2Ω2

)
∂v

∂ν̃
dΓ.

(4.13)

Since ∂v
∂xi

= ν̃i
∂v
∂ν̃ , i = 1, . . . , N and m · ν̃ ≤ 0 on Γ0, we have

− 1

2
(1 + ‖∇v‖2Ω2

)

∫
Γ0

(m · ν̃)|∇v|2dΓ + (1 + ‖∇v‖2Ω2
)

∫
Γ0

(m · ∇v)
∂v

∂ν̃
dΓ

=
1

2
(1 + ‖∇v‖2Ω2

)

∫
Γ0

(m · ν̃)
∣∣∂v
∂ν̃

∣∣2dΓ ≤ 0.

Moreover, since v = 0 on Γ0, (4.13) can be rewritten as

d

dt
Φ1(t) =− θ‖ut‖2Ω1

− (1− θ)(1 + ‖∇u‖2Ω1
)‖∇u‖2Ω1

+
1

2

∫
Γ2

(m · ν)|ut|2dΓ− 1

2
(1 + ‖∇u‖2Ω1

)

∫
Γ2

(m · ν)|∇u|2dΓ

+

∫
Γ2

[(m · ∇u) +
(N

2
− θ
)
u](1 + ‖∇u‖2Ω1

)
∂u

∂ν
dΓ

− µ1

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

− µ2

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]z(x, 1, t)dx



18 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

− θ‖vt‖2Ω2
− (1− θ)(1 + ‖∇v‖2Ω2

)‖∇v‖2Ω2
. (4.14)

It follows from Young’s inequality that∣∣− µ1

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]utdx

∣∣
≤ 2[R2 +

(N
2
− θ
)2
λ2

1]µ2
1η1‖∇u‖2Ω1

+
1

4η1
‖ut‖2Ω1

,

(4.15)

∣∣− µ2

∫
Ω1

[(m · ∇u) +
(N

2
− θ
)
u]z(x, 1, t)dx

∣∣
≤ 2[R2 +

(N
2
− θ
)2
λ2

1]µ2
2η2‖∇u‖2Ω1

+
1

4η2
‖z(x, 1, t)‖2Ω1

,

(4.16)

and∣∣ ∫
Γ2

[(m · ∇u) +
(N

2
− θ
)
u](1 + ‖∇u‖2Ω1

)
∂u

∂ν
dΓ
∣∣

≤ 2R2η3‖∇u‖2Γ2
+ 2
(N

2
− θ
)2
λ2η3‖∇u‖2Ω1

+
1

4η3

∥∥(1 + ‖∇u‖2Ω1
)
∂u

∂ν

∥∥2

Γ2

≤ 2R2η3‖∇u‖2Γ2
+ 2
(N

2
− θ
)2
λ2η3‖∇u‖2Ω1

+
r2

4η3
‖ut + k(0)u− k(t)u0 + k′ ∗ u‖2Γ2

≤ 2R2η3‖∇u‖2Γ2
+ 2
(N

2
− θ
)2
λ2η3‖∇u‖2Ω1

+
r2

η3
‖ut‖2Γ2

+
r2

η3
k2(t)‖u0‖2Γ2

+
r2

η3
k2(t)‖u‖2Γ2

− r2

η3
k(0)

∫
Γ2

(k′ � u)dΓ,

(4.17)

where ηi(i = 1, 2, 3) are sufficiently small positive constants and we have used
inequality (2.2) and the following identity

(1 + ‖∇u‖2Ω1
)
∂u

∂ν
=− r[ut + k(0)u− k(t)u0 + k′ ∗ u]

=− r[ut + k(t)u− k(t)u0 − k′ ◦ u].

Besides, λ and λ1 are the optimal constants of trace inequality and the first eigen-
value of −∆ with Dirichlet boundary condition, respectively, i.e. ‖u‖2Γ2

≤ λ‖∇u‖2Ω1

and ‖u‖2Ω1
≤ λ1‖∇u‖2Ω1

, respectively.
Substituting (4.15)-(4.17) into (4.14), we can derive

d

dt
Φ1(t) ≤

( 1

4η1
− θ
)
‖ut‖2Ω1

(R
2

+
r2

η3

)
‖ut‖2Γ2

− (1− θ)‖∇u‖4Ω1

−
{1− θ

2
− 2[R2 +

(N
2
− θ
)2
λ2

1](µ2
1η1 + µ2

2η2)

− 2
(N

2
− θ
)2
λ2η3

}
‖∇u‖2Ω1

−
(δ

2
− 2R2η3

)
‖∇u‖2Γ2

+
1

4η2
‖z(x, 1, t)‖2Ω1

− [
1− θ

2λ2k(t)
− r2

η3
k(t)]k(t)‖u‖2Γ2

+
r2

η3
k2(t)‖u0‖2Γ2

− r2

η3
k(0)

∫
Γ2

(k′ � u)(t)dΓ

− θ‖vt‖2Ω2
− (1− θ)(1 + ‖∇v‖2Ω2

)‖∇v‖2Ω2
.

