Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 104, pp. 1–31.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.104

GLOBAL SOLUTIONS AND BLOW-UP FOR WAVE EQUATIONS WITH

VARIABLE COEFFICIENTS AND BOUNDARY SUPERCRITICAL SOURCE

TAE GAB HA

Abstract. In this study, we consider a wave equation with variable coefficients, boundary

damping, and supercritical source terms. This study aims to study the local and global existence
and classify the decay rate of energy depending on the growth near zero in the damping term.

Furthermore, we demonstrate that the weak solution blows up in finite time with positive and

nonpositive initial energies.

1. Introduction

In this study, we investigate the local and global existence, energy decay rates, and finite time
blow-up of the solution to the wave equation

utt − µ(t)Lu = f(x, t) in Ω× (0,+∞),

u = 0 on Γ0 × (0,+∞),

µ(t)
∂u

∂νL
+ q(ut) = h(u) on Γ1 × (0,+∞),

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

(1.1)

where

Lu = div(A(x)∇u) =
n∑

i,j=1

∂

∂xi

(
aij(x)

∂u

∂xj

)
.

Here, A(x) = (aij(x)) is a symmetric and positive matrix and ∂u
∂νL

=
∑n

i,j=1 aij(x)
∂u
∂xj

νi, where

ν = (ν1, . . . , νn) is the outward unit normal to Γ. Ω is a bounded domain of Rn(n ≥ 3) with
smooth boundary Γ = Γ0 ∪ Γ1. Here, Γ0 and Γ1 are closed and disjoint with meas(Γ0) ̸= 0.

Problem (1.1) has been widely studied over the past few decades. The condition h(s)s ≤ 0
indicates that h represents the attractive force. When h(s)s ≥ 0 as in the present case, h represents
the source term. This scenario is more delicate than the attractive force because the solution of
(1.1) can blow up. The damping-source interplay in the system (1.1) arises naturally in numerous
contexts, such as classical mechanics, fluid dynamics, and quantum field theory (cf. [29, 41]). The
interaction between the two competitive forces, that is, the damping and source terms, makes the
problem attractive from a mathematical perspective.

For the present case when h is polynomial nonlinear source term such as h(u) = |u|γu, the
stability of (1.1) has been studied by many authors (see [7, 8, 14, 15, 16, 17, 18, 20, 21, 22, 24,
25, 26, 27, 28, 30, 31, 36, 37, 38, 39, 43] and a list on references therein), where h(u) is subcritical
or critical source. However, very few results addressed wave equations influenced by supercritical
sources (cf. [1, 3, 11, 12, 42]). For example, [42] proved the local and global existence, uniqueness
and Hadamard well-posedness for the wave equation when source terms can be supercritical or
super-supercritical. However, the author did not consider the energy decay and blow-up of the
solutions. Guo et al. [11] considered a system of nonlinear wave equations with supercritical sources

2020 Mathematics Subject Classification. 35L05, 35L20, 35A01, 35B40, 35B44.
Key words and phrases. Wave equation with variable coefficients; supercritical source; blow-up;

existence of solutions; energy decay rates.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted June 5, 2025. Published November 4, 2025.

1



2 T. G. HA EJDE-2025/104

and damping terms. They proved global existence and exponential and algebraic uniform decay
rates of energy, moreover, blow-up result for weak solutions with nonnegative initial energy. But
as far as I know, the only problem with considering supercritical source is the constant coefficients
case, that is, A = I and dimension n = 3.

In the case of variable coefficients, that is A ̸= I, boundary stability of the wave equation was
considered in [4, 9, 15, 18, 22]. The wave equations with variable coefficients arise in mathematical
modeling of inhomogeneous media in solid mechanics, electromagnetic, fluid flows through porous
media. For the variable coefficients problem, the main tool is Riemannian geometry method, which
was introduced by [46] and has been widely used in the literature, see [6, 9, 32, 33, 34, 44, 45]
and a list of references therein. However, there were very few results considered the source term.
For example, [4] proved the uniform decay rate of the energy to the viscoelastic wave equation
with variable coefficients and acoustic boundary conditions without damping term. Recently,
[26] studied the general decay rate of the energy for the wave equation with variable coefficients
and Balakrishnan-Taylor damping and source term without imposing any restrictive growth near
zero on the damping term. However, above-mentioned literature did not consider Riemannian
geometry, and only treated a subcritical source. [19] proved the uniform energy decay rates of the
wave equation with variable coefficients applying the Riemannian geometry method and modified
multiplier method. But, it was considered a subcritical source. More recently, the author proved
the existence of a solution, the energy decay rate, and the blow-up of the solution for a wave
equation with an interior supercritical source ([23]). The method and process of the proof in this
study are similar to [23]. However, there are some differences between this study and that of [23].
Reference [23] dealt with an interior supercritical source without considering Riemannian geometry.
Hence, the parameter analysis results for the source term were different (see Figures 1 and 2).
Moreover, because this study considers the boundary conditions and Riemannian geometry, the
analysis is more complicated, although the method and process of the proof are similar to [23]. To
the best of our knowledge, there is no existing literature addressing the variable coefficient problem
that includes both damping and source terms on Riemannian geometry while also considering the
boundary supercritical source.

Our main motivation is constituted by three dimensional case, in which the source term can
be supercritical on variable coefficient problem. The differences from previous literature are as
follows:

(i) Supercritical source for n ≥ 3.
(ii) Variable coefficient problem having source term on Riemannian geometry.
(iii) Blow-up result with positive initial energy as well as nonpositive initial energy.

To overcome difficulties and prove the above statements, first, we refine the energy space and
a constant used in potential well method, because we do not guarantee H1

0 (Ω) ↪→ Lγ+2(Γ1) since
the source term is supercritical. Also we have a hypothesis on damping term for proving existence
of solutions and energy decay rates (see Remark 2.2). Second, we use the Faedo-Galerkin method
because nonlinear semigroup arguments considered in the previous literatures cannot be used since
this paper deals with an operator −µ(t)L, which depends on t. Third, we refine the key point
constants used to prove blow-up result. So, this paper has improved and generalized previous
literatures.

The goal of this paper is to prove the existence result using the Faedo-Galerkin method and
truncated approximation method, and classify the energy decay rate applying the method devel-
oped in [46, 35]. Moreover, we prove the blow-up of the weak solution with positive initial energy
as well as nonpositive initial energy. This paper is organized as follows: In Section 2, we recall the
notation, hypotheses and some necessary preliminaries and introduce our main result. In Section
3, we prove the local existence of weak solutions, and show the global existence of weak solution
in each two conditions in Section 4. In Section 5, we prove the uniform decay rate under suit-
able conditions on the initial data and boundary damping by the differential geometric approach.
In Section 6, we prove the blow-up of the weak solution with positive initial energy as well as
nonpositive initial energy. by using contradiction method.



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 3

2. Preliminaries

We begin this section by introducing notation and stating our main results. We define the
Hilbert space H1

0 (Ω) = {u ∈ H1(Ω);u = 0 on Γ0} with the norm ∥u∥H1
0 (Ω) = ∥∇u∥L2(Ω) and

H = {u ∈ H1
0 (Ω);u ∈ Lγ+2(Γ1)} with the norm ∥u∥H = ∥u∥H1

0 (Ω) + ∥u∥Lγ+2(Γ1). ∥ · ∥p and ∥ · ∥p,Γ
are denoted by the Lp(Ω) norm and the Lp(Γ) norm, respectively, and ⟨u, v⟩ =

∫
Ω
u(x)v(x)dx

and ⟨u, v⟩Γ =
∫
Γ
u(x)v(x) dΓ. Moreover, we require some notations for Riemannian geometry, as

mentioned in [46] and the references therein. For reader comprehension, we have repeated these
here.

Let A(x) = (aij(x)) be a symmetric and positive definite matrix for all x ∈ Rn (n ≥ 3) and
aij(x) be smooth functions on Rn satisfying

c1

n∑
i=1

ξ2i ≤
n∑

i,j=1

aij(x)ξiξj , ∀x ∈ Rn, 0 ̸= ξ = (ξ1, . . . , ξn)
T ∈ Rn, (2.1)

where c1 is a positive constant. Set

G(x) =
(
gij(x)

)
= A−1(x).

For each x ∈ Rn, we define the inner product g(·, ·) = ⟨·, ·⟩g and the norm | · |g on the tangent
space Rn

x = Rn as

g(X,Y ) = ⟨X,Y ⟩g =

n∑
i,j=1

gij(x)αiβj , |X|g = ⟨X,X⟩
1
2
g ,

∀X =

n∑
i=1

αi
∂

∂xi
, Y =

n∑
i=1

βi
∂

∂xi
∈ Rn

x .

(2.2)

Subsequently, (Rn, g) is a Riemannian manifold with Riemann metric g. ∇gu and Dg are denoted
by the gradient of u and Levi-Civita connection in the Riemannian metric g, respectively. It
follows that

∇gu =

n∑
i=1

( n∑
j=1

aij(x)
∂u

∂xj

) ∂

∂xi
= A(x)∇u, |∇gu|2g =

n∑
i,j=1

aij(x)
∂u

∂xi

∂u

∂xj
. (2.3)

Let H be the vector field on (Rn, g). Subsequently, the covariant differential DgH of H deter-
mines a bilinear form on Rn

x × Rn
x , for each x ∈ Rn as

DgH(X,Y ) = ⟨DgXH,Y ⟩g, ∀X,Y ∈ Rn
x , (2.4)

where DgXH is the covariant derivative of the vector field H with respect to X.

(H1) Hypothesis on Ω. Let Ω ⊂ Rn be a bounded domain, n ≥ 3 with a smooth boundary
Γ = Γ0 ∪Γ1. Here Γ0 and Γ1 are closed and disjoint, where meas(Γ0) ̸= 0. A vector field H exists
on the Riemannian manifold (Rn, g) such that

DgH(X,X) ≥ σ|X|2g, ∀X ∈ Rn
x , x ∈ Ω, (2.5)

where σ is a positive constant and

H · ν ≤ 0 on Γ0 and H · ν ≥ δ > 0 on Γ1, (2.6)

where ν represents the unit outward normal vector to Γ. Moreover we assume that

µ(0)
∂u0
∂νL

+ g(u1) = h(u0) on Γ1. (2.7)

(H2) Hypothesis on µ and f . Let µ ∈ W 1,∞(0,∞) ∩W 1,1(0,∞) be a function satisfying the
conditions

µ(t) ≥ µ0 > 0 and µ′(t) ≤ 0 a.e. in [0,∞), (2.8)

where µ0 is a positive constant. We assume that

f ∈ H1(0,∞;L2(Ω)). (2.9)



4 T. G. HA EJDE-2025/104

(H3) Hypothesis on q. Let q : R → R be a nondecreasing C1 function such that q(0) = 0 and
suppose that there exist positive constants c3, c4, and a strictly increasing and odd function β of
C1 class on [−1, 1], such that

|β(s)| ≤ |q(s)| ≤ |β−1(s)| if |s| ≤ 1, (2.10)

c3|s|ρ+1 ≤ |q(s)| ≤ c4|s|ρ+1 if |s| > 1, (2.11)

where β−1 denotes the inverse function of β, and ρ ≥ 2(n−2)γ−2
n−(n−2)γ .

(H4) Hypothesis on γ. Let γ be a constant satisfying the condition

1

n− 2
< γ ≤ n− 1

n− 2
. (2.12)

Lemma 2.1 ([46]). Let u, v ∈ C1(Ω) and H, X be vector fields on (Rn, g). Then

H(u) = ⟨∇gu,H⟩g, ⟨H(x), A(x)X(x)⟩g = H(x) ·X(x), (2.13)

div(uH) = udiv(H) +H(u),

∫
Ω

div(H) dx =

∫
Γ

H · ν dΓ, (2.14)∫
Ω

u div(∇gv) dx =

∫
Γ

u
∂v

∂νL
dΓ−

∫
Ω

⟨∇gu,∇gv⟩g dx, (2.15)

⟨∇gu,∇g(H(u))⟩g = DgH(∇gu,∇gu) +
1

2
div(|∇gu|2gH)− 1

2
|∇gu|2g div(H). (2.16)

Hypothesis (2.5) was introduced by Yao [46] for the exact controllability of the wave equation
with variable coefficients. The existence of such a vector field depends on the sectional curvature
of the Riemannian manifold (Rn, g). There are several methods and examples in [46] to find out
a vector field H that is satisfied the hypothesis (2.5). Specially, if A = I, the constant coefficient
case, the condition (2.5) is automatically satisfied by choosing H = x− x0 for any x0 ∈ Rn.

Remark 2.2. In view of the critical Sobolev imbedding H1/2(Γ1) ↪→ L
2(n−1)
n−2 (Γ1), the mapping

k(u) = |u|γu is not locally Lipschitz from H1
0 (Ω) into L2(Γ1) for the supercritical values 1

n−2 <

γ ≤ n−1
n−2 . However, by the hypothesis on ρ

(
ρ ≥ 2(n−2)γ−2

n−(n−2)γ

)
, k(u) is locally Lipschitz from H1

0 (Ω)

into L
ρ+2
ρ+1 (Γ1).

