Blow-up and Bounds of Solutions for a Class of Semi-Linear Pseudo-
Parabolic Equations with -Laplacian Viscoelastic Term
TOUIL NADJI1, ABITA RAHMOUNE2
1Department of Technical Sciences,
University Amar Telidji,
Laghouat,
ALGERIA
2Department of Technical Sciences,
Laboratory of Pure and Applied Mathematics,
Laghouat University,
ALGERIA
Abstract: - In a bounded domain subject to Dirichlet boundary conditions, this paper discusses the phenomenon
of finite time blow-up of solutions for a particular class of evolution equations that affects the pseudo -
Laplacian viscoelastic term. We give the equation by:
Our findings show that, regardless of the initial energy and sizable initial values, the classical solutions of this
equation blow-up in finite time in two cases. Subject to certain conditions on p, q, g, and the initial given data,
we have established a new criterion for blow-up and provided lower and upper bounds on the solutions if blow-
up occurs.
Key-Words: - Pseudo-parabolic equation, -Laplacian viscoelastic term, memory term, blow-up time,
bounds of the blow-up time, critical exponents, variable nonlinearity.
Received: November 28, 2022. Revised: October 12, 2023. Accepted: November 8, 2023. Published: December 12, 2023.
1 Introduction
The pseudo-parabolic equation in the form of
is commonly used to describe various physical and
biological phenomena, such as the propagation of
nonlinear dispersive long waves, [1], population
aggregation, [2], heat conduction with two
temperatures, [3], and nonstationary processes in
semiconductors, [4], fluid dynamics,
electrorheological fluids, quantum mechanics
theory, [5], [6], [7], [8]. It originated from the study
of beams and heats. In reference, [9], the authors
provide a comprehensive overview of the system:
(1)
where if
if .
By exploiting the potential well method and the
comparison principle, they obtained global existence
and finite-time blow-up results for the solutions
with initial data at a high energy level. In recent
years, a great deal of attention has been given to the
study of mathematical nonlinear models with
variable-exponent nonlinearity. For instance,
modeling physical phenomena such as flows of
electrorheological fluids or fluids with temperature-
dependent viscosity, nonlinear viscoelasticity,
filtration processes through porous media and image
processing. More details on these problems can be
found in, [10], [11], [12]. Regarding parabolic
problems with nonlinearities of variable-exponent
type, many works have appeared. Let us mention
some of them. For instance, in the, [13], the author
studied the following problem:
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
157
Volume 18, 2023
(2)
where is a bounded domain with a smooth
boundary , and the source term is of the form:
(3)
with and the continuous
function are given functions satisfying
specific conditions. He established the local
existence of positive solutions and proved that
solutions with sufficiently large initial data blow up
in finite time. Parabolic problems with sources of
the form (3) appear in several branches of applied
mathematics and have been used to model chemical
reactions, heat transfer or population dynamics. The
nonlinear parabolic problems of the diffusion
equation with nonstandard -growth conditions
in the form:
(4)
for diverse choices of point functions ,
such as one might reasonably expect, this equation
arises naturally as the equation of motion in all sorts
of physical situations such as heat transfer, flows in
porous media, propagation of magnetic fields in
media with finite conductivities, and in chemical
kinetics or biochemical kinetics, to name just a few.
In the case where for
the choices of the function problem (4) occurs
in many mathematical models in fluid mechanics,
elasticity theory recently in image processing, [14],
[15], porous medium, [16], [17], the unidirectional
propagation of nonlinear, dispersive, long waves
and the aggregation of population, [18], and the
references therein. A series of papers related to
problems in the so-called rheological and
electrorheological fluids, which lead to spaces with
variable exponents, have appeared recently in, [18].
These topics are novel and attractive. It appears
from nonlinear elasticity theory, electrorheological
fluids, etc. These fluids possess the impressive
property that their viscosity depends on the electric
field in the fluids. For a general statement of the
underlying physics, [19], and for the mathematical
presents, [20]. The results detailed in those papers
were collected in the books, [21], [22]. Let be a
bounded domain in with a smooth
boundary . A class of pseudo parabolic
equations with -Laplacian viscoelastic terms
subject to homogenous Dirichlet boundary
conditions are written in the form of partial integro
differential equations by:
(5)
where and
are two measurable functions,
is a bounded domain, is Lipschitz
continuous, , with and
is a bounded function. The function
is a continuous function on
. This particular
model involves parabolic equations that are
nonlinear concerning the gradient of the solution
and have varying degrees of nonlinearity. The most
common case is the evolution -Laplace equation,
where the exponent is dependent on the external
electromagnetic field. For further information,
please refer to sources such as, [23], as well as their
respective references. The viscoelastic model has
become increasingly popular for analyzing the
dynamics of viscoelastic structures in recent years.
