Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 84, pp. 1–20.

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

BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS

FOR A p(x)-LAPLACIAN EQUATION WITH DELAY TERM AND

VARIABLE EXPONENTS

STANISLAV ANTONTSEV, JORGE FERREIRA, ERHAN PIŞKIN,

HAZAL YÜKSEKKAYA, MOHAMMAD SHAHROUZI

Communicated by J. I. Diaz

Abstract. In this article, we consider a nonlinear p(x)-Laplacian equation
with time delay and variable exponents. Firstly, we prove the blow up of

solutions. Then, by applying an integral inequality due to Komornik, we

obtain the decay result.

1. Introduction

In this work, we study the wave equation

utt −∆p(x)u+ µ1ut(x, t)|ut|m(x)−2(x, t)

+ µ2ut(x, t− τ)|ut|m(x)−2(x, t− τ)

= bu|u|q(x)−2 in Ω× (0,∞)

u(x, t) = 0 in ∂Ω× [0,∞),

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

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

(1.1)

with delay term. Here, Ω ⊂ Rn is a bounded domain with sufficiently smooth
boundary ∂Ω. τ > 0 is a time delay term, µ1 is a positive constant, µ2 is a real
number and b ≥ 0 is a constant. The term ∆p(·)u = (|∇u|p(x)−2∇u) is called
p(·)-Laplacian. The functions u0, u1, f0 are the initial data that will be specified
later.
p(·), q(·) and m(·) are the variable exponents; these are given as measurable

functions on Ω such that:

2 ≤ p− ≤ p(x) ≤ p+ ≤ p∗,
2 ≤ q− ≤ q(x) ≤ q+ ≤ q∗,

2 ≤ m− ≤ m(x) ≤ m+ ≤ m∗,
(1.2)

where

p− = ess infx∈Ω p(x), p+ = ess supx∈Ω p(x),

2010 Mathematics Subject Classification. 35B40, 35B44, 35L05.

Key words and phrases. Blow up; decay; variable exponents; delay term.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted June 7, 2021. Published October 8, 2021.
1



2 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

q− = ess infx∈Ω, q+ = ess supx∈Ω q(x),

m− = ess infx∈Ωm(x), m+ = ess supx∈Ωm(x)

and

p∗ =

{
ess infx∈Ω np(x)/(n− p(x)) if p+ < n,

+∞ if p+ > n.
(1.3)

Many phenomena in engineering and physics lead up to problems that deal with
evolution equations, which are modeled by partial differential equations. Up to
now, there are many results about partial differential equations with time delay
effects. Our main goal in this work is to study the equation with p(·)-Laplacian
and the delay term µ2ut(x, t − τ) which make the problem more interesting than
those studied in the literature. Equation (1.1) is a very general equation.

Time delay appears in many practical problems such as thermal, biological,
economic, chemical, physical phenomena and it can be a source of instability [11].
Mathematically, these properties have practical and theoretical importance. On the
other hand, the delay term is a source that may destabilize the asymptotic stability
of solutions for an evolutionary system. This result is well justified in mathematical
analysis and physics examples, such as non-instant transmission phenomena and
biological models [32].

The problems with variable exponents arise in many branches of sciences such as
nonlinear elasticity theory, electrorheological fluids and image processing [4, 5, 29].
Many works about wave equation with constant delay or delay effects with time-
varying have been published.

Equations with delay. Feng and Li [7], studied the equation

utt + ∆2u− divF (∇u)− σ(t)

∫ t

0

g(t− s)∆2u(s)ds+ µ1|ut|m−1ut

+ µ2|ut(x, t− τ)|m−1ut(x, t− τ) = 0,

(1.4)

where Ω ⊆ Rn (n ≥ 1) is a bounded domain with smooth boundary ∂Ω. They
proved the general rates of energy decay of the initial value problem and the bound-
ary value problem by using the energy perturbation method.

Messaoudi and Kafini [11] considered the equation

utt − div(|∇u|m−2∇u) + µ1ut(x, t) + µ2ut(x, t− τ) = b|u|p−2u. (1.5)

Under suitable conditions, they proved the blow-up of solutions of (1.5) in a finite
time.

Nicaise and Pignotti [20] considered the wave equation with time delay,

utt −∆u+ µ1ut + µ2ut(t− τ) = 0, (1.6)

and they established stability results under the assumption 0 < µ2 < µ1.
Park[22] treated the Kirchhoff models with time delay and perturbation of p-

Laplacian type

utt + ∆2u−∆pu− a0∆ut + a1ut(x, t− τ) + f(u) = g(x), (1.7)

where ∆pu = div(|∇u|p−2∇u) is the usual p-Laplacian operator and a0 > 0, a1 ∈ R,
τ > 0 is time delay. He established the existence of global attractors and the finite
dimensionality of the attractors by establishing some functionals.



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 3

Equations without delay. Pişkin [23] studied the quasilinear hyperbolic equation

utt − div(|∇u|m∇u)−∆ut + |ut|q−1ut = |u|p−1u, (1.8)

where m > 0, p, q ≥ 1. He investigated the global existence, decay and blow up of
solutions. He proved the decay estimates of the energy function by using Nakao’s
inequality and obtained the blow up of solutions and lifespan estimates in three
different ranges of the initial energy.

Wu and Xue [31] considered the quasi-linear wave equation

utt −∆ut − div(|∇u|m∇u) + a|ut|αut = b|u|p−1u, (1.9)

where a, b, α, m, p ≥ 0. By using multiplier methods, they gave the precise uniform
estimation of the decay rate, when the initial data are in a potential well.

Variable exponent nonlinearity. Recently, much attention has been paid to the
study of nonlinear hyperbolic, parabolic and elliptic equations with variable expo-
nents as nonlinearities. The so-called equations with non-standard growth condi-
tions. Actually, only few work regarding hyperbolic problems with nonlinearities
of variable-exponent type have appeared [15].