For n > 3, if 2
n−2 ≤ γ ≤ n−1

n−2 , then the inequality ρ ≥ γ always holds true under the assumption

ρ ≥ 2(n−2)γ−2
n−(n−2)γ (see Figure 1).

Figure 1. Admissible range of the damping parameter ρ and the exponent of
the source γ.



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 5

Definition 2.3. A function u(x, t) is called a weak solution of (1.1) on Ω × (0, T ) if u ∈
Cw(0, T ;H) ∩ C1

w(0, T ;L
2(Ω)), ut|Γ1 ∈ Lρ+2(0, T ; Γ1) and satisfies (1.1) in the distribution sense,

i.e., ∫ T

0

∫
Ω

−utϕt dx dt+
∫
Ω

utϕdx
∣∣∣T
0
+

∫ T

0

µ(t)

∫
Ω

⟨∇gu,∇gϕ⟩g dx dt

+

∫ T

0

∫
Γ1

q(ut)ϕdΓ dt−
∫ T

0

∫
Γ1

h(u)ϕdΓ dt

=

∫ T

0

∫
Ω

f(x, t)ϕdx dt,

for any ϕ ∈ Cw(0, T ;H) ∩ C1
w(0, T ;L

2(Ω)), ϕ|Γ1
∈ Lρ+2(0, T ; Γ1) and u(x, 0) = u0(x) ∈ H,

ut(x, 0) = u1(x) ∈ L2(Ω).

One can easily check that H = H1
0 (Ω) when

1
n−2 < γ ≤ 2

n−2 . Moreover, if n = 3, then we can

replace H by H1
0 (Ω), since H

1
0 (Ω) ↪→ Lγ+2(Γ1) (see Figure 2).

Figure 2. Admissible range of parameters a and b with respect to the trace
imbedding H1

0 (Ω) ↪→ Laγ+b(Γ1).

The energy associated with problem (1.1) when h(u) = |u|γu is

E(t) =
1

2
∥ut(t)∥22 +

1

2
µ(t)∥ |∇gu(t)|g∥22 −

1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1
.

We now state our main results.

Theorem 2.4. Suppose that (H1)–(H4) hold and let h(u) = |u|γu. Then for the given initial
data (u0, u1) ∈ H ×2 (Ω), there exist T > 0 and a weak solution of problem (1.1). Moreover, the
following energy identity holds for all 0 ≤ t ≤ T ,

E(t) +

∫ t

0

∫
Γ1

q(us)us dΓ ds−
1

2

∫ t

0

u′(s)∥ |∇gu(s)|g∥22 ds−
∫ t

0

∫
Ω

f(x, s)usdx ds = E(0). (2.17)

Furthermore, if one of the 2 assumptions hold: ρ ≥ γ or

E(0) < d0 and ∥ |∇gu0|g∥2 < λ0 (2.18)

with f = 0, where

λ0 =
( µ0

Kγ+2
0

)1/γ

d0 =
γµ0

2(γ + 2)
λ20, K0 = sup

u∈H,u ̸=0

( ∥u∥γ+2,Γ1

∥ |∇gu|g∥2

)
.

Then the solution u(x, t) of (1.1) is global.



6 T. G. HA EJDE-2025/104

Theorem 2.5. Suppose that the hypotheses in Theorem 2.4 and (2.18) with ρ ≤ γ and f = 0
hold. Then we have following energy decay rates:

Case 1: β is linear. Then

E(t) ≤ C1e
−ωt,

where ω is a positive constant.
Case 2: β has polynomial growth near zero, that is, β(s) = sρ+1. Then

E(t) ≤ C2

(1 + t)2/ρ
.

Case 3: β does not necessarily have polynomial growth near zero. Then

E(t) ≤ C3

(
F−1

(1
t

))2

,

where F (s) = sβ(s) and Ci (i = 1, 2, 3) are positive constants that depends only on E(0).

Theorem 2.6. Suppose that hypotheses (H1)–(H4) hold with ρ < γ and f = 0. Moreover, assume
that

(u0, u1) ∈ {(u0, u1) ∈ H × L2(Ω); ∥ |∇gu0|g∥2 > λ0, −1 < E(0) < d0}
and

β−1(1) ≤
( (γ + 2)(µ0γλ

2
0 − 2(γ + 2)E1)

2

8(γ + 1)meas(Γ1)(µ0λ20 − 2E1)

) γ+1
γ+2

, (2.19)

where

E1 =

{
0 if E(0) < 0,

positive constant with E(0) < E1 < d0, E1 < E(0) + 1 if E(0) ≥ 0.

Then the weak solution of problem (1.1) blows up in finite time.

We have summarized our results as follows.

(1) Local existence is obtained for the regions I, II, and III in Figure 1.
(2) Global existence is obtained for the region I and III, or the region II with the condition

(2.18) in Figure 1.
(3) Energy decay rate is obtained for region II in Figure 1 with the condition (2.18).
(4) For the region II in Figure 1 with the condition (2.19), we obtain the blow-up in finite

time.

3. Proof of Theorem 2.4: local existence

The proof is based on three steps according to the following condition of the source term h:

(1) Existence of the global solution when h is globally Lipschitz from H1
0 (Ω) to L

2(Γ1).
(2) Existence of the local solution when h is locally Lipschitz from H1

0 (Ω) to L
2(Γ1).

(3) Existence of the local solution when h is locally Lipschitz from H1
0 (Ω) to L

ρ+2
ρ+1 (Γ1).

Since the mapping |u|γu is locally Lipschitz from H1
0 (Ω) to L

ρ+2
ρ+1 (Γ1) (see Remark 2.2), the

existence of the local solution can be guaranteed even if h(u) = |u|γu.

3.1. Globally Lipschitz source. We first deal with the case where the source h is globally
Lipschitz from H1

0 (Ω) to L
2(Γ1). In this case, we have the following result.

Proposition 3.1. Assume that (H1)-(-H3) hold. In addition, assume that (u0, u1) ∈ H × L2(Ω)
and h : H1

0 (Ω) → L2(Γ1) is globally Lipschitz continuous. Then problem (1.1) has a unique global
solutions u ∈ Cw(0, T ;H) ∩ C1

w(0, T ;L
2(Ω)) for arbitrary T > 0.

Our goal in this subsection is to show the local existence result for problem (1.1). We construct
an approximate solution by using the Faedo-Galerkin method. Let {wj}j∈N be a basis in H1

0 (Ω)
and define Vm = span{w1, w2, . . . , wm}. Let um0 and um1 be sequences of Vm such that um0 → u0



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 7

strongly in H1
0 (Ω) and u

m
1 → u1 strongly in L2(Ω). We search for a function, for each η ∈ (0, 1)

and m ∈ N, such that

uηm(t) =

m∑
j=1

δjm(t)wj

satisfying the approximate perturbed equation∫
Ω

uηmtt wdx+ µ(t)

∫
Ω

⟨∇gu
ηm,∇gw⟩gdx+ η

∫
Γ1

uηmt w dΓ

+

∫
Γ1

q(uηmt )w dΓ−
∫
Γ1

h(uηm)w dΓ

=

∫
Ω

f(t)wdx, w ∈ Vm,

uηm0 =

m∑
j=1

⟨u0, wj⟩wj , uηm1 =

m∑
j=1

⟨u1, wj⟩wj .

(3.1)

Since (3.1) is a normal system of ordinary differential equations, there exist solutions uηm to
problem (3.1). A solution u to problem (1.1) on some interval [0, tm), tm ∈ (0, T ] will be obtained
as the limit of uηm as m → ∞ and η → 0. Next, we show that tm = T and the local solution is
uniformly bounded independent of m, η and t. For this purpose, let us replace w by uηmt in (3.1)
we obtain

d

dt

[1
2
∥uηmt ∥22 +

1

2
µ(t)∥ |∇gu

ηm|g∥22 +
1

γ + 2
∥uηm∥γ+2

γ+2,Γ1

]
+ η∥uηmt ∥22,Γ1

+

∫
Γ1

q(uηmt ) uηmt dΓ

=
1

2
µ′(t)∥ |∇gu

ηm|g∥22 +
∫
Γ1

h(uηm)uηmt dΓ +

∫
Γ1

|uηm|γuηmuηmt dΓ +

∫
Ω

f(t)uηmt dx.

(3.2)
Now we estimate

∫
Γ1
q(uηmt ) uηmt dΓ,

∫
Γ1

|uηm|γuηmuηmt dΓ and
∫
Γ1
h(uηm)uηmt dΓ. From the

hypotheses on q, we have∫
Γ1

q(uηmt ) uηmt dΓ =

∫
|uηm

t |≤1

q(uηmt ) uηmt dΓ +

∫
|uηm

t |>1

q(uηmt ) uηmt dΓ

≥
∫
|uηm

t (t)|>1

q(uηmt ) uηmt dΓ

≥ c3

∫
Γ1

|uηmt |ρ+2 dΓ− c3

∫
|uηm

t |≤1

|uηmt |ρ+2 dΓ

≥ c3∥uηmt ∥ρ+2
ρ+2,Γ1

− c3 meas(Γ1).

(3.3)

From ρ ≥ 2(n−2)γ−2
n−(n−2)γ , we have the imbedding H1

0 (Ω) ↪→ L
2(n−1)
n−2 (Γ1) ↪→ L(γ+1) ρ+2

ρ+1 (Γ1), so that

by Young’s inequality with ρ+1
ρ+2 + 1

ρ+2 = 1 we deduce that∫
Γ1

|uηm|γuηmuηmt dΓ ≤ C(ϵ0)∥uηm∥γ+1

(γ+1) ρ+2
ρ+1 ,Γ1

+ ϵ0∥uηmt ∥ρ+2
ρ+2,Γ1

≤ C(ϵ0)∥ |∇gu
ηm|g∥γ+1

2 + ϵ0∥uηmt ∥ρ+2
ρ+2,Γ1

≤ C(ϵ)
(
1 + ∥ |∇gu

ηm|g∥2
)2

+ ϵ0∥uηmt ∥ρ+2
ρ+2,Γ1

≤ C(ϵ)
(
1 + ∥ |∇gu

ηm|g∥22
)
+ ϵ0∥uηmt ∥ρ+2

ρ+2,Γ1
.

(3.4)

Under the assumption that h is globally Lipschitz from H1
0 (Ω) into L

2(Γ1) we have

∥h(uηm)∥2,Γ1
≤ ∥h(uηm)−h(0)∥2,Γ1

+ ∥h(0)∥2,Γ1
≤ Lh∥∇uηm∥2+ ∥h(0)∥2,Γ1

≤ C4(∥∇uηm∥2+1),

where Lh is the Lipschitz constant and C4 is for some positive constant, so that by Hölder’s and
Young’s inequalities and from the fact (2.1) and the imbedding Lρ+2(Γ1) ↪→ L2(Γ1), we deduce



8 T. G. HA EJDE-2025/104

that ∣∣∫
Γ1

h(uηm)uηmt dΓ
∣∣ ≤ C(ϵ)∥h(uηm)∥22,Γ1

+ ϵ∥uηmt ∥22,Γ1

≤ C(ϵ)
( 1

c1
∥ |∇gu

ηm|g∥22 + 1
)
+ ϵ∥uηmt ∥22,Γ1

,

≤ C(ϵ)
( 1

c1
∥ |∇gu

ηm|g∥22 + 1
)
+ ϵC2

ρ+2,2∥u
ηm
t ∥2ρ+2,Γ1

≤ C(ϵ)
( 1

c1
∥ |∇gu

ηm|g∥22 + 1
)
+ ϵ2ρ+1C2

ρ+2,2

(
1 + ∥uηmt ∥ρ+2

ρ+2,Γ1

)
,

(3.5)

where Cρ+2,2 is an imbedding constant. Replacing (3.3), (3.4) and (3.5) in (3.2) we obtain

d

dt

[1
2
∥uηmt ∥22 +

1

2
µ(t)∥ |∇gu

ηm|g∥22 +
1

γ + 2
∥uηm∥γ+2

γ+2,Γ1

]
+ η∥uηmt ∥22,Γ1

+
(
c3 − ϵ0 − ϵ2ρ+1C2

ρ+2,2

)
∥uηmt ∥ρ+2

ρ+2,Γ1

≤ c3 meas(Γ1) + C(ϵ0) + C(ϵ) + ϵ2ρ+1C2
ρ+2,2

+
(1
2
µ′(t) + C(ϵ0) +

C(ϵ)

c1

)
∥ |∇gu

ηm|g∥22 +
1

2
∥f(t)∥22 +

1

2
∥uηmt ∥22.