(4.18)



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 19

Choosing ηi (i = 1, 2, 3) small enough such that

α1 =
1− θ

2
− 2[R2 +

(N
2
− θ
)2
λ2

1](µ2
1η1 + µ2

2η2)− 2
(N

2
− θ
)2
λ2η3 > 0,

δ

2
− 2R2η3 > 0,

it follows from limt→∞ k(t) = 0 that our result holds for t0 > 0 large enough. �

Next, we define the functional

Φ2(t) := τ

∫
Ω1

∫ 1

0

e−τρ|z(x, ρ, t)|2dρdx.

Then we have the following lemma.

Lemma 4.4. The functional Φ2 satisfies

d

dt
Φ2(t) ≤ −C(τ)[

∫
Ω1

∫ 1

0

|z(x, ρ, t)|2dρdx+ ‖z(x, 1, t)‖2Ω1
] + ‖ut‖2Ω1

,

where C(τ) is a positive constant only depending on τ .

Proof. We use the method introduced by [30] to prove this lemma. Taking the
derivative of Φ2(t) directly, and using (2.5), we have

d

dt
Φ2(t) =2τ

∫
Ω1

∫ 1

0

e−τρz(x, ρ, t)zt(x, ρ, t)dρdx

=− 2

∫
Ω1

∫ 1

0

e−τρz(x, ρ, t)zρ(x, ρ, t)dρdx

=−
∫

Ω1

∫ 1

0

d

dρ
[e−τρ |z(x, ρ, t)|2]dρdx− τ

∫
Ω1

∫ 1

0

e−τρ |z(x, ρ, t)|2 dρdx

=−
∫

Ω1

e−τ |z(x, 1, t)|2 − |z(x, 0, t)|2 dx− τ
∫

Ω1

∫ 1

0

e−τρ |z(x, ρ, t)|2 dρdx

≤− C(τ)[

∫
Ω1

∫ 1

0

|z(x, ρ, t)|2 dρdx+ ‖z(x, 1, t)‖2Ω1
] + ‖ut‖2Ω1

,

where C(τ) is a positive constant only depending on τ . �

Proof of Theorem 2.3. We define the Lyapunov functional

L(t) := M1E(t) +M2Φ1(t) +M3Φ2(t), (4.19)

where Mi (i = 1, 2, 3) are positive constants which will be determined later.



20 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

Differentiating L(t) directly and using Lemma 4.1-Lemma 4.4, we have

d

dt
L(t) ≤−

[(
µ1 −

ζ

2τ
− µ2

2

)
M1 −

( 1

4η1
− θ
)
M2 −M3

]
‖ut‖2Ω1

− [(1− θ)M2]‖∇u‖4Ω1
−M2α1‖∇u‖2Ω1

−M3C(τ)

∫
Ω1

∫ 1

0

|z(x, ρ, t)|2 dρdx

−
[( ζ

2τ
− µ2

2

)
M1 −

M2

4η2
+M3C(τ)

]
‖z(x, 1, t)‖2Ω1

−
[M1r

2
−
(r2

η3
+
R

2

)
M2

]
‖ut‖2Γ2

+
(M1r

2
+
M2r

2

η3

)
k2(t)‖u0‖2Γ2

− M1r

2

∫
Γ2

(k′′ � u)(t)dΓ +
M1r

2
k′(t)‖u‖2Γ2

−M2θ‖vt‖2Ω2
−M2(1− θ)(1 + ‖∇v‖2Ω2

)‖∇v‖2Ω2
.