Equations with delay. Messaoudi and Kafini [14] studied the equation

utt −∆u+ µ1ut(x, t)|ut|m(x)−2(x, t) + µ2ut(x, t− τ)|ut|m(x)−2(x, t− τ)

= bu|u|p(x)−2.
(1.10)

They obtained decay estimates and global nonexistence of solutions.

Equations without Delay. Antontsev [1, 2] considered the equation

∂ttu− div(a(x, t)|∇u|p(x,t)∇u)− α∆ut = b(x, t)u|u|σ(x,t)−2, (1.11)

in Ω, a bounded domain of Rn, where α > 0 is a constant and a, b, p, σ are given
by functions. For certain solutions with non-positive initial energy, he proved the
blow-up results. Antontsev [1, 3] studied equation (1.11) and proved the local and
the global existence of weak solutions.

Messaoudi et al. [8] studied the equation

utt −∆u+ ut|ut|m(·)−2 = u|u|p(·)−2. (1.12)

They proved a global result and obtained the stability result by applying an integral
inequality due to Komornik.

In [16], the authors considered the equation

utt − div(|∇u|r(·)−2∇u) + |ut|m(·)−2ut = 0, (1.13)

where the exponents m(·) and r(·) are given by measurable functions on Ω. They
proved the decay results for the solution under suitable assumptions. Also, the
authors gave two numerical applications to illustrate the theoretical results.

In the presence of the strong damping term −∆ut, equation (1.13) takes the
form

utt − div(|∇u|r(·)−2∇u)−∆ut + |ut|m(·)−2ut = 0, (1.14)

where Ω is a bounded domain. Messaoudi [17] studied the nonlinear wave equation
(1.14) with variable exponents. He established several decay results depending of
the range of the variable exponents m and r. In recent years, some other authors
investigated hyperbolic type equations with variable exponents; see [10, 21, 24, 25,
26, 27, 30].



4 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

Our purpose is to study the blow up of solutions with negative initial energy
and the decay results for the nonlinear wave equation (1.1) with time-dependent
delay and variable exponents. Our result extends the equation (1.5), from constant-
exponent nonlinearities to variable-exponent nonlinearities.

This article is organized as follows: In Section 2, the definitions of variable
exponent in Sobolev and Lebesgue spaces are introduced. In Section 3, we prove
the blow up of solutions. Finally, in Section 4, the decay results will be obtained.

2. Preliminaries

In this part, we begin by introducing some preliminary facts about Lebesgue
Lp(·)(Ω) and Sobolev W 1,p(·)(Ω) spaces with variable exponents; see [1, 5, 6, 9, 13,
14, 28].

Let p : Ω → [1,∞) be a measurable function. We define the variable-exponent
in Lebesgue space with a variable exponent p(·) by

Lp(·)(Ω) =
{
u : Ω→ R : measurable in Ω,

∫
Ω

|u|p(·)dx <∞
}
,

with a Luxemburg-type norm

‖u‖p(·) = inf
{
λ > 0 :

∫
Ω

|u
λ
|p(x)dx ≤ 1

}
.

Equipped with this norm, Lp(·)(Ω) is a Banach space. (see [5])
Now, we define the variable-exponent Sobolev space

W 1,p(·)(Ω) =
{
u ∈ Lp(·)(Ω) : ∇u exists and |∇u| ∈ Lp(·)(Ω)

}
.

The variable exponent Sobolev space with respect to the norm

‖u‖1,p(·) = ‖u‖p(·) + ‖∇u‖p(·)

is a Banach space. The space W
1,p(·)
0 (Ω) is defined as the closure of C∞0 (Ω) in

W 1,p(·)(Ω). For u ∈W 1,p(·)
0 (Ω), we can define an equivalent norm

‖u‖1,p(·) = ‖∇u‖p(·) .

The dual space of W
1,p(·)
0 (Ω) is defined as W

−1,p′(·)
0 (Ω), in the same way that the

usual Sobolev spaces, where 1
p(·) + 1

p′(·) = 1.

We also suppose that p(·), q(·) and m(·) satisfy the log-Hölder continuity condi-
tion:

|q(x)− q(y)| ≤ − A

log |x− y|
, for a.e. x, y ∈ Ω, with |x− y| < δ, (2.1)

where A > 0 and 0 < δ < 1.

Lemma 2.1 (Poincaré inequality [1]). Assume that q(·) satisfies (2.1) and let Ω
be a bounded domain of Rn. Then

‖u‖p(·) ≤ c‖∇u‖p(·) for all u ∈W 1,p(·)
0 (Ω),

where c = c(p−, p+, |Ω|) > 0.

Lemma 2.2 ([5]). If p(·) ∈ C(Ω) and q : Ω→ [1,∞) is a measurable function such
that ess inf fx∈Ω(p∗(x) − q(x)) > 0, with p∗ defined in (1.3), is satisfied, then the

embedding W
1,p(·)
0 (Ω) ↪→ Lq(·)(Ω) is continuous and compact.



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 5

Lemma 2.3 ([1]). If p+ < ∞ and p : Ω → [1,∞) is a measurable function, then
C∞0 (Ω) is dense in Lp(·)(Ω).

Lemma 2.4 (Hölder’s inequality [1]). Let p, q, s ≥ 1 be measurable functions de-
fined on Ω and

1

s(y)
=

1

p(y)
+

1

q(y)
, for a.e. y ∈ Ω.

that is satisfied. If f ∈ Lp(·)(Ω) and g ∈ Lq(·)(Ω), then fg ∈ Ls(·)(Ω) and

‖fg‖s(·) ≤ 2‖f‖p(·)‖g‖q(·).

Lemma 2.5 (Unit ball property [1]). Let p ≥ 1 be a measurable function on Ω.
Then

‖f‖p(·) ≤ 1 if and only if %p(·)(f) ≤ 1,

where

%p(·)(f) =

∫
Ω

|f(x)|p(x)dx.