(3.6)

By integrating (3.6) over (0, t) with t ∈ (0, tm) we have

1

2
∥uηmt ∥22 +

1

2
µ(t)∥ |∇gu

ηm|g∥22 +
1

γ + 2
∥uηm∥γ+2

γ+2,Γ1
+ η

∫ t

0

∥uηms ∥22,Γ1
ds

+
(
c3 −−ϵ0 − ϵ2ρ+1C2

ρ+2,2

)∫ t

0

∥uηms ∥ρ+2
ρ+2,Γ1

ds

≤
(
c3 meas(Γ1) + C(ϵ0) + C(ϵ) + ϵ2ρ+1C2

ρ+2,2

)
T

+
1

2
∥u1∥22 +

1

2
µ(0)∥ |∇gu0|g∥22 +

1

γ + 2
∥u0∥γ+2

γ+2,Γ1

+
(1
2
∥µ′∥L∞(0,T ) + C(ϵ0) +

C(ϵ)

c1

)∫ t

0

∥ |∇gu
ηm(s)|g∥22 ds

+
1

2

∫ t

0

∥f(s)∥22 ds+
1

2

∫ t

0

∥uηms (s)∥22 ds.

(3.7)

Therefore, choosing ϵ0 = c3/4 and ϵ = c3
2ρ+3C2

ρ+2,2
and then by Gronwall’s lemma we obtain

∥uηmt ∥22 + ∥ |∇gu
ηm|g∥22 + ∥uηm∥γ+2

γ+2,Γ1
+

∫ t

0

∥uηms (s)∥22,Γ1
ds+

∫ t

0

∥uηms (s)∥ρ+2
ρ+2,Γ1

ds ≤ C5, (3.8)

where C5 is a positive constant which is independent of m, η, and t. Estimate (3.8) implies that

uηm is uniformly bounded in L∞(0, T ;H), (3.9)

uηmt is uniformly bounded in L∞(0, T ;L2(Ω)). (3.10)

We note that from (3.8), taking the hypotheses on q into account we also obtain∫ t

0

∫
Γ1

|q(uηms (s))|2 dΓ ds ≤ C6, (3.11)

where C6 is a positive constant independent of m, η and t.
From (3.8)-(3.11), there exists a subsequence of {uηm}, which we still denote by {uηm}, such

that

uηm → uη weak star in L∞(0, T ;H),

uηmt → uηt weakly star in L∞(0, T ;L2(Ω)),

uηm → uη weakly in L2(0, T ;H),



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 9

uηmt → uηt weakly in L2(0, T ;L2(Ω)),

uηmt → uηt weakly in L2(0, T ;L2(Γ1)),

uηmtt → uηtt weakly in L2(0, T ;H−1(Ω)),

q(uηmt ) → ψ weakly in L2(0, T ;L2(Γ1)).

Since H1/2(Γ) ↪→ L2(Γ) is compact, we have, thanks to Aubin-Lions Theorem that

uηm → uη strongly in L2(0, T ;L2(Γ1)),

and consequently, by using the Lions lemma, we deduce

h(uηm) → h(uη) weakly in L2(0, T ;L2(Γ1)).

The above convergences permit us to pass to the limit in the (3.1). Since {wj} is a basis of
H1

0 (Ω) and Vm is dense in H1
0 (Ω), after passing to the limit we obtain∫ T

0

∫
Ω

uηtt, vdxθ(t) dt+

∫ T

0

µ(t)

∫
Ω

⟨∇gu
η,∇gv⟩gdxθ(t) dt+ η

∫ T

0

∫
Γ1

uηt v dΓθ(t) dt

+

∫ T

0

∫
Γ1

ψv dΓθ(t) dt−
∫ T

0

∫
Γ1

h(uη)vdxθ(t) dt

=

∫ T

0

∫
Ω

f(x, t)vdxθ(t) dt,

(3.12)

for all θ ∈ D(0, T ) and v ∈ H1
0 (Ω).

Since estimates (3.8) and (3.11) are also independent of η, we can pass to the limit as η → 0 in
uη obtaining a function u by the same argument used to obtain uη from uηm, such that

uη → u weakly in L2(0, T ;H), (3.13)

uηt → ut weakly in L2(0, T ;L2(Ω)), (3.14)

uηt → ut weakly in L2(0, T ;L2(Γ1)), (3.15)

uηtt → utt weakly in L2(0, T ;H−1(Ω)), (3.16)

q(uηt ) → ψ weakly in L2(0, T ;L2(Γ1)), (3.17)

h(uη) → h(u) weakly in L2(0, T ;L2(Γ1)). (3.18)

By the above convergences in (3.12), we have∫ T

0

∫
Ω

utt, vdxθ(t) dt+

∫ T

0

µ(t)

∫
Ω

⟨∇gu,∇gv⟩gdxθ(t) dt

+

∫ T

0

∫
Γ1

ψv dΓθ(t) dt−
∫ T

0

∫
Γ1

h(u)vdxθ(t) dt

=

∫ T

0

∫
Ω

f(x, t)vdxθ(t) dt.

(3.19)

From (3.19) and taking v ∈ D(Ω), we conclude that

utt − µ(t)Lu = f in D′(Ω× (0, T )) (3.20)

and since (3.17) and (3.18), it holds that

µ(t)
∂u

∂ν
+ ψ = h(u) in L2(0, T ;L2(Γ1)).

Our goal is to show that ψ = q(ut). Indeed, considering w = uηm in (3.1) and then integrating
over (0, T ), we have∫ T

0

⟨uηmtt , uηm⟩ dt+
∫ T

0

µ(t)∥ |∇gu
ηm|g∥22 dt+ η

∫ T

0

⟨uηmt , uηm⟩Γ1
dt



10 T. G. HA EJDE-2025/104

+

∫ T

0

⟨q(uηmt ), uηm⟩Γ1 dt−
∫ T

0

⟨h(uηm), uηm⟩Γ1 dt

=

∫ T

0

⟨f, uηm⟩ dt.

Then from convergences (3.13)-(3.18) we obtain

lim
m→∞,η→0

∫ T

0

µ(t)∥ |∇gu
ηm|g∥22 dt

= −
∫ T

0

⟨utt, u⟩ dt−
∫ T

0

⟨ψ, u⟩Γ1
dt+

∫ T

0

⟨h(u), u⟩Γ1
dt+

∫ T

0

⟨f, u⟩ dt.
(3.21)

By combining (3.20) and (3.21), we have

lim
m→∞,η→0

∫ T

0

µ(t)∥ |∇gu
ηm|g∥22 dt =

∫ T

0

µ(t)∥ |∇gu|g∥22 dt,

which implies that

|∇gu
ηm|g → |∇gu|g strongly in L2(0, T ;L2(Ω)). (3.22)

Next, considering w = uηmt in (3.1) and then integrating over (0, T ), we have∫ T

0

⟨uηmtt , u
ηm
t ⟩ dt+

∫ T

0

µ(t)

∫
Ω

⟨∇gu
ηm,∇gu

ηm
t ⟩g dx dt+ η

∫ T

0

∥uηmt ∥22,Γ1
dt

+

∫ T

0

⟨q(uηmt ), uηmt ⟩Γ1
dt−

∫ T

0

⟨h(uηm), uηmt ⟩Γ1
dt

=

∫ T

0

⟨f, uηmt ⟩ dt.

From (3.14)-(3.18) and (3.22), we arrive at

lim
m→∞,η→0

∫ T

0

⟨q(uηmt ), uηmt ⟩Γ1
dt =

∫ T

0

⟨ψ, ut⟩Γ1
dt. (3.23)

On the other hand, since q is a nondecreasing monotone function, we obtain∫ T

0

⟨q(uηmt )− q(φ), uηmt − φ⟩Γ1 dt ≥ 0

for all φ ∈ L2(Γ1). Thus, it implies that∫ T

0

⟨q(uηmt ), φ⟩Γ1
dt+

∫ T

0

⟨q(φ), uηmt − φ⟩Γ1
dt ≤

∫ T

0

⟨q(uηmt ), uηmt ⟩Γ1
dt.

By considering (3.15), (3.17) and (3.23), we obtain∫ T

0

⟨ψ − q(φ), ut − φ⟩Γ1
dt ≥ 0, (3.24)

which implies that ψ = q(ut).
We now show the uniqueness of the solution. Let u1 and u2 be two solutions of problem (1.1).

Then z = u1 − u2 satisfies∫
Ω

zttwdx+ µ(t)

∫
Ω

⟨∇gz,∇gw⟩gdx+

∫
Γ1

(q(u1t )− q(u2t ))w dΓ =

∫
Γ1

(h(u1)− h(u2)w dΓ dt,

for all w ∈ H. By replacing w = zt in above identity and observing that q is monotonically
nondecreasing and h : H1

0 (Ω) → L2(Γ1) is globally Lipschitz, it holds that

d

dt

[1
2
∥zt∥22 +

1

2
µ(t)∥ |∇gz|g∥22

]
≤ C7∥ |∇gz|g∥22,

Where C7 is for some positive constant. By integrating from 0 to t and using Gronwall’s Lemma,
we conclude that ∥zt∥2 = ∥ |∇gz|g∥2 = 0.



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 11

3.2. Locally Lipschitz source. In this subsection, we loosen the globally Lipschitz condition on
the source by allowing h to be locally Lipschitz continuous. More precisely, we have the following
result.

Proposition 3.2. Assume that (H!)-(H4) hold. In addition, assume that (u0, u1) ∈ H × L2(Ω)
and h : H1

0 (Ω) → L2(Γ1) is locally Lipschitz continuous satisfying c5|s|γ+1 ≤ |h(s)| ≤ c6|s|γ+1,
where c5, c6 are for some positive constants. Then problem (1.1) has a unique local solution
u ∈ Cw(0, T ;H) ∩ C1

w(0, T ;L
2(Ω)) for some T > 0.

Proof. We define

hK(u) =

{
h(u) if ∥ |∇gu|g∥2 ≤ K,

h
(

Ku
∥ |∇gu|g∥2

)
if ∥ |∇gu|g∥2 > K,

where K is a positive constant. With this truncated hK , we consider the problem

utt − µ(t)Lu = f(x, t) in Ω× (0,+∞),

u = 0 on Γ0 × (0,+∞),

µ(t)
∂u

∂νL
+ q(ut) = hK(u) on Γ1 × (0,+∞),

u(x, 0) = u0(x), ut(x, 0) = u1(x).

(3.25)

Since hK : H1
0 (Ω) → L2(Γ1) is globally Lipschitz continuous for each K (see [10]), then by

Proposition 3.1, the truncated problem (3.25) has a unique global solution uK ∈ Cw(0, T ;H) ∩
C1

w(0, T ;L
2(Ω)) for any T > 0. Moreover by [30] there exists a sequence of functions ulK , which

converges to uK in the class Cw(0, T ;H
2(Ω)) ∩ C1

w(0, T ;H
1
0 (Ω)). For simplifying the notation in

the rest of the proof, we shall express ulK as u.
By the regularity of u, we can multiply (3.25) by ut and integrate on Ω×(0, t), where 0 < t < T .

Then we obtain by using the fact µ′(s) < 0 for all s > 0,

1

2
(∥ut∥22 + µ(t)∥ |∇gu|g∥22) +

1

γ + 2
∥u∥γ+2

γ+2,Γ1
+

∫ t

0

∫
Γ1

q(us(x, s))us(x, s) dΓ ds

≤ 1

2
(∥u1∥22 + µ(0)∥ |∇gu0|g∥22) +

1

γ + 2
∥u0∥γ+2

γ+2,Γ1
+

∫ t

0

∫
Ω

f(x, s)us(x, s)dx ds

+
(
1 +

1

c5

)∫ t

0

∫
Γ1

|hK(u(x, s))| |us(x, s)| dΓ ds.

(3.26)

We note that hK : H1
0 (Ω) → L

ρ+2
ρ+1 (Γ1) is globally Lipschitz with Lipschitz constant Lh(K) (see

[10, 13]). Hence we estimate the last term on the right-hand side of (3.26) as(
1 +

1

c5

)∫ t

0

∫
Γ1

|hK(u(x, s))| |us(x, s)| dΓ ds

≤
(
1 +

1

c5

)∫ t

0

∥hK(u(s))∥ ρ+2
ρ+1 ,Γ1

∥us∥ρ+2,Γ1
ds

≤ ϵ1

∫ t

0

∥us(s)∥ρ+2
ρ+2,Γ1

ds+ C(ϵ1)

∫ t

0

∥hK(u(s))∥
ρ+2
ρ+1
ρ+2
ρ+1 ,Γ1

ds

≤ ϵ1

∫ t

0

∥us(s)∥ρ+2
ρ+2,Γ1

ds+ C(ϵ1)
( 2
√
c1
Lh(K)

) ρ+2
ρ+1

∫ t

0

∥ |∇gu|g∥22 ds

+ tC(ϵ1)
(( 2

√
c1
Lh(K)

) ρ+2
ρ+1

+ 2−(ρ+1)|h(0)|
ρ+2
ρ+1 meas(Γ1)

)
.