(4.20)

Choosing Mi > 0, (i = 1, 2, 3) such that(
µ1 −

ζ

2τ
− µ2

2

)
M1 −

( 1

4η1
− θ
)
M2 −M3 > 0,( ζ

2τ
− µ2

2

)
M1 −

M2

4η2
+M3C(τ) > 0,

M1r

2
−
(r2

η3
+
R

2

)
M2 > 0,

and since E(t) is equivalent to

‖ut‖2Ω1
+ ‖vt‖2Ω2

+ ‖∇u‖2Ω1
+ ‖∇v‖2Ω2

+
1

4
‖∇u‖4Ω1

+
1

4
‖∇v‖4Ω2

+ k(t)‖u‖2Γ2
−
∫

Γ2

(k′ � u)dΓ +

∫
Ω1

∫ 1

0

|z(x, ρ, t)|2dρdx,

we know that there exist positive constants β1, β2, β3, such that

d

dt
L(t) ≤ −β1E(t) + β2k

2(t)‖u0‖2Γ2
− β3

∫
Γ2

(k′ � u)(t)dΓ, ∀t ≥ t0. (4.21)

Multiplying (4.21) by ξ(t) and using (H) and (4.1), we obtain

ξ(t)
d

dt
L(t) ≤− β1ξ(t)E(t) + β2ξ(t)k

2(t)‖u0‖2Γ2
− β3ξ(t)

∫
Γ2

(k′ � u)(t)dΓ

≤− β1ξ(t)E(t) + β2ξ(t)k
2(t)‖u0‖2Γ2

+ β3

∫
Γ2

(k′′ � u)(t)dΓ

≤− β1ξ(t)E(t) + β2ξ(t)k
2(t)‖u0‖2Γ2

+ β3[−2

r

d

dt
E(t) + k2(t)‖u0‖2Γ2

].

(4.22)

Noting that ξ′(t) ≤ 0, we have

d

dt
[ξ(t)L(t) +

2β3

r
E(t)] ≤ −β1ξ(t)E(t) + [β2ξ(0) + β3]k2(t)‖u0‖2Γ2

. (4.23)



EJDE-2021/95 A TRANSMISSION PROBLEM OF KIRCHHOFF TYPE 21

Now, define the functional

L(t) := ξ(t)L(t) +
2β3

r
E(t), ∀t ≥ t0.

Then it is easy to verify that L(t) is equivalent to E(t) and there exist constants
γ1, γ2 > 0 such that

d

dt
L(t) ≤ −γ1ξ(t)L(t) + γ2k

2(t)‖u0‖2Γ2
, ∀t ≥ t0. (4.24)

Case 1: If u0 = 0 on Γ2, inequality (4.24) becomes

d

dt
L(t) ≤ −γ1ξ(t)L(t).

Integrating this inequality from 0 to t, we have

L(t) ≤ L(0)e−γ1
∫ t
0
ξ(s)ds, ∀t ≥ t0. (4.25)

It follows from the equivalence relation between L(t) and E(t) that there exists a
constant C > 0 such that

E(t) ≤ CE(0)e−γ1
∫ t
0
ξ(s)ds, ∀t ≥ t0.

Case 2: If u0 6= 0 on Γ2, we set

F(t) = L(t)− γ2‖u0‖2Γ2
e−γ1

∫ t
0
ξ(s)ds

∫ t

0

k2(s)eγ1
∫ s
0
ξ(r)drds, ∀t ≥ t0. (4.26)

Then by calculating directly, and using (4.24) we obtain

d

dt
F(t) ≤ −γ1ξ(t)F(t), ∀t ≥ t0.

Integrating this inequality over (0, t), we have

F(t) ≤ F(0)e−γ1
∫ t
0
ξ(s)ds, ∀t ≥ t0. (4.27)

Combining (4.26) and (4.27), we have

L(t) ≤
[
L(0) + γ2‖u0‖2Γ2

∫ t

0

k2(s)eγ1
∫ s
0
ξ(r)drds

]
e−γ1

∫ t
0
ξ(s)ds, ∀t ≥ t0. (4.28)

It follows from the equivalence relation between L(t) and E(t) that there exists a
constant C > 0 such that

E(t) ≤ C
[
E(0) + γ2‖u0‖2Γ2

∫ t

0

k2(s)eγ1
∫ s
0
ξ(r)drds

]
e−γ1

∫ t
0
ξ(s)ds, ∀t ≥ t0.

The proof of Theorem 2.3 is complete. �

Acknowledgments. This work was supported by the Natural Science Foundation
of Shandong Province of China (No. ZR2019MA072) and the Fundamental Research
Funds for the Central Universities (No. 201964008). The authors would like to
thank all the reviewers for their insightful and constructive comments.



22 Z. Q. LIU, C. C. GAO, Z. B. FANG EJDE-2021/95

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Zhiqing Liu

School of Mathematics and Physics, Qingdao University of Science and Technology,
Qingdao 266061, China

Email address: Lzhiqing1005@163.com

Cunchen Gao

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Email address: ccgao123@126.com

Zhong Bo Fang (corresponding author)

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
Email address: fangzb7777@hotmail.com


	1. Introduction
	2. Preliminaries and main results
	3. Well-posedness
	4. Decay estimates
	Acknowledgments

	References