Lemma 2.6 ([1]). If p ≥ 1 is a measurable function on Ω. Then

min
{
‖u‖p

−

p(·), ‖u‖
p+

p(·)
}
≤ %p(·)(u) ≤ max

{
‖u‖p

−

p(·), ‖u‖
p+

p(·)
}

for any u ∈ Lp(·)(Ω) and for a.e. x ∈ Ω.

3. Blow up

In this part, we deal with the blow up of the solution for problem (1.1) with
negative initial energy, when b > 0. Now, we introduce, as in [20], the new variable

z(x, ρ, t) = ut(x, t− τρ), x ∈ Ω, ρ ∈ (0, 1), t > 0,

which implies that

τzt(x, ρ, t) + zρ(x, ρ, t) = 0, x ∈ Ω, ρ ∈ (0, 1), t > 0.

Using the above transformation, problem (1.1) can be written in the equivalent
form

utt − div(|∇u|p(x)−2∇u) + µ1ut(x, t)|ut(x, t)|m(x)−2

+ µ2z(x, 1, t)|z(x, 1, t)|m(x)−2 = bu|u|q(x)−2, in Ω× (0,∞)

τzt(x, ρ, t) + zρ(x, ρ, t) = 0 Ω× (0, 1)× (0,∞)

z(x, ρ, 0) = f0(x,−ρτ) Ω× (0, 1)

u(x, t) = 0 ∂Ω× [0,∞)

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

(3.1)

Similar to [14] we can write the following definition.

Definition 3.1. Fix T > 0. We call (u, z) a strong solution of (3.1) if

u ∈W 2,∞([0, T );L2(Ω)) ∩W 1,∞([0, T );H1
0 (Ω)) ∩ L∞([0, T );H2(Ω) ∩H1

0 (Ω)),

ut ∈ Lm(·)(Ω× (0, T )),

z ∈W 1,∞([0, 1]× [0, T );L2(Ω)) ∩ L∞([0, 1];Lm(·)(Ω) ∩ [0, T ))



6 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

and (u, z) satisfies the initial data and (3.1) in the sense that∫
Ω

utt(·, t)vdx−
∫

Ω

div(|∇u(·, t)|p(·)−2∇u(·, t))vdx

+ µ1

∫
Ω

|ut(·, t)|m(·)−2ut(·, t)vdx+ µ2

∫
Ω

|z(·, 1, t)|m(·)−2z(·, 1, t)vdx

= b

∫
Ω

|u(·, t)|q(·)−2u(·, t)vdx

(3.2)

and

τ

∫
Ω

zt(·, ρ, t)wdx+

∫
Ω

zρ(·, ρ, t)wdx = 0, (3.3)

for a.e. t ∈ [0, T ) and for (v, w) ∈ H1
0 (Ω) ∩ L2(Ω).

The energy functional associated with problem (3.1) is defined as

E(t) =
1

2
‖ut‖2 +

∫
Ω

1

p(x)
|∇u|p(x)dx

+

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ− b

∫
Ω

|u|q(x)

q(x)
dx

(3.4)

for t ≥ 0, where ξ is a continuous function that satisfies

τ |µ2|(m(x)− 1) < ξ(x) < τ(µ1m(x)− |µ2|), x ∈ Ω. (3.5)

The following lemma gives that, under the condition µ1 > |µ2|, E(t) is non-
increasing.

Lemma 3.2. Let (u, z) be a solution of (3.1). Then there exists C0 > 0 such that

E′(t) ≤ −C0

∫
Ω

(|ut|m(x) + |z(x, 1, t)|m(x))dx ≤ 0. (3.6)

Proof. Multiplying the first equation in (3.1) by ut, integrating over Ω, then mul-
tiplying the second equation of (3.1) by 1

τ ξ(x)|z|m(x)−2z, and integrating over
Ω× (0, 1), we obtain

d

dt
[
1

2

∫
Ω

u2
tdx+

∫
Ω

1

p(x)
|∇u|p(x)dx+

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ

− b
∫

Ω

|u|q(x)

q(x)
dx]

= −µ1

∫
Ω

|ut|m(x)dx− 1

τ

∫
Ω

∫ 1

0

ξ(x)|z(x, ρ, t)|m(x)−2zzρ(x, ρ, t)dρdx

− µ2

∫
Ω

utz(x, 1, t)|z(x, 1, t)|m(x)−2dx.

(3.7)

The last two terms of the right-hand side of (3.7) can be estimated as follows,

− 1

τ

∫
Ω

∫ 1

0

ξ(x)|z(x, ρ, t)|m(x)−2zzρ(x, ρ, t)dρdx

= −1

τ

∫
Ω

∫ 1

0

∂

∂ρ

(ξ(x)|z(x, ρ, t)|m(x)

m(x)

)
dρdx

=
1

τ

∫
Ω

ξ(x)

m(x)

(
|z(x, 0, t)|m(x) − |z(x, 1, t)|m(x)

)
dx



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 7

=

∫
Ω

ξ(x)

τm(x)
|ut|m(x)dx−

∫
Ω

ξ(x)

τm(x)
|z(x, 1, t)|m(x).

We use Young’s inequality, q = m(x)
m(x)−1 and q′ = m(x) for the last term, and then

we obtain

‖ut|‖z(x, 1, t)|m(x)−1 ≤ 1

m(x)
|ut|m(x) +

m(x)− 1

m(x)
|z(x, 1, t)|m(x).

As a result,

− µ2

∫
Ω

utz|z(x, 1, t)|m(x)−2dx

≤ |µ2|
(∫

Ω

1

m(x)
|ut(t)|m(x)dx+

∫
Ω

m(x)− 1

m(x)
|z(x, 1, t)|m(x)dx

)
.

Therefore,

dE(t)

dt
≤ −

∫
Ω

[µ1 − (
ξ(x)

τm(x)
+
|µ2|
m(x)

)]|ut(t)|m(x)dx

−
∫

Ω

(
ξ(x)

τm(x)
− |µ2|(m(x)− 1)

m(x)
)|z(x, 1, t)|m(x)dx.