(3.27)

From the hypothesis on q, we have∫ t

0

∫
Γ1

q(us(x, s))us(x, s) dΓ ds ≥ c3

∫ t

0

∥us(s)∥ρ+2
ρ+2,Γ1

ds− tc3 meas(Γ1). (3.28)



12 T. G. HA EJDE-2025/104

By replacing (3.27) and (3.28) in (3.26) and choosing ϵ1 ≤ c3, we obtain

∥ut∥22 + ∥ |∇gu|g∥22 + ∥u∥γ+2
γ+2,Γ1

+

∫ t

0

∥us(s)∥ρ+2
ρ+2,Γ1

ds

≤ C9 + C1(Lh(K))T + C2(Lh(K))

∫ t

0

∥us(s)∥22 + ∥ |∇gu(s)|g∥22 ds
(3.29)

for all t ∈ [0, T ], where

C9 =
1

2C8
(∥u1∥22 + µ(0)∥ |∇gu0|g∥22 + ∥u0∥γ+2

γ+2,Γ1
+ ∥f∥L2(0,T ;L2(Ω))),

C1(Lh(K)) =
1

C8
C(ϵ1)

(( 2
√
c1
Lh(K)

) ρ+2
ρ+1

+ 2−(ρ+1)|h(0)|
ρ+2
ρ+1 meas(Γ1)

)
+

1

C8
c3 meas(Γ1),

C2(Lh(K)) =
1

C8

(
C(ϵ1)

( 2
√
c1
Lh(K)

) ρ+2
ρ+1

+
1

2

)
,

for C8 = min{ 1
γ+2 ,

µ0

2 , c3 − ϵ1}. Thus by Gronwall’s inequality, (3.29) becomes

∥ut∥22 + ∥ |∇gu|g∥22 + ∥u∥γ+2
γ+2,Γ1

+

∫ t

0

∥us(s)∥ρ+2
ρ+2,Γ1

ds ≤ (C9 + C1(Lh(K))T )eC2(Lh(K))t,

for all t ∈ [0, T ]. If we choose

T = min
{ 1

C1(Lh(K))
,

1

C2(Lh(K))
ln 2

}
, (3.30)

then

∥ut∥22+∥ |∇gu|g∥22+∥u∥γ+2
γ+2,Γ1

+

∫ t

0

∥us(s)∥ρ+2
ρ+2,Γ1

ds ≤ 2(C9+1) ≤ K2 for all t ∈ [0, T ], (3.31)

provided we choose K2 ≥ 2(C9 + 1). Consequently, (3.31) gives us that ∥ |∇gu|g∥2 ≤ K for all
t ∈ [0, T ]. Therefore, by the definition of hK , we have that hK(u) = h(u) on [0, T ]. By the
uniqueness of solutions, the solution of the truncated problem (3.25) accords with the solution of
the original, non-truncated problem (1.1) for t ∈ [0, T ], which means that the proof is complete. □

3.3. Completion of the proof for the local existence. To establish the existence of solutions,
we need to extend the result in Proposition 3.2 where the source h is locally Lipschitz from H1

0 (Ω)

into L
ρ+2
ρ+1 (Γ1). For the construction of the Lipschitz approximation for the source, we employ

another truncated function introduced in [40]. Let δn ∈ C∞
0 (R) be a cut off function such that

0 ≤ δn ≤ 1, δn(s) =

{
1, if |s| ≤ n,

0, if |s| ≥ 2n,

and |δ′n(s)| ≤ C
n for some constant C independent from n. We also define

hn(u) = h(u)δn(u). (3.32)

Then the truncated function hn satisfies the following lemma. The proof is a routine series of
estimates as in [1, 13], so we omit it.

Lemma 3.3. The following statements hold.

(1) hn : H1
0 (Ω) → L2(Γ1) is globally Lipschitz continuous.

(2) hn : H1−ϵ
0 (Ω) → L

ρ+2
ρ+1 (Γ1) is locally Lipschitz continuous with Lipschitz constant indepen-

dent of n.

With the truncated source hn defined in (3.32), by Proposition 3.2 and Lemma 3.3, we have a
unique local solution un ∈ Cw(0, T ;H) ∩ C1

w(0, T ;L
2(Ω)) satisfying the following approximation



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 13

of (1.1)

utt − µ(t)Lu = f(x, t) in Ω× (0,+∞),

u = 0 on Γ0 × (0,+∞),

µ(t)
∂u

∂νL
+ q(ut) = hn(u) on Γ1 × (0,+∞),

u(x, 0) = u0(x), ut(x, 0) = u1(x).

(3.33)

From Lemma 3.3, the life span T of each solution un, given in (3.30), is independent of n since

the local Lipschitz constant of the mapping hn : H1
0 (Ω) → L

ρ+2
ρ+1 (Γ1) is independent of n. Also

we known that T depends on K, where K2 ≥ 2(C9 + 1), however, since ∥un1∥22 + ∥ |∇gu
n
0 |g∥22 +

∥un0∥
γ+2
γ+2,Γ1

→ ∥u1∥22 + ∥ |∇gu0|g∥22 + ∥u0∥γ+2
γ+2,Γ1

, we can choose K sufficiently large so that K is

independent of n. By (3.31),

∥unt ∥22 + ∥ |∇gu
n|g∥22 + ∥un∥γ+2

γ+2,Γ1
≤ K2 (3.34)

for all t ∈ [0, T ]. Therefore, there exists a function u and a subsequence of {un}, which we still
denote by {un}, such that

un → u weak star in L∞(0, T ;H), (3.35)

unt → ut weak star in L∞(0, T ;L2(Ω)). (3.36)

By (3.34), (3.35) and (3.36), we infer that

∥ut∥22 + ∥ |∇gu|g∥22 + ∥u∥γ+2
γ+2,Γ1

≤ K2 (3.37)

for all t ∈ [0, T ]. Moreover, by Aubin-Lions Theorem, we have

un → u strongly in L∞(0, T ;H1−ϵ(Ω)), (3.38)

for 0 < ϵ < 1. Since un is a solution of (3.33), it holds that∫ T

0

∫
Ω

untt, ϕ dx dt+

∫ T

0

µ(t)

∫
Ω

⟨∇gu
n,∇gϕ⟩g dx dt+

∫ T

0

∫
Γ1

q(unt )ϕdΓ dt

=

∫ T

0

∫
Ω

f(x, t)ϕdx dt+

∫ T

0

∫
Γ1

hn(u
n)ϕdΓ dt,

(3.39)

for any ϕ ∈ Cw(0, T ;H) ∩ C1
w(0, T ;L

2(Ω)), ϕ ∈ Lρ+2(0, T ; Γ1).
Now we show that

lim
n→∞

∫ T

0

∫
Γ1

hn(u
n)ϕdΓ dt =

∫ T

0

∫
Γ1

h(u)ϕdΓ dt. (3.40)

Indeed, we have∣∣∣∫ T

0

∫
Γ1

(hn(u
n)− h(u))ϕdΓ dt

∣∣∣
≤

∫ T

0

∫
Γ1

|hn(un)− hn(u)| |ϕ| dΓ dt+
∫ T

0

∫
Γ1

|hn(u)− h(u)| |ϕ| dΓ dt.
(3.41)

By (2) in Lemma 3.3 and (3.38), we obtain∫ T

0

∫
Γ1

|hn(un)− hn(u)| |ϕ| dΓ dt

≤
(∫ T

0

∫
Γ1

|hn(un)− hn(u)|
ρ+2
ρ+1 dΓ dt

) ρ+1
ρ+2

(∫ T

0

∫
Γ1

|ϕ|ρ+2 dΓ dt
) 1

ρ+2

≤ C(K)∥ϕ∥Lρ+2(0,T ;Γ1)

(∫ T

0

∥un − u∥
ρ+2
ρ+1

H1−ϵ(Ω) dt
) ρ+1

ρ+2 → 0.

(3.42)



14 T. G. HA EJDE-2025/104

Since δn(u(x)) → 1 a.e. in Ω, we have hn(u) → h(u) a.e. Then we also have |hn(u)−h(u)|
ρ+2
ρ+1 ≤

2
ρ+2
ρ+1 |h(u)|

ρ+2
ρ+1 and h(u) ∈ L

ρ+2
ρ+1 (Γ1), for u ∈ H1

0 (Ω). Thus by the Lebesgue Dominated Conver-
gence Theorem, we have∫ T

0

∫
Γ1

|hn(u)− h(u)| |ϕ| dΓ dt

≤
(∫ T

0

∫
Γ1

|hn(u)− h(u)|
ρ+2
ρ+1 dΓ dt

) ρ+1
ρ+2

(∫ T

0

∫
Γ1

|ϕ|ρ+2 dΓ dt
) 1

ρ+2

≤ ∥ϕ∥Lρ+2(0,T ;Γ1)

(∫ T

0

∫
Γ1

|h(u)|
ρ+2
ρ+1 |δn(u)− 1|

ρ+2
ρ+1 dΓ dt

) ρ+1
ρ+2 → 0.

(3.43)

Convergences (3.42) and (3.43), (3.41) gives us (3.40).
On the other hand, by using similar arguments from (3.21) to (3.24), we obtain

q(unt ) → q(ut) weakly in L2(0, T ;L2(Γ1)). (3.44)

Convergences (3.36), (3.37), (3.40) and (3.44) permit us to pass to the limit in (3.39) and
conclude the following result.

Proposition 3.4. Assume that (H!)-(H4) hold. In addition, assume that (u0, u1) ∈ H × L2(Ω)

and h : H1
0 (Ω) → L

ρ+2
ρ+1 (Γ1) is locally Lipschitz continuous. Then problem (1.1) has a local solution

u ∈ Cw(0, T ;H) ∩ C1
w(0, T ;L

2(Ω)) for some T > 0.

Let h(u) = |u|γu, then h : H1
0 (Ω) → L

ρ+2
ρ+1 (Γ1) is locally Lipschitz continuous (see Remark 2.2).

Thus by Proposition 3.4, the proof of the local existence statement in Theorem 2.4 is complete.

3.4. Energy identity. It is well known that to prove the uniqueness of weak solutions, we will
justify the energy identity (2.17). The energy identity can be derived formally by multiplying
(1.1) by ut. But, such a calculation is not justified, since ut is not sufficiently regular to be the
test function in as required in Definition ??. To overcome this problem, we employ the operator
T ϵ = (I− ϵL)−1 to smooth function in space, which is mentioned in Appendix A of [13]. We recall
important properties of T ϵ which play an essential role when establishing the energy identity.

Lemma 3.5 ([13]). Let uϵ = T ϵu. Then following statements hold.

(1) If u ∈ L2(Ω), then ∥uϵ∥2 ≤ ∥u∥2 and uϵ → u in L2(Ω) as ϵ→ 0.
(2) If u ∈ H1

0 (Ω), then ∥∇uϵ∥2 ≤ ∥∇u∥2 and uϵ → u in H1
0 (Ω) as ϵ→ 0.

(3) If u ∈ Lp(Γ1) with 1 < p <∞, then ∥uϵ∥p,Γ1
≤ ∥u∥p,Γ1

and uϵ → u in Lp(Γ1) as ϵ→ 0.

We will now justify the energy identity (2.17). We aapply the operator T ϵ on every term of
(1.1) and multiply by uϵt. Then we obtain by integrating in space and time∫ t

0

∫
Ω

uϵssu
ϵ
sdx ds+

∫ t

0

µ(s)

∫
Ω

⟨∇gu
ϵ,∇gu

ϵ
s⟩gdx ds+

∫ t

0

∫
Γ1

T ϵ(q(us))u
ϵ
s dΓ ds

=

∫ t

0

∫
Ω

f(x, s)uϵsdx ds+

∫ t

0

∫
Γ1

T ϵ(h(u))uϵs dΓ ds.

(3.45)

Since u ∈ H1
0 (Ω) and ut ∈ L2(Ω), we have by Lemma 3.5, uϵ → u in H1

0 (Ω) and u
ϵ
t → ut in L

2(Ω).
Therefore using this convergences, we have

lim
ϵ→0

(∫ t

0

∫
Ω

uϵssu
ϵ
sdx ds+

∫ t

0

µ(s)

∫
Ω

⟨∇gu
ϵ,∇gu

ϵ
s⟩gdx ds

)
=

1

2

(
∥ut∥22 + ∥ |∇gu|g∥22 − ∥u1∥22 − ∥ |∇gu0|g∥22

)
.