Consequently, for all x ∈ Ω, relation (3.5) gives

f1(x) = µ1 − (
ξ(x)

τm(x)
+
|µ2|
m(x)

) > 0,

f2(x) =
ξ(x)

τm(x)
− |µ2|(m(x)− 1)

m(x)
> 0.

Since that m(x), and hence, ξ(x) is bounded, we infer that f1(x) and f2(x) are
also bounded. We define

C0(x) = min{f1(x), f2(x)} > 0 for any x ∈ Ω,

and take C0 = infx∈Ω C0(x), then C0(x) ≥ C0 > 0. Therefore,

E′(t) ≤ −C0

[ ∫
Ω

|ut(t)|m(x)dx+

∫
Ω

|z(x, 1, t)|m(x)dx
]
≤ 0.

�

To establish the blow up, we suppose that E(0) < 0 in addition to (1.3). Setting

H(t) = −E(t); (3.8)

we have
H ′(t) = −E′(t) ≥ 0,

0 < H(0) ≤ H(t) ≤ b
∫

Ω

|u|q(x)

q(x)
dx ≤ b

p−
%(u),

(3.9)

where

%(u) = %q(·)(u) =

∫
Ω

|u|q(x)dx.

Lemma 3.3 ([15]). Assume that the conditions of Lemma 2.2 hold. Then, exists
a constant C > 1, depending only of Ω, such that

%s/q
−

(u) ≤ C
(
‖∇u‖p

−

p(·) + %(u)
)
. (3.10)



8 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

Then, we have the following inequalities:

‖u‖sq− ≤ C(‖∇u‖p
−

p(·) + ‖u‖q
−

q−), (3.11)

%s/q
−

(u) ≤ C
(
|H(t)|+ ‖ut‖22 + %(u) +

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ

)
, (3.12)

‖u‖sq− ≤ C
(
|H(t)|+ ‖ut‖22 + ‖u‖q

−

q− +

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ

)
, (3.13)

for any u ∈W 1,p(·)
0 (Ω) and p− ≤ s ≤ q−. Let (u, z) be a solution of (3.1). Then

%(u) ≥ C‖u‖q
−

q− , (3.14)∫
Ω

|u|m(x)dx ≤ C
(
%m
−/q−(u) + %m

+/q−(u)
)
. (3.15)

The blow up of problem (3.1) is given by the following theorem.

Theorem 3.4. Let u0 ∈ W 1,p(·)
0 (Ω), u1 ∈ L2(Ω). Assume that the condition (2.1)

holds and
2 ≤ max{m+, p+} < q− ≤ q(x) ≤ q+ ≤ p∗(x),

where

p∗(x) =

{
np(x)

ess infx∈Ω(n−m(x)) if p+ < n,

+∞ if p+ > n.

Moreover, suppose that E(0) < 0. Then, the solution of (3.1) blows up in finite
time.

Proof. We define

L(t) = H1−α(t) + ε

∫
Ω

uut(x, t)dx (3.16)

for a small ε to be chosen later, and

0 < α ≤ min
{q− − 2

2q−
,

q− −m+

q−(m+ − 1)

}
. (3.17)

Differentiation of (3.16), using the first equation in (3.1), gives

L′(t) = (1− α)H−α(t)H ′(t) + ε

∫
Ω

u2
tdx− ε

∫
Ω

|∇u|p(x)dx

+ εb

∫
Ω

|u|q(x)dx− εµ1

∫
Ω

uut(x, t)|ut(x, t)|
m(x)−2

dx

− εµ2

∫
Ω

uz(x, 1, t)|z(x, 1, t)|
m(x)−2

dx.

From the definition of H(t) and for 0 < a < 1, we obtain

L′(t) ≥ C0(1− α)H−α(t)[

∫
Ω

|ut(t)|m(x)dx+

∫
Ω

|z(x, 1, t)|
m(x)

dx]

+ ε
(

(1− a)q−H(t) +
(1− a)q−

2
‖ut‖2

)
+ ε(1− a)q−

∫
Ω

1

p(x)
|∇u|p(x)dx

+ ε(1− a)q−
∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 9

+ ε

∫
Ω

u2
tdx− ε

∫
Ω

|∇u|p(x) + εab

∫
Ω

|u|q(x)dx

− εµ1

∫
Ω

uut(x, t)|ut(x, t)|
m(x)−2

dx

− εµ2

∫
Ω

uz(x, 1, t)|z(x, 1, t)|
m(x)−2

dx.

Therefore,

L′(t) ≥ C0(1− α)H−α(t)[

∫
Ω

|ut(t)|m(x)dx+

∫
Ω

|z(x, 1, t)|
m(x)

dx]

+ ε(1− a)q−H(t) + ε
(1− a)q− + 2

2
‖ut‖2

+ ε
( (1− a)q−

p+
− 1
)∫

Ω

|∇u|p(x)dx

+ ε(1− a)q−
∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ+ εab%(u)

− εµ1

∫
Ω

uut(x, t)|ut(x, t)|
m(x)−2

dx

− εµ2

∫
Ω

uz(x, 1, t)|z(x, 1, t)|
m(x)−2

dx.

(3.18)

From Young’s inequality, we obtain∫
Ω

|ut|
m(x)−1

|u|dx

≤ 1

m−

∫
Ω

δm(x)|u|
m(x)

dx+
m+ − 1

m+

∫
Ω

δ−
m(x)

m(x)−1 |ut|
m(x)

dx

(3.19)

and ∫
Ω

|z(x, 1, t)|
m(x)−1

|u|dx

≤ 1

m−

∫
Ω

δm(x)|u|
m(x)

dx+
m+ − 1

m+

∫
Ω

δ−
m(x)

m(x)−1 |z(x, 1, t)|
m(x)

dx.