(3.46)

Since ut, q(ut) ∈ L2(Γ1), we easily check that

lim
ϵ→0

∫ t

0

∫
Γ1

T ϵ(q(us))u
ϵ
s dΓ ds =

∫ t

0

∫
Γ1

(q(us))us dΓ ds. (3.47)



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 15

Recall that ut ∈ Lρ+2(Γ1) and h(u) ∈ L
ρ+2
ρ+1 (Γ1). By Lemma 3.5, we have uϵt → ut in L

ρ+2(Γ1)

and T ϵ(h(u)) → h(u) in L
ρ+2
ρ+1 (Γ1). Thus by Lebesgue Dominated Convergence Theorem, we

obtain

lim
ϵ→0

∫ t

0

∫
Γ1

T ϵ(h(u))uϵs dΓ ds =

∫ t

0

∫
Γ1

(h(u))us dΓ ds. (3.48)

Convergences (3.46)-(3.48) permit us to pass to the limit in (3.45), consequently, the energy
identity (2.17) holds.

3.5. Strong time continuity. Our objective of this subsection is to prove the strong time con-
tinuity of the weak solution u, as stated in Propositions 3.1, 3.2, and 3.4.

Proposition 3.6. Assume that (H!)-(H4) hold. Let u be a local weak solution to problem (1.1).
Then the local weak solution u enjoys strong time continuity, i.e.,

u ∈ C(0, T ;H) ∩ C1(0, T ;L2(Ω)).

Proof. To establish strong continuity, i.e., u ∈ C(0, T ;H)∩C1(0, T ;L2(Ω)), it is sufficient to show
that the corresponding norm functions, t 7→ ∥u(t)∥H and t 7→ ∥ut(t)∥2, are continuous on [0, T ].

The energy identity (2.17) holds for any s, t ∈ [0, T ],

E(t)− E(s) =

∫ t

s

(1
2
µ′(τ)∥ |∇gu(τ)|g∥22dτ −

∫
Γ1

q(uτ )uτ dΓdτ +

∫
Ω

f(x, τ)uτdx
)
dτ.

From the a priori estimates derived during the proof of local existence, we have established that u ∈
L∞(0, T ;H), ut ∈ L∞(0, T ;L2(Ω)), and ut ∈ Lρ+2(0, T ; Γ1). These regularity results, combined
with the conditions on µ, f and q, ensure that the integrand on the right-hand side belongs
to L1(0, T ). A function defined by the integral of an L1 function is absolutely continuous, and
therefore continuous. Thus, we conclude that E(t) is a continuous function on [0, T ].

Let us define the functional

Ψ(t) := ∥ut(t)∥22 + µ(t)∥ |∇gu(t)|g∥22.

Using the definition of the energy E(t), we can write Ψ(t) as

Ψ(t) = 2E(t) +
2

γ + 2
∥u(t)∥γ+2

γ+2,Γ1
.

Since E(t) is continuous, the continuity of Ψ(t) depends on the continuity of the norm function
t 7→ ∥u(t)∥γ+2,Γ1

. To prove this, we use a standard compactness argument. We know that
u ∈ L∞(0, T ;H1

0 (Ω)) and ut ∈ L∞(0, T ;L2(Ω)). By the Aubin-Lions lemma,

u ∈ C(0, T ;L2(Ω)).

Similarly, by the compact Sobolev trace imbedding H1
0 (Ω) ↪→ Lq(Γ1) for q <

2(n−1)
n−2 , we have

u ∈ C(0, T ;Lq(Γ1))

for any such q. Therefore since u ∈ L∞(0, T ;Lγ+2(Γ1)), a standard interpolation argument shows
that the mapping t 7→ u(t) is also strongly continuous in the intermediate space Lγ+2(Γ1). Thus,
the norm function t 7→ ∥u(t)∥γ+2,Γ1

is continuous. Since Ψ(t) is a sum and product of continuous
functions, we conclude that Ψ(t) is continuous on [0, T ].

By Proposition 3.4, we have ut ∈ Cw(0, T ;L
2(Ω)) and u ∈ Cw(0, T ;H). On the other hand, the

property of weak convergence implies that norms are lower semi-continuous. For any s ∈ [0, T ]:

∥ut(s)∥22 ≤ lim inf
t→s

∥ut(t)∥22 and ∥ |∇gu(s)|g∥22 ≤ lim inf
t→s

∥ |∇gu(t)|g∥22.

This implies that

Ψ(s) = ∥ut(s)∥22 + µ(s)∥ |∇gu(s)|g∥22 ≤ lim inf
t→s

Ψ(t).

However, we proved that Ψ(t) is continuous, which means

lim
t→s

Ψ(t) = Ψ(s).



16 T. G. HA EJDE-2025/104

This requires lim supt→s Ψ(t) = Ψ(s) as well. The only way for the inequality from lower semi-
continuity and the equality from continuity to both hold is if the limits of the individual norm
terms also hold, i.e.,

lim
t→s

∥ut(t)∥22 = ∥ut(s)∥22 and lim
t→s

∥ |∇gu(t)|g∥22 = ∥ |∇gu(s)|g∥22.

The combination of weak continuity and norm continuity implies strong continuity. Therefore,

u ∈ C1(0, T ;L2(Ω)) and u ∈ C(0, T ;H1
0 (Ω).

Since u is strongly continuous in both H1
0 (Ω) and L

γ+2(Γ1), it is strongly continuous in the sum
space H. This completes the proof that u ∈ C(0, T ;H) ∩ C1(0, T ;L2(Ω)). □

4. Proof of Theorem 2.4: global existence

In this section we prove that a local weak solution u on [0, T ] can be extended to [0,∞). From
the standard continuation argument of ODE theory, it suffices to show that ∥ut∥22 + ∥ |∇gu|g∥22 +
∥u∥γ+2

γ+2,Γ1
is bounded independent of t. We now consider the following two cases:

4.1. ρ ≥ γ. Using the energy identity (2.17), we obtain

d

dt

[1
2
∥ut∥22 +

1

2
µ(t)∥ |∇gu|g∥22 +

1

γ + 2
∥u∥γ+2

γ+2,Γ1

]
= −

∫
Γ1

q(ut)ut dΓ +
1

2
µ′(t)∥ |∇gu|g∥22 + 2

∫
Γ1

|u|γuut dΓ +

∫
Ω

f(t)utdx.

(4.1)

By the same argument as (3.3), we have∫
Γ1

q(ut)ut dΓ ≥ c3∥ut∥ρ+2
ρ+2,Γ1

− c3 meas(Γ1). (4.2)

Using the Hölder and Young inequalities with γ+1
γ+2 + 1

γ+2 = 1 and the imbedding Lρ+2(Γ1) ↪→
Lγ+2(Γ1), we deduce that

2

∫
Γ1

|u|γuut dΓ ≤ C(ϵ2)∥u∥γ+2
γ+2,Γ1

+ ϵ2C
γ+2
ρ+2,γ+2∥ut∥

γ+2
ρ+2,Γ1

≤ C(ϵ2)∥u∥γ+2
γ+2,Γ1

+ ϵ22
ρ+1Cγ+2

ρ+2,γ+2

(
1 + ∥ut∥ρ+2

ρ+2,Γ1

)
,

(4.3)

where Cρ+2,γ+2 is an imbedding constant. By replacing (4.2) and (4.3) in (4.1) and using the
Young inequality and (2.8), we obtain

d

dt

[1
2
∥ut∥22 +

1

2
µ(t)∥ |∇gu|g∥22 +

1

γ + 2
∥u∥γ+2

γ+2,Γ1

]
≤ 1

2
∥ut∥22 + C(ϵ2)∥u∥γ+2

γ+2,Γ1
+ (c3 meas(Γ1) + ϵ22

ρ+1Cγ+2
ρ+2,γ+2 +

1

2
∥f∥22)

+ (ϵ22
ρ+1Cγ+2

ρ+2,γ+2 − c3)∥ut∥ρ+2
ρ+2,Γ1

.

(4.4)

Let

Ẽ(t) =
1

2
∥ut∥22 +

1

2
µ(t)∥ |∇gu|g∥22 +

1

γ + 2
∥u∥γ+2

γ+2,Γ1
.

Choosing ϵ2 = c3
2ρ+1Cγ+2

ρ+2,γ+2

, we rewrite (4.4) as

Ẽ′(t) ≤ C10 + C11Ẽ(t),

where C10 and C11 are positive constants. Now applying Gronwall’s inequality, we have that

Ẽ(t) ≤ (C12Ẽ(0)+C13)e
C12t, where C12 and C13 are positive constants. Consequently, since Ẽ(0)

is bounded we conclude that ∥ut∥22 + ∥ |∇gu|g∥22 + ∥u∥γ+2
γ+2,Γ1

is bounded.



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 17

4.2. Potential well. First of all, we will find a stable region. We set

0 < K0 := sup
u∈H,u ̸=0

( ∥u∥γ+2,Γ1

∥ |∇gu|g∥2

)
<∞

and the functional

J(u) =
µ0

2
∥ |∇gu|g∥22 −

1

γ + 2
∥u∥γ+2

γ+2,Γ1
, u ∈ H. (4.5)

We also define the function, for λ > 0,

j(λ) =
µ0

2
λ2 − 1

γ + 2
Kγ+2

0 λγ+2, (4.6)

then

λ0 =
( µ0

Kγ+2
0

)1/γ

is the absolute maximum point of j and

j(λ0) =
γµ0

2(γ + 2)
λ20 = d0.

The energy associated with problem (1.1) is

E(t) =
1

2
∥ut(t)∥22 +

1

2
µ(t)∥ |∇gu(t)|g∥22 −

1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1
, (4.7)

for u ∈ H. By (2.8) and (4.5)-(4.7), we deduce

E(t) ≥ J(u(t)) ≥ µ0

2
∥ |∇gu(t)|g∥22 −

Kγ+2
0

γ + 2
∥ |∇gu(t)|g∥γ+2

2 = j(∥ |∇gu(t)|g∥2). (4.8)

Lemma 4.1. Let u be a weak solution for problem (1.1). Suppose that E(0) < d0 and ∥ |∇gu0|g∥2 <
λ0. Then

∥ |∇gu(t)|g∥2 < λ0 for all t ≥ 0.

Proof. It is easy to verify that j is increasing for 0 < λ < λ0, decreasing for λ > λ0, j(λ) → −∞
as λ → +∞. Then since d0 > E(0) ≥ j(∥ |∇gu0|g∥2) ≥ j(0) = 0, there exist λ′0 < λ0 < λ̃0, which
satisfy

j(λ′0) = j(λ̃0) = E(0). (4.9)

Considering that E(t) is non-increasing, we have

E(t) ≤ E(0) for all t ≥ 0. (4.10)

From (4.8) and (4.9), we deduce that

j(∥ |∇gu0|g∥2) ≤ E(0) = j(λ′0). (4.11)

Since ∥ |∇gu0|g∥2 < λ0, λ
′
0 < λ0 and j is increasing in [0, λ0), from (4.11) it holds that

∥ |∇gu0|g∥2 ≤ λ′0. (4.12)

Next, we prove that

∥ |∇gu(t)|g∥2 ≤ λ′0 for all t ≥ 0. (4.13)

We argue by contradiction. Suppose that (4.13) does not hold. Then there exists time t∗ which
satisfies

∥ |∇gu(t
∗)|g∥2 > λ′0. (4.14)

If ∥ |∇gu(t
∗)|g∥2 < λ0, from (4.8), (4.9) and (4.14) we can write

E(t∗) ≥ j(∥ |∇gu(t
∗)|g∥2) > j(λ′0) = E(0),

which contradicts (4.10).
If ∥ |∇gu(t

∗)|g∥2 ≥ λ0, then, in view of (4.12), there exists λ̄0 which satisfies

∥ |∇gu0|g∥2 ≤ λ′0 < λ̄0 < λ0 ≤ ∥ |∇gu(t
∗)|g∥2. (4.15)



18 T. G. HA EJDE-2025/104

Consequently, from the continuity of the function t 7→ ∥ |∇gu(t)|g∥2 there exists t̄ ∈ (0, t∗) satis-
fying

∥ |∇gu(t̄)|g∥2 = λ̄0. (4.16)

Then from (4.8), (4.9), (4.15) and (4.16), we obtain

E(t̄) ≥ j(∥ |∇gu(t̄)|g∥2) = j(λ̄0) > j(λ′0) = E(0),

which also contradicts (4.10). This completes the proof. □

From (4.8) and Lemma 4.1, we arrive at

E(t) ≥ J(u(t)) > ∥ |∇gu(t)|g∥22
(µ0

2
− Kγ+2

0

γ + 2
λγ0

)
= µ0∥ |∇gu(t)|g∥22

(1
2
− 1

γ + 2

)
(4.17)

and, consequently,

J(t) ≥ 0 (J(t) = 0 iff u = 0) and ∥ |∇gu(t)|g∥22 ≤ 2(γ + 2)

µ0γ
E(t). (4.18)

By (4.17), we obtain

J(u(t)) >
µ0γ

2(γ + 2)
∥ |∇gu(t)|g∥22. (4.19)

Hence
1

2
∥ut(t)∥22 +

µ0γ

2(γ + 2)
∥ |∇gu(t)|g∥22 <

1

2
∥ut(t)∥22 + J(u(t)) ≤ E(t) ≤ E(0).

Therefore, there exists a positive constant C14 independent of t such that

∥ut(t)∥22 + ∥ |∇gu(t)|g∥22 ≤ C14E(0). (4.20)

Moreover, if we define the functional

I(u(t)) = µ0∥ |∇gu(t)|g∥22 − ∥u(t)∥γ+2
γ+2,Γ1

,

then from the relationship I(u(t)) = (γ + 2)J(u(t)) − µ0γ
2 ∥ |∇gu(t)|g∥22 and the strict inequality

(4.19), we obtain
I(u(t)) > 0 for all t ≥ 0. (4.21)

Consequently, from (4.20) and (4.21) we have

∥ut(t)∥22 + ∥ |∇gu(t)|g∥22 + ∥u(t)∥γ+2
γ+2,Γ1

≤ (1 + µ0)C14E(0).

This completes the proof of the global existence of solutions of (1.1).

5. Proof of Theorem 2.5: energy decay

In this section we prove the uniform decay rates for the solution of the problem

utt − µ(t)Lu = 0 in Ω× (0,+∞),

u = 0 on Γ0 × (0,+∞),

µ(t)
∂u

∂νL
+ q(ut) = |u|γu on Γ1 × (0,+∞),

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

(5.1)

We consider the additional hypothesis on H:

σ ≤ div(H) ≤ σ(γ + 4)

γ + 2
. (5.2)

Unless otherwise stated, the constant C is a generic positive constant, different in various occur-
rences. We define the energy associated with problem (5.1),

E(t) =
1

2
∥ut∥22 +

1

2
µ(t)∥ |∇gu|g∥22 −

1

γ + 2
∥u∥γ+2

γ+2,Γ1
.

Then

E′(t) =
1

2
µ′(t)∥ |∇gu|g∥22 −

∫
Γ1

q(ut)ut dΓ ≤ 0,



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 19

it follows that E(t) is a non-increasing function.
First, we recall technical lemmas which will play an essential role when establishing the asymp-

totic behavior.

Lemma 5.1 ([35]). Let E : R+ → R+ be a non-increasing function and ϕ : R+ → R+ a strictly
increasing function of class C1 such that

ϕ(0) = 0 and ϕ(t) → +∞ as t→ +∞.

Assume that there exists σ ≥ 0 and ω > 0 such that∫ +∞

S

E1+σ(t)ϕ′(t) dt ≤ 1

ω
Eσ(0)E(S)

for all S ≥ 0. Then E has the decay properties

• if σ = 0, then E(t) ≤ E(0)e1−ωϕ(t) for all t ≥ 0,

• if σ > 0, then E(t) ≤ E(0)
(

1+σ
1+ωσϕ(t)

) 1
σ

for al t ≥ 0.

Lemma 5.2 ([35]). Let E : R+ → R+ be a non-increasing function and ϕ : R+ → R+ a strictly
increasing function of class C1 such that

ϕ(0) = 0 and ϕ(t) → +∞ as t→ +∞.

Assume that there exist σ > 0, σ′ ≥ 0 and C > 0 such that∫ +∞

S

E1+σ(t)ϕ′(t) dt ≤ CE1+σ(S) +
C

(1 + ϕ(S))σ′E
σ(0)E(S), 0 ≤ S < +∞.

Then, there exists C > 0 such that

E(t) ≤ E(0)
C

(1 + ϕ(t))(1+σ′)/σ
, ∀t > 0.

Let us now multiply equation (5.1) by Ep(t)ϕ′(t)Mu, where

Mu = 2H(u) + (div(H)− σ)u,

p ≥ 0 and ϕ : R → R is a concave nondecreasing function of class C2, such that ϕ(t) → +∞ as
t→ +∞, and then integrate the obtained result over Ω× [S, T ]. Then we have

0 =

∫ T

S

Ep(t)ϕ′(t)

∫
Ω

Mu
(
utt − µ(t)Lu

)
dx dt

=

∫ T

S

Ep(t)ϕ′(t)

∫
Ω

uttMu dx dt−
∫ T

S

Ep(t)ϕ′(t)

∫
Ω

(div(H)− σ)uµ(t)Ludx dt

− 2

∫ T

S

Ep(t)ϕ′(t)

∫
Ω

H(u)µ(t)Ludx dt.

(5.3)

We note that∫ T

S

Ep(t)ϕ′(t)

∫
Ω

uttMu dx dt

=
[
Ep(t)ϕ′(t)

∫
Ω

utMudx
]T
S
−
∫ T

S

(pEp−1(t)E′(t)ϕ′(t) + Ep(t)ϕ′′(t))

∫
Ω

utMu dx dt

− 2

∫ T

S

Ep(t)ϕ′(t)

∫
Ω

utH(ut) dx dt−
∫ T

S

Ep(t)ϕ′(t)

∫
Ω

(div(H)− σ)|ut|2 dx dt

and

−
∫ T

S

Ep(t)ϕ′(t)

∫
Ω

(div(H)− σ)uµ(t)Ludx dt

= −
∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Γ1

(div(H)− σ)u
∂u

∂νL
dΓ dt



20 T. G. HA EJDE-2025/104

+

∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Ω

(div(H)− σ)|∇gu|2g dx dt .

Using Lemma 2.1 and that H(u) ∂u
∂νL

= |∇gu|2g on Γ0, we have

− 2

∫ T

S

Ep(t)ϕ′(t)

∫
Ω

H(u)µ(t)Ludx dt

= −
∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Γ1

(
2
∂u

∂νL
H(u)− |∇gu|2g(H · ν)

)
dΓ dt

−
∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Γ0

|∇gu|2g(H · ν) dΓ dt+ 2

∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Ω

DgH(∇gu,∇gu) dx dt

−
∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Ω

|∇gu|2g div(H) dx dt.

By placing above identities in (5.3), we obtain

σ

∫ T

S

Ep(t)ϕ′(t)

∫
Ω

|ut|2 dx dt+ 2

∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Ω

DgH(∇gu,∇gu) dx dt

− σ

∫ T

S

Ep(t)ϕ′(t)µ(t)

∫
Ω

|∇gu|2g dx dt− (div(H)− σ)

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|u|γ+2 dΓ dt

= −
[
Ep(t)ϕ′(t)

∫
Ω

utMudx
]T
S
+

∫ T

S

(pEp−1(t)E′(t)ϕ′(t) + Ep(t)ϕ′′(t))

∫
Ω

utMu dx dt

+ 2

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|u|γuH(u) dΓ dt

+

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

q(ut)Mu+
(
|ut|2 − µ(t)|∇gu|2g

)
(H · ν) dΓ dt

:= I1 + I2 + I3 + I4.

(5.4)

Now we estimate terms on the right-hand side of (5.4).

Estimate for I1 := −
[
Ep(t)ϕ′(t)

∫
Ω
utMudx

]T
S
. Using the Young inequality and the inequality∫

Ω

|u|2dx ≤ c∗Ω

∫
Ω

|∇gu|2gdx, c∗Ω > 0, ∀u ∈ H1
0 (Ω),

we obtain ∣∣∫
Ω

utMudx
∣∣ ≤ CE(t), (5.5)

consequently,

I1 ≤ −C
[
Ep(t)ϕ′(t)E(t)

]T
S
≤ CEp+1(S). (5.6)

Estimate o I2 :=
∫ T

S

(
pEp−1(t)E′(t)ϕ′(t) + Ep(t)ϕ′′(t)

) ∫
Ω
utMu dx dt. From (5.5), we have

|I2| ≤ C

∫ T

S

|pEp−1(t)E′(t)ϕ′(t) + Ep(t)ϕ′′(t)|E(t) dt

≤ CEp(S)

∫ T

S

−E′(t) dt+ CEp+1(S)

∫ T

S

−ϕ′′(t) dt

≤ CEp+1(S).

(5.7)

Estimate for I3 := 2
∫ T

S
Ep(t)ϕ′(t)

∫
Γ1

|u|γuH(u) dΓ dt. By the Young inequality with ρ+1
ρ+2 +

1
ρ+2 =

1 and from the fact k(u) = |u|γu is locally Lipschitz from H1
0 (Ω) into L

ρ+2
ρ+1 (Γ1), (2.1) and (4.18),



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 21

we obtain ∫
Γ1

|u|γuH(u) dΓ ≤ C(ϵ3)

∫
Γ1

|u|
(γ+1)(ρ+2)

ρ+1 dΓ + ϵ3

∫
Γ1

|H(u)|ρ+2 dΓ

≤ C(ϵ3)L
ρ+2
ρ+1
γ ∥∇u∥

ρ+2
ρ+1

2 + ϵ3 sup
x∈Ω

|H|ρ+2
g

∫
Γ1

|∇gu|ρ+2
g dΓ

≤ C(ϵ3)L
ρ+2
ρ+1
γ c−1

1 ∥ |∇gu|g∥22 + ϵ3 sup
x∈Ω

|H|ρ+2
g

∫
Γ1

|∇gu|ρ+2
g dΓ

≤ C(ϵ3)L
ρ+2
ρ+1
γ

2(γ + 2)

µ0γc1
E(t) + ϵ3 sup

x∈Ω

|H|ρ+2
g

∫
Γ1

|∇gu|ρ+2
g dΓ;

(5.8)

consequently,

I3 ≤ C(ϵ3)E
p+1(S) + ϵ3 sup

x∈Ω

|H|ρ+2
g

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|∇gu|ρ+2
g dΓ dt. (5.9)

Estimate for I4 :=
∫ T

S
Ep(t)ϕ′(t)

∫
Γ1
q(ut)Mu+

(
|ut|2−µ(t)|∇gu|2g

)
(H · ν) dΓ dt. From the Young

inequality with ρ+1
ρ+2 + 1

ρ+2 = 1, we have

2

∫
Γ1

q(ut)H(u) dΓ ≤ C(ϵ4)

∫
Γ1

|q(ut)|
ρ+2
ρ+1 dΓ + ϵ4

∫
Γ1

|H(u)|ρ+2 dΓ

≤ C(ϵ4)

∫
Γ1

|q(ut)|
ρ+2
ρ+1 dΓ + ϵ4 sup

x∈Ω

|H|ρ+2
g

∫
Γ1

|∇gu|ρ+2
g dΓ.

Arguments similar to those for (5.8) yield

(div(H)− σ)

∫
Γ1

q(ut)u dΓ ≤ C

∫
Γ1

|q(ut)|
α

α−1 dΓ + C

∫
Γ1

|u|α dΓ ≤ C

∫
Γ1

|q(ut)|
α

α−1 dΓ + CE(t),

where α = (γ+1)(ρ+2)
ρ+1 . Then we obtain

I4 ≤ CEp+1(S) + C(ϵ4)

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|q(ut)|
ρ+2
ρ+1 dΓ dt+ C

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|q(ut)|
α

α−1 dΓ dt

+ C

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|ut|2 dΓ dt+ ϵ4 sup
x∈Ω

|H|ρ+2
g

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|∇gu|ρ+2
g dΓ dt

− δµ0

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|∇gu|2g dΓ dt.

(5.10)
By replacing (5.6), (5.7), (5.9) and (5.10) in (5.4) and choosing ϵ3, ϵ4 small enough, from (5.2)

we obtain ∫ T

S

Ep+1(t)ϕ′(t) dt ≤ CEp+1(S) + C

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|ut|2 dΓ dt︸ ︷︷ ︸
:=I5

+ C

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|q(ut)|
ρ+2
ρ+1 dΓ dt︸ ︷︷ ︸

:=I6

+ C

∫ T

S

Ep(t)ϕ′(t)

∫
Γ1

|q(ut)|
α

α−1 dΓ dt︸ ︷︷ ︸
:=I7

.

(5.11)

Now we estimate the last three terms on the right-hand side of (5.11).