(3.20)

As in [19], the estimates (3.19) and (3.20) remain valid if δ is time-dependent. Thus,
taking δ such that

δ−
m(x)

m(x)−1 = kH−α(t),

where k ≥ 1 is specified later, we obtain∫
Ω

δ−
m(x)

m(x)−1 |ut|
m(x)

dx = kH−α(t)

∫
Ω

|ut|m(x)dx, (3.21)∫
Ω

δ−
m(x)

m(x)−1 |z(x, 1, t)|
m(x)

dx = kH−α(t)|z(x, 1, t), |
m(x)

dx (3.22)∫
Ω

δm(x)|u|
m(x)

dx =

∫
Ω

k1−m(x)Hα(m(x)−1)(t)|u|m(x)dx

≤
∫

Ω

k1−m−Hα(m+−1)(t)

∫
Ω

|u|m(x)dx.

(3.23)



10 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

From (3.14) and (3.15), we have

Hα(m+−1)(t)

∫
Ω

|u|m(x)dx

≤ C[(%(u))m
−/q−+α(m+−1) + (%(u))m

+/q−+α(m+−1)].

(3.24)

From (3.17), we conclude that

s = m− + αq−(m+ − 1) ≤ q−, s = m+ + αq−(m+ − 1) ≤ q−.
Therefore,

Hα(m+−1)(t)

∫
Ω

|u|m(x)dx ≤ C(‖∇u‖p
−

p(·) + %(u)). (3.25)

Combining (3.19)-(3.25), we obtain

L′(t) ≥ (1− α)H−α(t)
[
C0 − ε

(m+ − 1

m+

)
ck
] ∫

Ω

|ut(t)|m(x)dx

+ (1− α)H−α(t)
[
C0 − ε

(m+ − 1

m+

)
ck
] ∫

Ω

|z(x, 1, t)|
m(x)

dx

+ ε
( (1− a)q− − p+

p+
− C

m−km−−1

)∫
Ω

|∇u|p(x)dx

+ ε(1− a)q−H(t) + ε
(1− a)q− + 2

2
‖ut‖2

+ ε
(
ab− C

m−km−−1

)
%(u)

+ ε(1− a)q−
∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ.

(3.26)

Let us choose a small enough such that

(1− a)q− + 2

2
> 0,

and k so large that

(1− a)q− − p+

p+
− C

m−km−−1
> 0 and ab− C

m−km−−1
> 0.

Once k and a are fixed, we choose ε small enough such that

C0 − ε
(m+ − 1

m+

)
ck > 0,

L(0) = H1−α(0) + ε

∫
Ω

u0(x)u1(x)dx > 0.

Hence, (3.26) becomes

L′(t) ≥ εη
[
H(t) + ‖ut‖2 + ‖∇u‖p

−

p(·) + %
p(·)(u)

+

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ

] (3.27)

for a constant η > 0. Eventually,

L(t) ≥ L(0) > 0, ∀t ≥ 0.



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 11

Now, for some constants σ, Γ > 0, we denote

L′(t) ≥ ΓLσ(t).

For this reason, we estimate∣∣ ∫
Ω

uut(x, t)dx
∣∣ ≤ ‖u‖2‖ut‖2 ≤ C‖u‖q−‖ut‖2,

which indicates ∣∣ ∫
Ω

uut(x, t)dx
∣∣1/(1−α) ≤ C‖u‖1/(1−α)

q− ‖ut‖1/(1−α)
2

and by Young’s inequality∣∣ ∫
Ω

uut(x, t)dx
∣∣1/(1−α) ≤ C[‖u‖µ/(1−α)

q− + ‖ut‖Θ/(1−α)
2 ],

where 1/µ+1/Θ = 1. The choice of Θ = 2(1−α) will make µ/(1−α) = 2/(1−2α) ≤
q−. Thus, ∣∣ ∫

Ω

uut(x, t)dx
∣∣1/(1−α) ≤ C[‖u‖sq− + ‖ut‖22],

where s = µ/(1− α). From (3.13), we obtain∣∣ ∫
Ω

uut(x, t)dx
∣∣1/(1−α)

≤ C
[
|H(t)|+ ‖ut(t)‖2 + %(u) +

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ

]
.

(3.28)

Moreover, we have

L1/(1−α)(t) =
[
H(1−α)(t) + ε

∫
Ω

uut(x, t)dx
]1/(1−α)

≤ 2α/(1−α)
[
H(t) + |

∫
Ω

uut dx|1/(1−α)
]

≤ C
[
|H(t)|+ ‖ut(t)‖2 + %(u) +

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ

]
.

Therefore, for some Ψ > 0, from (3.27), we obtain

L′(t) ≥ ΨL1/(1−α)(t).

A simple integration over (0, t) gives

Lα/(1−α)(t) ≥ 1

L−α/(1−α)(0)−Ψαt/(1− α)

which implies that the solution blows up in a finite time T ∗, with

T ∗ ≤ 1− α
Ψα[L(0)]α/(1−α)

.

The proof is complete. �



12 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

4. Decay of solutions

In this part, we prove our decay result, when b = 0. Now, we introduce the
variable

z(x, ρ, t) = ut(x, t− τρ), x ∈ Ω, ρ ∈ (0, 1), t > 0;

thus,

τzt(x, ρ, t) + zρ(x, ρ, t) = 0, x ∈ Ω, ρ ∈ (0, 1), t > 0.

Consequently, problem (1.1) is transformed into

utt − div(|∇u|p(x)−2∇u) + µ1ut(x, t)|ut(x, t)|m(x)−2

+ µ2z(x, 1, t)|z(x, 1, t)|m(x)−2 = 0 in Ω× (0,∞),

τzt(x, ρ, t) + zρ(x, ρ, t) = 0 in Ω× (0, 1)× (0,∞)

z(x, ρ, 0) = f0(x,−ρτ) in Ω× (0, 1)

u(x, t) = 0 on ∂Ω× [0, 1)

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

(4.1)

We define the modified energy functional for problem (4.1) by

E(t) =
1

2
‖ut‖2 +

∫
Ω

1

p(x)
|∇u|p(x)dx

+

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρ,

(4.2)

where ξ is the continuous function given in (3.5) and t ≥ 0.
Similar to Lemma 3.2, we easily establish, for µ1 > |µ2| and for some C0 > 0,

that

E′(t) ≤ −C0

∫
Ω

(
|ut|m(x) + |z(x, 1, t)|m(x)

)
dx ≤ 0. (4.3)

Lemma 4.1 (Komornik, [12]). Let E : R+ → R+ be a non-increasing function and
assume that there are constants σ, ω > 0 such that∫ ∞

s

E1+σ(t)dt ≤ 1

Ω
Eσ(0)E(s) = cE(s), ∀s > 0.

Then

E(t) ≤

{
cE(0)/(1 + t)1/σ if σ > 0,

cE(0)e−ωt if σ = 0.

for all t ≥ 0.

To prove our main result, we need of the following lemmas.

Lemma 4.2 ([14]). The functional

F (t) = τ

∫ 1

0

∫
Ω

e−ρτξ(x)|z(x, ρ, t)|m(x) dx dρ

satisfies

F ′(t) ≤
∫

Ω

ξ(x)|ut|m(x)dx− τe−τ
∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x) dx dρ

along the solution of (4.1).



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 13

Lemma 4.3 ([18]). Let u be a solution of (4.1). Then, for some C > 0,

%p(x)(∇u) ≥ C‖∇u‖p
+

p− . (4.4)

Theorem 4.4. Let u0 ∈ W
1,p(·)
0 (Ω), u1 ∈ L2(Ω) be given and assume that m(·)

and p(·) belong to C(Ω̄). Suppose that condition (2.1) is satisfied and

2 ≤ p− ≤ p(x) ≤ p+ ≤ m− ≤ m(x) ≤ m+ ≤ p−
∗
, ∀x ∈ C(Ω),

where

p−
∗

=

{
ess infx∈Ω

np−

(n−p−) if p− < n,

+∞ if p− ≥ n.
Then, there exist two constants c, α > 0 independent of t such that for any global
solution of (4.1), it holds

E(t) ≤

{
ce−αt if m(x) = 2,

cE(0)/(1 + t)2/(m+−2) if m+ > 2.

Proof. We multiply the first equation of (4.1) by uEr(t), for r > 0 that will be
specified later, and integrate over Ω× (s, T ), s < T . So, we obtain∫ T

s

Er(t)

∫
Ω

[
uutt − udiv(|∇u|p(x)−2∇u) + µ1uut|ut|m(x)−2

+ µ2uz(x, 1, t)|z(x, 1, t)|m(x)−2
]
dx dt = 0,

which implies that∫ T

s

Er(t)

∫
Ω

( d
dt

(uut)− u2
t + |∇u|p(x) + µ1uut(x, t)|ut(x, t)|m(x)−2

+ µ2uz(x, 1, t)|z(x, 1, t)|m(x)−2
)
dx dt = 0.

(4.5)

By using the definition of E(t), given in (4.2), and the relation

d

dt

(
Er(t)

∫
Ω

uut dx
)

= rEr−1(t)E′(t)

∫
Ω

uut dx+ Er(t)
d

dt

∫
Ω

uut dx,

equation (4.5) becomes

2

∫ T

s

Er+1(t)dt

≤ −
∫ T

s

d

dt

(
Er(t)

∫
Ω

uut dx
)
dt+ r

∫ T

s

Er−1(t)E′(t)

∫
Ω

uut dx dt

+ 2

∫ T

s

Er(t)

∫
Ω

u2
t dx dt− µ1

∫ T

s

Er(t)

∫
Ω

uut|ut|m(x)−2 dx dt

− µ2

∫ T

s

Er(t)

∫
Ω

uz(x, 1, t)|z(x, 1, t)|m(x)−2 dx dt

+ 2

∫ T

s

Er(t)

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρdt.

(4.6)

Now, we estimate the right-hand side terms of equation (4.6).



14 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

The first term is estimated as follows:

∣∣− ∫ T

s

d

dt
(Er(t)

∫
Ω

uut dx)dt
∣∣

=
∣∣Er(s)∫

Ω

uut(x, s)dx− Er(T )

∫
Ω

uut(x, T )dx
∣∣

≤ 1

2
Er(s)

[ ∫
Ω

u2(x, s)dx+

∫
Ω

u2
t (x, s)dx

]
+

1

2
Er(T )

[ ∫
Ω

u2(x, T )dx+

∫
Ω

u2
t (x, T )dx

]
≤ Er(s)

[1

2
c∗

∫
Ω

|∇u(x, s)|2dx+ E(s)
]

+ Er(T )
[1

2
c∗

∫
Ω

|∇u(x, T )|2dx+ E(T )
]

where c∗ is the embedding constant. So, we obtain

∣∣− ∫ T

s

d

dt
(Er(t)

∫
Ω

uut dx)dt
∣∣

≤ Er(s)[c‖∇u(s)‖2p− + E(s)] + Er(T )[c‖∇u(T )‖2p− + E(T )]

≤ Er+1(s) + cEr(s)(‖∇u(s)‖p
+

p−)2/p+

+ Er+1(T ) + cEr(T )(‖∇u(T )‖p
+

p−)
2

p+ ,

where c is a generic positive constant that may change their value from a line to
another. Then, we use (4.4), and recalling that E(t) is non-increasing, we obtain

∣∣− ∫ T

s

d

dt
(Er(t)

∫
Ω

uut dx)dt
∣∣

≤ Er+1(s) + cEr(s)(%p(x)(∇u(s)))2/p+

+ Er+1(T ) + cEr(T )(%p(x)(∇u(T )))
2

p+

≤ Er+1(s) + c(E(s))
r+ 2

p+ + Er+1(T ) + c(E(T ))
r+ 2

p+

≤ Er+1(s) + c(E(s))
r+ 2

p+ .