22 T. G. HA EJDE-2025/104

Case 1: β is linear. Since β is linear, we can rewrite the hypothesis on q as

c7|s| ≤ |q(s)| ≤ c8|s| if |s| ≤ 1,

c3|s| ≤ c3|s|ρ+1 ≤ |q(s)| ≤ c4|s|ρ+1 if |s| > 1,

for some positive constants c7, c8. Then we obtain∫
|ut|≤1

|ut|2 dΓ ≤ c−1
7

∫
|ut|≤1

utq(ut) dΓ ≤ −c−1
7 E′(t), (5.12)∫

|ut|>1

|ut|2 dΓ ≤ c−1
3

∫
|ut|>1

utq(ut) dΓ ≤ −c−1
3 E′(t), (5.13)∫

|ut|≤1

|q(ut)|
ρ+2
ρ+1 dΓ ≤

∫
|ut|≤1

|q(ut)|2 dΓ ≤ c8

∫
|ut|≤1

utq(ut) dΓ ≤ −c8E′(t), (5.14)∫
|ut|>1

|q(ut)|
ρ+2
ρ+1 dΓ =

∫
|ut|>1

|q(ut)|
1

ρ+1 |q(ut)| dΓ ≤ c
1

ρ+1

4

∫
|ut|>1

utq(ut) dΓ ≤ −c
1

ρ+1

4 E′(t), (5.15)∫
|ut|≤1

|q(ut)|
α

α−1 dΓ ≤ c8

∫
|ut|≤1

utq(ut) dΓ ≤ −c8E′(t) (5.16)

and, since ρ ≤ γ, it holds that ρ+1
α−1 ≤ 1. Consequently,∫

|ut|>1

|q(ut)|
α

α−1 dΓ =

∫
|ut|>1

|q(ut)|
1

α−1 |q(ut)| dΓ

≤ c
1

α−1

4

∫
|ut|>1

utq(ut) dΓ

≤ −c
1

α−1

4 E′(t).

(5.17)

From (5.12)-(5.17), we obtain

I5 + I6 + I7 ≤ CEp+1(S). (5.18)

Combining (5.11) and (5.18), it follows that∫ T

S

Ep+1(t)ϕ′(t) dt ≤ CEp(0)E(S),

which implies by Lemma 5.1 with p = 0

E(t) ≤ E(0)e1−
ϕ(t)
C .

Let us set ϕ(t) := mt, where m is for some positive constant, then ϕ(t) satisfies all the required
properties and we obtain that the energy decays exponentially to zero.

Case 2: β has polynomial growth near zero. Assume that β(s) = sρ+1. Let p = ρ
2 , then we

rewrite (5.11) as∫ T

S

E
ρ
2+1(t)ϕ′(t) dt

≤ CE(S) + C

∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
Γ1

|ut|2 dΓ dt+ C

∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
Γ1

|q(ut)|
ρ+2
ρ+1 dΓ dt

+ C

∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
Γ1

|q(ut)|
α

α−1 dΓ dt.

(5.19)

By the hypotheses on q and the Hölder inequality with 2
ρ+2 + ρ

ρ+2 = 1, we have∫
|ut|≤1

|ut|2 dΓ ≤
∫
|ut|≤1

(utq(ut))
2

ρ+2 dΓ ≤ C
(∫

|ut|≤1

utq(ut)dΓ
) 2

ρ+2 ≤ C
(
−E′(t)

) 2
ρ+2 ,∫

|ut|>1

|ut|2 dΓ ≤ c
2(ρ+2)
3

∫
|ut|>1

(utq(ut))
2

ρ+2 dΓ ≤ C
(
−E′(t)

) 2
ρ+2 .



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 23

Then ∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
Γ1

|ut|2 dΓ dt

=

∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
|ut|≤1

|ut|2 dΓ dt+
∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
|ut|>1

|ut|2 dΓ dt

≤ C

∫ T

S

E
ρ
2 (t)ϕ′(t)

(
−E′(t)

) 2
ρ+2 dt

≤ ϵ5

∫ T

S

E
ρ
2+1(t)ϕ′(t) dt+ C(ϵ5)E(S).

(5.20)

As for (5.14) and (5.15) we have∫
|ut|≤1

|q(ut)|
ρ+2
ρ+1 dΓ ≤

∫
|ut|≤1

|q(ut)|2 dΓ ≤
∫
|ut|≤1

(utq(ut))
2

ρ+2 dΓ ≤ C
(
−E′(t)

) 2
ρ+2 ,∫

|ut|>1

|q(ut)|
ρ+2
ρ+1 dΓ ≤ −c

1
ρ+1

4 E′(t).

Then ∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
Γ1

|q(ut)|
ρ+2
ρ+1 dΓ dt ≤ ϵ6

∫ T

S

E
ρ
2+1(t)ϕ′(t) dt+ C(ϵ6)E(S). (5.21)

As for (5.16) and (5.17) we have∫
|ut|≤1

|q(ut)|
α

α−1 dΓ ≤
∫
|ut|≤1

|q(ut)|2 dΓ ≤ C
(
−E′(t)

) 2
ρ+2 ,∫

|ut|>1

|q(ut)|
α

α−1 dΓ ≤ −c
1

α−1

4 E′(t).

Then ∫ T

S

E
ρ
2 (t)ϕ′(t)

∫
Γ1

|q(ut)|
α

α−1 dΓ dt ≤ ϵ7

∫ T

S

E
ρ
2+1(t)ϕ′(t) dt+ C(ϵ7)E(S). (5.22)

By replacing (5.20)-(5.22) in (5.19) and choosing ϵ5, ϵ6, ϵ7 sufficiently small, we obtain∫ T

S

E
ρ
2+1(t)ϕ′(t) dt ≤ CE(S),

which implies by Lemma 5.1 and choosing ϕ(t) = mt,

E(t) ≤ CE(0)

(1 + t)2/ρ
.

Case 3: β does not necessarily have polynomial growth near zero. We use the method of
partitions of boundary modified the arguments in [35]. For every t ≥ 1, we consider the following
partitions of boundary depending ϕ′(t):

Γ1
1 = {x ∈ Γ1 : |ut(t)| ≤ ϕ′(t)}, Γ2

1 = {x ∈ Γ1 : ϕ′(t) < |ut(t)| ≤ 1},
Γ3
1 = {x ∈ Γ1 : |ut(t)| > 1}

if ϕ′(t) ≤ 1; or

Γ4
1 = {x ∈ Γ1 : |ut(t)| ≤ 1 < ϕ′(t)}, Γ5

1 = {x ∈ Γ1 : 1 < |ut(t)| ≤ ϕ′(t)},
Γ6
1 = {x ∈ Γ1 : |ut(t)| > ϕ′(t) > 1}

if ϕ′(t) > 1. Then Γ1 = Γ1
1 ∪ Γ2

1 ∪ Γ3
1 (or Γ1 = Γ4

1 ∪ Γ5
1 ∪ Γ6

1). Let us estimate I5, I6 and I7 on
these partitions.

(i) Part on Γi
1, i = 3, 5, 6. By same arguments as (5.13), (5.15) and (5.17) we obtain∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|ut|2 dΓ dt ≤ CEp+1(S), (5.23)



24 T. G. HA EJDE-2025/104∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|q(ut)|
ρ+2
ρ+1 dΓ dt ≤ CEp+1(S), (5.24)∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|q(ut)|
α

α−1 dΓ dt ≤ CEp+1(S). (5.25)

(ii) Part on Γ2
1. Using that β is increasing, ϕ′ is non-increasing and (2.10), we obtain∫ T

S

Ep(t)ϕ′(t)

∫
Γ2
1

|ut|2 dΓ dt ≤
∫ T

S

Ep(t)

β(ϕ′(t))

∫
Γ2
1

utq(ut) dΓ dt

≤ 1

β(ϕ′(T ))

∫ T

S

Ep(t)
(
−E′(t)

)
dt

≤ 1

β(ϕ′(T ))
Ep+1(S),

(5.26)

∫ T

S

Ep(t)ϕ′(t)

∫
Γ2
1

|q(ut)|
ρ+2
ρ+1 dΓ dt ≤

∫ T

S

Ep(t)

∫
Γ2
1

|u′(t)| |q(ut)|2 dΓ dt

≤ β−1(1)

∫ T

S

Ep(t)

∫
Γ2
1

utq(ut) dΓ dt

≤ β−1(1)Ep+1(S)

(5.27)

and∫ T

S

Ep(t)ϕ′(t)

∫
Γ2
1

|q(ut)|
α

α−1 dΓ dt ≤
∫ T

S

Ep(t)

∫
Γ2
1

|u′(t)| |q(ut)|2 dΓ dt ≤ β−1(1)Ep+1(S). (5.28)

(iii) Part on Γi
1, i = 1, 4. Using that E(t) is non-increasing and β−1 is increasing, we have∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|ut|2 dΓ dt ≤
Ep(S)

(β−1(ϕ′(T )))2

∫ T

S

∫
Γi
1

ϕ′(t)(β−1(ϕ′(t)))2 dΓ dt

≤ meas(Γ1)

(β−1(ϕ′(T )))2
Ep(S)

∫ T

S

ϕ′(t)(β−1(ϕ′(t)))2 dt.

(5.29)

From (2.10), we obtain∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|q(ut)|
ρ+2
ρ+1 dΓ dt ≤

∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|q(ut)|2 dΓ dt

≤
∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

(β−1(|u′(t)|))2 dΓ dt

≤ meas(Γ1)E
p(S)

∫ T

S

ϕ′(t)(β−1(ϕ′(t)))2 dt

(5.30)

and ∫ T

S

Ep(t)ϕ′(t)

∫
Γi
1

|q(ut)|
α

α−1 dΓ dt ≤ meas(Γ1)E
p(S)

∫ T

S

ϕ′(t)(β−1(ϕ′(t)))2 dt. (5.31)

Therefore, from (5.23)-(5.31), we deduce that

I5 + I6 + I7 ≤ CEp+1(S) + CEp(S)

∫ T

S

ϕ′(t)(β−1(ϕ′(t)))2 dt. (5.32)

To estimate the last term of the right-hand side of (5.32), we need the following additional
assumption over ϕ (see [35, p.434])∫ ∞

1

ϕ′(t)(β−1(ϕ′(t)))2 dt converges.



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 25

Then by replacing (5.32) in (5.11) we obtain∫ T

S

Ep+1(t)ϕ′(t) dt ≤ CEp+1(S) + CEp(S)

∫ +∞

S

ϕ′(t)(β−1(ϕ′(t)))2 dt

≤ CEp+1(S) + CEp(S)

∫ +∞

ϕ(S)

(
β−1(

1

(ϕ−1)′(s)
)
)2

ds.

(5.33)

We define ψ(t) = 1+
∫ t

1
1

β( 1
s )
ds, t ≥ 1. Then ψ is strictly increasing and convex (cf. [35], [39]).

We now take ϕ(t) = ψ−1(t), then we can rewrite (5.33) as∫ T

S

Ep+1(t)ϕ′(t) dt ≤ CEp+1(S) +
C

ϕ(S)
Ep(S),

which implies, by applying Lemma 5.2 with p = 1,

E(t) ≤ C

ϕ2(t)
∀t > 0.

Let s0 be a number such that β( 1
s0
) ≤ 1. Since β is nondecreasing, we have

ψ(s) ≤ 1 + (s− 1)
1

β( 1s )
≤ 1

F ( 1s )
∀s ≥ s0,

where F (s) = sβ(s), consequently, having in mind that ϕ = ψ−1, the above inequality yields

s ≤ ϕ
( 1

F ( 1s )

)
= ϕ(t) with t =

1

F ( 1s )
.

Then we conclude that
1

ϕ(t)
≤ F−1(

1

t
).

Therefore the proof is complete.

6. Proof of Theorem 2.6: blow-up

First we introduce a following lemma that is essential role for proving the blow-up.

Lemma 6.1. Under the hypotheses given in Theorem 2.6 the weak solution to problem (1.1)
satisfies

∥ |∇gu(t)|g∥2 > λ0 for all 0 < t < Tmax.

Proof. We recall the function, for λ > 0,

j(λ) =
µ0

2
λ2 − 1

γ + 2
Kγ+2

0 λγ+2,

where K0 = supu∈H,u ̸=0

(
∥u∥γ+2,Γ1

∥ |∇gu|g∥2

)
. Then

λ0 =
( µ0

Kγ+2
0

)1/γ

is the absolute maximum point of j and

j(λ0) =
γµ0

2(γ + 2)
λ20 = d0.

The energy associated with problem (1.1) is

E(t) =
1

2
∥ut(t)∥22 +

1

2
µ(t)∥ |∇gu(t)|g∥22 −

1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1
.

From the definition of j, we have

E(t) ≥ µ0

2
∥ |∇gu(t)|g∥22 −

1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1
≥ j(∥ |∇gu(t)|g∥2) for all t ≥ 0. (6.1)

Note that j is increasing for 0 < λ < λ0, decreasing for λ > λ0, and j(λ) → −∞ as λ→ +∞.
Now we consider the initial energy E(0) divided into two cases: E(0) ≥ 0 and E(0) < 0.