(4.7)

In the above estimate, for p− > 2, we applied the Hölder inequality

∫
Ω

|∇u|2dx ≤ |Ω|
p−−2

p−
(∫

Ω

|∇u|p
−
dx
)2/p−

.

Estimate (4.7) for the case p− = 2 is also true.



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 15

Similarly, we deal with the term∣∣r ∫ T

s

Er−1(t)E′(t)

∫
Ω

uut dx dt
∣∣

≤ −c
∫ T

s

Er−1(t)E′(t)[E(T ) + cE
2

p+ (t)]dt

≤ −c[
∫ T

s

Er(t)E′(t) +

∫ T

s

(E(t))
r+ 2

p+−1
E′(t)dt]

≤ c[Er+1(s) + (E(s))
r+ 2

p+ ].

(4.8)

To treat the other term, we define

Ω+ = {x ∈ Ω : |ut(x, t)| ≥ 1}, Ω− = {x ∈ Ω : |ut(x, t)| < 1},

and use Hölder’s and Young’s inequalities. Then we have∣∣2 ∫ T

s

Er(t)

∫
Ω

u2
t dx dt

∣∣
=
∣∣∣2 ∫ T

s

Er(t)
[ ∫

Ω+

u2
tdx+

∫
Ω−

u2
tdx
]
dt
∣∣∣

≤ c
∫ T

s

Er(t)
[( ∫

Ω+

|ut|m
−
dx
)2/m−

+
(∫

Ω−

|ut|m
+

dx
)2/m+]

dt

≤ c
∫ T

s

Er(t)
[( ∫

Ω

|ut|m(x)dx
)2/m−

+
(∫

Ω

|ut|m(x)dx
)2/m+]

dt

≤ c
∫ T

s

Er(t)[(−E′(t))2/m− + (−E′(t))2/m+

]dt

≤ cε
∫ T

s

(E(t))rm
−/(m−−2)dt+ cε

∫ T

s

(−E′(t))dt

+ cε

∫ T

s

Er+1(t)dt+ cε

∫ T

s

(−E′(t))2(r+1)/m+

dt.

where cε = 1
r+1

( ε(r+1)
r

)−r
.

Choose r such that r = m+/2− 1 will make rm−

m−−2 = r + 1 + m+−m−
m−−2 . Now, we

consider two cases, m− > 2 and m− = 2.
For m− > 2, we have∣∣2∫ T

s

Er(t)

∫
Ω

u2
t dx dt

∣∣
≤ cε

∫ T

s

Er+1(t)dt+ cε(E(0))
m+−m−

m−−2

∫ T

s

Er+1(t)dt+ cεE(s)

≤ c̃ε
∫ T

s

Er+1(t)dt+ cεE(s),

(4.9)

where c̃ is a positive constant.
For m− = 2, we obtain∣∣2∫ T

s

Er(t)

∫
Ω

u2
t dx dt

∣∣



16 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

=
∣∣∣2 ∫ T

s

Er(t)
[ ∫

Ω+

u2
tdx+

∫
Ω−

u2
tdx
]
dt
∣∣∣

≤ c
∫ T

s

Er(t)
[ ∫

Ω+

|ut|m(x)dx+
(∫

Ω−

|ut|m
+

dx
)2/m+]

dt

≤ c
∫ T

s

Er(t)
[ ∫

Ω

|ut|m(x)dx+ (

∫
Ω

|ut|m(x)dx)2/m+
]
dt

≤ c
∫ T

s

Er(t)(−E′(t))dt+ c

∫ T

s

Er(t)(−E′(t))2/m+

dt

≤ cEr+1(s) + cε

∫ T

s

Er+1(t)dt+ cε

∫ T

s

(−E′(t))2(r+1)/m+

dt.

Therefore, with the choice of r = m+/2− 1, we obtain∣∣2∫ T

s

Er(t)

∫
Ω

u2
t dx dt

∣∣ ≤ cEr+1(s) + cε

∫ T

s

Er+1(t)dt+ cεE(s)

≤ cε
∫ T

s

Er+1(t)dt+ (cε + cEr(0))E(s)

≤ cε
∫ T

s

Er+1(t)dt+ c̃εE(s),

(4.10)

where c̃ε = cε + cEr(0).

Because of m+ ≥ p+ and r = m+

2 − 1, we have r + 2
p+ − 1 ≥ 0. As a result, the

estimates (4.7) and (4.8) become∣∣− ∫ T

s

d

dt

(
Er(t)

∫
Ω

uut dx
)
dt
∣∣ ≤ Er+1(s) + c(E(s))

r+ 2

p+

≤ [Er(0) + c(E(0))
r+ 2

p+−1
]E(s)

= c̃E(s),

(4.11)

and ∣∣r ∫ T

s

Er−1(t)E′(t)

∫
Ω

uut dx dt
∣∣ ≤ c[Er+1(s) + (E(s))

r+ 2

p+ ]

≤ c[Er(0) + (E(0))
r+ 2

p+−1
]E(s)

= c̃E(s),

(4.12)

respectively.
For the next term, by using Young’s inequality, we have∣∣− µ1

∫ T

s

Er(t)

∫
Ω

u|ut|m(x)−1 dx dt
∣∣

≤ ε
∫ T

s

Er(t)

∫
Ω

|u|m(x) dx dt+ c

∫ T

s

Er(t)

∫
Ω

cε(x)|ut|m(x) dx dt

≤ ε
∫ T

s

Er(t)
[ ∫

Ω+

|u|m
−
dx+

∫
Ω−

|u|m
+

dx
]
dt

+ c

∫ T

s

Er(t)

∫
Ω

cε(x)|ut|m(x) dx dt,



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 17

where we used Young’s inequality with

p(x) =
m(x)

m(x)− 1
and p′(x) = m(x),

and thus

cε(x) = (m(x)− 1)m(x)m(x)/(1−m(x))ε1/(1−m(x)).