26 T. G. HA EJDE-2025/104

Case 1: E(0) ≥ 0. There exist λ′1 < λ0 < λ1 such that

j(λ1) = j(λ′1) = E(0). (6.2)

By considering that E(t) is non-increasing, we have

E(t) ≤ E(0) for all t > 0. (6.3)

From (6.1) and (6.2) we deduce that

j(∥ |∇gu0|g∥2) ≤ E(0) = j(λ1). (6.4)

Since ∥ |∇gu0|g∥2 > λ0, λ0 < λ1, and j(λ) is decreasing for λ0 < λ, from (6.4) we obtain

∥ |∇gu0|g∥2 ≥ λ1. (6.5)

Now we prove that

∥ |∇gu(t)|g∥2 ≥ λ1 for all 0 < t < Tmax (6.6)

by using the contradiction method. Suppose that (6.6) does not hold. Then there exists t∗ in
(0, Tmax) that satisfies

∥ |∇gu(t
∗)|g∥2 < λ1. (6.7)

If ∥ |∇gu(t
∗)|g∥2 > λ0, from (6.1), (6.2), and (6.7), we can write

E(t∗) ≥ j(∥ |∇gu(t
∗)|g∥2) > j(λ1) = E(0),

which contradicts (6.3).
If ∥ |∇gu(t

∗)|g∥2 ≤ λ0, we have, in view of (6.5), that there exists λ̄ which satisfies

∥ |∇gu(t
∗)|g∥2 ≤ λ0 < λ̄ < λ1 ≤ ∥ |∇gu0|g∥2. (6.8)

Consequently, from the continuity of the function t 7→ ∥ |∇gu(t)|g∥2 there exists t̄ ∈ (0, t∗) satisfy-
ing ∥ |∇gu(t̄)|g∥2 = λ̄. Then from the last identity and taking (6.1), (6.2), and (6.8) into account
we deduce that

E(t̄) ≥ j(∥ |∇gu(t̄)|g∥2) = j(λ̄) > j(λ1) = E(0),

which also contradicts (6.3).

Case 2: E(0) < 0. There is a λ2 > λ0 such that j(λ2) = E(0); consequently, by (6.1) we have

j(∥ |∇gu0|g∥2) ≤ E(0) = j(λ2).

From the fact j(λ) is decreasing for λ0 < λ, we obtain

∥ |∇gu0|g∥2 ≥ λ2.

By the same argument as in Case 1, we obtain

∥ |∇gu(t)|g∥2 ≥ λ2 for all 0 < t < Tmax.

Thus the proof is complete. □

Now we prove the blow-up result. To prove that Tmax is finite, we argue by contradiction.
Assume that the weak solution u(t) can be extended to the whole interval [0,∞). Let E1 be a
real number such that

E1 =

{
0 if E(0) < 0,

a positive constant satisfying E(0) < E1 < d0 and E1 < E(0) + 1 if E(0) ≥ 0.

By setting G(t) := E1 − E(t), we have

G′(t) = −E′(t) ≥ 0, (6.9)

which implies that G(t) is nondecreasing, consequently,

0 < G0 := E1 − E(0) < 1 (6.10)



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 27

and from Lemma 6.1, (2.8), and the definition of d0, we have

G0 ≤ G(t) ≤ E1 −
µ0

2
∥ |∇gu(t)|g∥22 +

1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1

< d0 −
µ0

2
λ20 +

1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1

≤ 1

γ + 2
∥u(t)∥γ+2

γ+2,Γ1
.

(6.11)

We define

M(t) = G1−χ(t) + τN(t), N(t) =

∫
Ω

utudx, (6.12)

where χ and τ are small positive constants to be chosen later. Then we have

M ′(t) = (1− χ)G−χ(t)G′(t) + τN ′(t). (6.13)

Now we analyze the last term on the right-hand side of (6.13).

Lemma 6.2. It holds

N ′(t) ≥ C15

(
∥ut∥22 + ∥u∥γ+2

γ+2,Γ1
+G(t)−G′(t)Gχ−χ

0 G−χ(t)
)

+ µ0

(θ
2
− 1

)
∥ |∇gu|g∥22 − θE1 −

ζ

ϵ8
,

(6.14)

where C15 is a positive constant, 0 < χ < γ−ρ
(ρ+2)(γ+2) , θ = γ + 2− ϵ8 with 0 < ϵ8 < min{1, γ}, and

ζ =
(γ+1)meas(Γ1)

(
β−1(1)

) γ+2
γ+1

γ+2 .

Proof. Using (1.1), we obtain

N ′(t) = ∥ut∥22 − µ(t)∥ |∇gu|g∥22 + ∥u∥γ+2
γ+2,Γ1

−
∫
Γ1

q(ut)u dΓ

≥
(
1 +

θ

2

)
∥ut∥22 + µ0

(θ
2
− 1

)
∥ |∇gu|g∥22 +

(
1− θ

γ + 2

)
∥u∥γ+2

γ+2,Γ1

+ θG(t)− θE1 −
∫
Γ1

q(ut)u dΓ,

(6.15)

where θ = γ + 2− ϵ8 with 0 < ϵ8 < min{1, γ}.
Now we estimate the last term on the right-hand side of (6.15). We note that

∣∣∫
Γ1

q(ut)u dΓ
∣∣ ≤ ∫

Γ1

|q(ut)| |u| dΓ =

∫
|ut|≤1

|q(ut)| |u| dΓ +

∫
|ut|>1

|q(ut)| |u| dΓ.

By using (2.10) and the imbedding Lγ+2(Γ1) ↪→ Lρ+2(Γ1), we have∫
|ut|≤1

|q(ut)| |u| dΓ ≤
(∫

|ut|≤1

|q(ut)|
ρ+2
ρ+1 dΓ

) ρ+1
ρ+2

(∫
|ut|≤1

|u|ρ+2 dΓ
) 1

ρ+2

≤
(∫

|ut|≤1

|β−1(1)|
ρ+2
ρ+1 dΓ

) ρ+1
ρ+2 ∥u∥ρ+2,Γ1

≤ β−1(1)
(
meas(Γ1)

) γ+1
γ+2 ∥u∥|γ+2,Γ1

≤
(γ + 1)meas(Γ1)

(
β−1(1)

) γ+2
γ+1

ϵ8(γ + 2)
+

ϵγ+1
8

γ + 2
∥u∥γ+2

γ+2,Γ1
.

(6.16)



28 T. G. HA EJDE-2025/104

On the other hand, by using (2.11), we obtain∫
|ut|>1

|q(ut)| |u| dΓ ≤ c4

∫
|ut|>1

|ut|ρ+1|u| dΓ

≤ c4

(∫
|ut|>1

|ut|ρ+2 dΓ
) ρ+1

ρ+2 ∥u∥ρ+2,Γ1

≤
(
C(ϵ9)

∫
|ut|>1

|ut|ρ+2 dΓ + ϵ9∥u∥
(γ+2)(χ+ 1

γ+2 )(ρ+2)

γ+2,Γ1

)
∥u∥−(γ+2)χ

γ+2,Γ1
,

(6.17)

where 0 < χ < γ−ρ
(ρ+2)(γ+2) and C(ϵ9), ϵ9 are for positive constants. Moreover χ < γ−ρ

(ρ+2)(γ+2)

implies that (χ+ 1
γ+2 )(ρ+ 2) < 1. Hence we obtain

∥u∥(γ+2)(χ+ 1
γ+2 )(ρ+2)

γ+2,Γ1
≤

{
∥u∥γ+2

γ+2,Γ1
if ∥u∥γ+2

γ+2,Γ1
> 1,

G−1
0 G0 if ∥u∥γ+2

γ+2,Γ1
≤ 1.

From (6.10) and (6.11) we have

∥u∥(γ+2)(χ+ 1
γ+2 )(ρ+2)

γ+2,Γ1
≤ G−1

0 ∥u∥γ+2
γ+2,Γ1

and, consequently, from (2.11), (6.9), (6.11), and (6.17),∫
|ut|>1

|q(ut)| |u| dΓ ≤
(
C(ϵ9)

∫
|ut|>1

|ut|ρ+2 dΓ + ϵ9G
−1
0 ∥u∥γ+2

γ+2,Γ1

)
∥u∥−(γ+2)χ

γ+2,Γ1

≤
(
C(ϵ9)G

′(t) + ϵ9G
−1
0 ∥u∥γ+2

γ+2,Γ1

)
G−χ(t)

≤ C(ϵ9)G
′(t)Gχ−χ

0 G−χ(t) + ϵ9G
−(χ+1)
0 ∥u∥γ+2

γ+2,Γ1
,

(6.18)

for 0 < χ < χ. From (6.16) and (6.18), we obtain that∣∣∫
Γ1

q(ut)u dΓ
∣∣ ≤ C(ϵ9)G

′(t)Gχ−χ
0 G−χ(t) +

( ϵγ+1
8

γ + 2
+ ϵ9G

−(χ+1)
0

)
∥u∥γ+2

γ+2,Γ1
+
ζ

ϵ8
, (6.19)

where ζ =
(γ+1)meas(Γ1)

(
β−1(1)

) γ+2
γ+1

γ+2 .

By replacing (6.19) in (6.15) and choosing ϵ9 small enough we obtain

N ′(t) ≥ C16

(
∥ut∥22 + ∥u∥γ+2

γ+2,Γ1
+G(t)−G′(t)Gχ−χ

0 G−χ(t)
)
+ µ0

(θ
2
− 1

)
∥ |∇gu|g∥22 − θE1 −

ζ

ϵ8
,

where C16 is a positive constant. Therefore (6.14) follows. □

The following Lemma estimates the last three terms on the right-hand side of (6.14).

Lemma 6.3. It holds

µ0

(θ
2
− 1

)
∥ |∇gu|g∥22 − θE1 −

ζ

ϵ8
> 0 for

ℓ−
√
ℓ2 − 4ϱζ

2ϱ
≤ ϵ8 ≤ ℓ+

√
ℓ2 − 4ϱζ

2ϱ
, (6.20)

where ϱ =
µ0λ

2
0

2 − E1 and ℓ =
γµ0λ

2
0

2 − (γ + 2)E1.

Proof. From Lemma 5.1 and the definition of θ, we have

µ0

(θ
2
− 1

)
∥ |∇gu|g∥22 − θE1 −

ζ

ϵ8
> µ0

(θ
2
− 1

)
λ20 − θE1 −

ζ

ϵ8

=
(
E1 −

µ0λ
2
0

2

)
ϵ8 −

ζ

ϵ8
+
γµ0λ

2
0

2
− (γ + 2)E1

= −ϱϵ
2
8 − ℓϵ8 + ζ

ϵ8
:= P (ϵ8).

(6.21)

We note that

ϱ =
µ0λ

2
0

2
− E1 >

µ0λ
2
0

2
− d0 =

1

γ + 2
Kγ+2

0 λγ+2
0 > 0,



EJDE-2025/104 WAVE EQUATION WITH VARIABLE COEFFICIENTS AND SUPERCRITICAL SOURCE 29

ℓ =
γµ0λ

2
0

2
− (γ + 2)E1 >

γµ0λ
2
0

2
− (γ + 2)d0 = 0.

Since (2.19) holds, we have ℓ2−4ϱζ ≥ 0. Therefore, P (ϵ8) represents a curve connecting horizontal

axis points
ℓ−
√

ℓ2−4ϱζ

2ϱ and
ℓ+
√

ℓ2−4ϱζ

2ϱ , and

P (ϵ8) ≥ 0 for
ℓ−

√
ℓ2 − 4ϱζ

2ϱ
≤ ϵ8 ≤ ℓ+

√
ℓ2 − 4ϱζ

2ϱ
.

Thus we obtain

µ0

(θ
2
− 1

)
∥ |∇gu|g∥22 − θE1 −

ζ

ϵ8
> 0 for

ℓ−
√
ℓ2 − 4ϱζ

2ϱ
≤ ϵ8 ≤ ℓ+

√
ℓ2 − 4ϱζ

2ϱ
. □

Figure 3. The figure of P (ϵ8)

Combining (6.13), (6.14), (6.20) and then choosing 0 < χ < min{ 1
2 , χ} and τ small enough, we

obtain
M ′(t) ≥ C17

(
∥ut∥22 + ∥u∥γ+2

γ+2,Γ1
+G(t)

)
,

where C17 is a positive constant, which implies that M(t) is a positive increasing function. By the
same arguments on [16, p.333], we have

M ′(t) ≥ C18M
1

1−χ (t) for all t ≥ 0,

where C18 is a positive constant and 1 < 1
1−χ < 2. Then we conclude that M(t) blows up in finite

time and u also blows up in finite time. Thus this is a contradiction, consequently, the proof is
complete.

Acknowledgments. This research was supported by the Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2022-
NR075641).

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Tae Gab Ha
Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk National Uni-

versity, Jeonju 54896, Korea

Email address: tgha@jbnu.ac.kr


	1. Introduction
	2. Preliminaries
	(H1) Hypothesis on 
	(H2) Hypothesis on  and  f
	(H3) Hypothesis on q
	(H4) Hypothesis on 

	3.  Proof of Theorem 2.4: local existence
	3.1. Globally Lipschitz source
	3.2. Locally Lipschitz source
	3.3. Completion of the proof for the local existence
	3.4. Energy identity
	3.5. Strong time continuity

	4. Proof of Theorem 2.4: global existence
	4.1. 
	4.2. Potential well

	5. Proof of Theorem 2.5: energy decay
	Case 1:  is linear
	Case 2:  has polynomial growth near zero
	Case 3:  does not necessarily have polynomial growth near zero

	6. Proof of Theorem 2.6: blow-up
	Acknowledgments

	References