Using the embedding, we obtain

∣∣− µ1

∫ T

s

Er(t)

∫
Ω

u|ut|m(x)−1 dx dt
∣∣

≤ ε
∫ T

s

Er(t)[c1‖∇u‖m
−

p− + c2‖∇u‖m
+

p− ] +

∫ T

s

Er(t)

∫
Ω

cε(x)|ut|m(x)dx,

where c1 and c2 are positive constants independent of ε.
From (4.2) and (4.4), we have

∣∣− µ1

∫ T

s

Er(t)

∫
Ω

u|ut|m(x)−1 dx dt
∣∣

≤ ε
∫ T

s

Er(t)
[
c1(%p(x)(∇u))

m−
p+ + c2(%p(x)(∇u))

m+

p+
]
dt

+ c

∫ T

s

Er(t)

∫
Ω

cε(x)|ut|m(x) dx dt

≤ εc′1
∫ T

s

Er+1(t)(E(t))
m−
p+ −1

dt+ εc′2

∫ T

s

Er+1(t)(E(t))
m+

p+ −1
dt

+ c

∫ T

s

Er(t)

∫
Ω

cε(x)|ut|m(x) dx dt

≤ c′ε
(

(E(0))
m−
p+ −1

+ (E(0))
m+

p+ −1
)∫ T

s

Er+1(t)dt

+

∫ T

s

Er(t)

∫
Ω

cε(x)|ut|m(x) dx dt,

(4.13)

where c′1, c′2 and c′ are positive constants independent of ε.
The next term of (4.6) can be estimated in a similar way to obtain

∣∣− µ2

∫ T

s

Er(t)

∫
Ω

u|z(x, 1, t)|m(x)−1 dx dt
∣∣

≤ ε
∫ T

s

Er(t)
[
c1(%p(x)(∇u))

m−
p+ + c2(%p(x)(∇u))

m+

p+
]
dt

+ c

∫ T

s

Er(t)

∫
Ω

cε(x)|z(x, 1, t)|m(x) dx dt

≤ c′ε
(

(E(0))
m−
p+ −1

+ (E(0))
m+

p+ −1
)∫ T

s

Er+1(t)dt

+ c

∫ T

s

Er(t)

∫
Ω

cε(x)|z(x, 1, t)|m(x) dx dt.

(4.14)



18 ANTONTSEV, FERREIRA, PIŞKIN, YÜKSEKKAYA, SHAHROUZI EJDE-2021/84

For the last term of (4.6), by using Lemma 4.2, we have

2

∫ T

s

Er(t)

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρdt

≤ 2

m−

∫ T

s

Er(t)

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x) dx dρdt

≤ − 2τ

m−

∫ T

s

Er(t)
d

dt

(∫ 1

0

∫
Ω

e−ρτξ(x)|z|m(x) dx dρ
)
dt

+
2

m−

∫ T

s

Er(t)

∫
Ω

ξ(x)|ut|m(x) dx dt

≤ − 2τ

m−
[Er(t)

∫ 1

0

∫
Ω

e−ρτξ(x)|z|m(x) dx dρ]t=Tt=s

+
2

m−

∫ T

s

Er(t)

∫
Ω

ξ(x)|ut|m(x) dx dt.

As ξ(x) is bounded, by using (4.2), we obtain

2

∫ T

s

Er(t)

∫ 1

0

∫
Ω

ξ(x)|z(x, ρ, t)|m(x)

m(x)
dx dρdt

≤ 2τe−τ

m−
Er(s)E(s) +

2c

m−
Er+1(T )

≤ 2τe−τ

m−
Er(0)E(s) +

2c

m−
Er(T )E(s) ≤ c∗E(s),

(4.15)

for some c∗ > 0.
By combining (4.6)-(4.15), we have∫ T

s

Er+1(t)dt ≤ cε
(

1 + (E(0))
m−
p+ −1

+ (E(0))
m+

p+ −1
)∫ T

s

Er+1(t)dt

+ cE(s) + c

∫ T

s

Er(t)

∫
Ω

cε(x)|z(x, 1, t)|m(x) dx dt.

(4.16)

We choose ε > 0 small such that

cε
(

1 + (E(0))
m−
p+ −1

+ (E(0))
m+

p+ −1
)
< 1.

Then, we have∫ T

s

Er+1(t)dt ≤ cE(s) + c

∫ T

s

Er(t)

∫
Ω

cε(x)|z(x, 1, t)|m(x) dx dt.

Once ε is fixed, cε(x) ≤M , since m(x) is bounded. So, we obtain∫ T

s

Er+1(t)dt ≤ cE(s) + cM

∫ T

s

Er(t)

∫
Ω

|z(x, 1, t)|m(x) dx dt

≤ cE(s)− C0M

∫ T

s

Er(t)E′(t)dt

≤ cE(s) +
C0M

r + 1
(Er+1(s)− Er+1(T ))

≤ cE(s).

(4.17)



EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 19

Thus, by taking T →∞, we obtain∫ ∞
s

E
m+

2 (t)dt ≤ cE(s).

Therefore, Komornik’s Lemma (with σ = r = m+/2 − 1) provides the desired
result. �

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Stanislav Antontsev

Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia

Email address: antontsevsn@mail.ru

Jorge Ferreira

Department of Exact Sciences, Federal Fluminense University, Rio de Janeiro, Brazil
Email address: ferreirajorge2012@gmail.com

Erhan Pişkin

Department of Mathematics, Dicle University, Diyarbakır, Turkey
Email address: episkin@dicle.edu.tr

Hazal Yüksekkaya
Department of Mathematics, Dicle University, Diyarbakır, Turkey

Email address: hazally.kaya@gmail.com

Mohammad Shahrouzi
Department of Mathematics, Jahrom University, P.O. Box 74137-66171 Jahrom, Iran

Email address: mshahrouzi@jahromu.ac.ir


	1. Introduction
	Equations with delay
	Equations without delay
	Variable exponent nonlinearity
	Equations with delay
	Equations without Delay

	2. Preliminaries
	3. Blow up
	4. Decay of solutions
	References