There is a common issue known as problem (5),
which appears in various mathematical models used
in engineering and physics. Over the last few
decades, equations containing viscoelastic terms
have received significant attention, and numerous
findings have been made regarding the existence,
uniqueness, and regularity of weak or classical
solutions. For more detail on this topic, we
recommend referring to source, [ 42 ], [25]. In a
recent investigation of a homogeneous Dirichlet
boundary value problem, the study, [25], found that
when is a constant, and and satisfy certain
conditions, a weak solution for (5) with positive
initial energy will blow-up in finite time. However,
the conditions on and are quite rigid. When
, the conventional Fourier law of heat flux
is typically substituted with the following equation
(6)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
158
Volume 18, 2023
where is the temperature, is the diffusion
coefficient, and the integral term represents the
memory effect in the material. Looking at it
mathematically, we expect the primary term in the
equation to have the most significant impact on the
integral term, allowing us to use the theory of
parabolic equation to solve problem (5). The
property of finite time blow-up is crucial for many
evolutionary equations. Exploring the blow-up of
solutions can be done through various methods.
Kaplan introduced the first eigenvalue method in
1963, Levine introduced the concavity method
during the 1970s, and the comparison method is
based on the comparison principle. Recently, for
constant and , the problem (5)
reduces to the following equation:
(7)
when is not equal to zero (), in, [26], the
author studied the blow-up results of (7), and found
a lower bound for the solution’s blow-up time if it
occurs. Additionally, he created a new blow-up
criterion and provided an upper bound for the
solution’s blow-up time based on certain conditions
involving , , and . Based on previous research,
we have found that the solution of problem (5)
blows up when given arbitrary positive initial
energy and appropriate large initial values, as long
as and . Additionally, we have
proven that the nonnegative solutions must blow-up
in a finite amount when given negative initial
energy.
2 Preliminaries
Let be a measurable function.
denotes the set of the real measurable functions on
such that:
The variable-exponent space equipped with
the Luxemburg-type norm:
is a Banach space. Throughout the paper, we use
to indicate the -norm for .
Next, we will define the variable-exponent Sobolev
space in the following manner
This space is a Banach space, which is defined by
its norm:
In addition, we have established that is
the closure of in It is known that
for the elements of the Poincaré
inequality holds,
(8)
and an equivalent norm of can be
defined by:
To state and prove our main result, we need to
establish the following hypotheses.
The measurable exponent functions and
provided meet the requirements.
where for a given measurable function on
assuming except that , and everifies the
log-Hölder continuity condition:
(9)
where satisfies
The memory kernel is a
function satisfying:
(10)
(11)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
159
Volume 18, 2023
3 Blow-up in Finite Time and Bounds
of Blow-Up Time
In this section, we will prove that the blow-up of
solutions to problem (5) with arbitrary positive
energy and suitable initial data, besides, we get a
new bounds for the blow-up time if the variable
exponents and the initial data satisfy some
conditions.
As it is well known that degenerate equations do
not have classical solutions, we give a precise
definition of the weak solution.
Definition 1 A function
is
called weak solution of problem (5), if and if only if
the equality
Holds for all
.
The proof of the first main result relies heavily on
the significance of these two lemmas:
Lemma 2 Suppose that a positive, twice-
differentiable function satisfies the inequality
where is some constant. If and
, then there exists
such
that tends to infinity as .
In the following, we prepare some lemmas needed
in the proof of the main results.
Lemma 3 (Sobolev-Poincarà inequality) If
satisfy For all , then the following
embedding
are continuous, and we get:
where the optimal constant of the Sobolev
embedding is denoted by , and the norm of
is represented by . The following
property is associated:
(12)
for any
Our main result is presented here.
We assert the local existence of a solution for
(5), even without proof. This can be gained through
the Faedo-Galerkin methods, in combination with
the fixed point theorem in Banach spaces.
Theorem 1 Assuming that both and are
valid. Problem (5) has a local solution, denoted as
, that satisfies
and
for .
3.1 First blow-up Result
One of the primary techniques for proving the blow-
up of solutions involves calculating the energy
function and using the concavity argument.
Let
(13)
for any positive such that
Theorem 2 Let us consider the assumptions of
Theorem 1 . If for any given such that
with
(14)
If and satisfy (9) and hold,
then the solution can exist for a finite
amount of time. However, if there exists a
such that
, this
means that the solution blows up in finite time in
-norm. and are given in (13).
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
160
Volume 18, 2023
Lemma 6 Under the assumptions of Theorem 5, the
corresponding energy to problem (5)
, is considered by
(15)
decreasing, that is
(16)
where
Proof. For a solution to problem (5), multiplying
Equation (5) by , integrating the result over ,
using the Green’s formula, we find:
(17)
A direct calculation of the last term on the left side
of (17) can views as follows:
(18)
Putting (18) in (17), we get:
Integrating the above identify over , we obtain
(19)
Proving Theorem 2 relies on the significance of the
following lemma 7.
Lemma 7 Under the assumptions of Theorem 2, the
solution of problem (5) satisfies the following
inequalities
(20)
and
(21)
where and as in (13).
Proof. Set
Integrating by parts, and using Eq. (5), we obtain
(22)
Applying Young and Hölder inequalities, the
second term in the right-hand side of (22) can be
estimated as follows
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
161
Volume 18, 2023
=
+
(23)
for any . Using (22) and (23), we conclude
(24)
by , it is clear to check that
(25)
which connect with (24) give
(26)
Inequality (26) confirms that
(27)
because
is positive, we have
By solving nonhomogeneous ordinary differential
equation, we can obtain
(28)
Substituting (28) into (27), it follows that
Here is the proof for the first main result:
We point out that the the main method employed in
this proof is based on the concavity technique,
taking into account the idea used in, [26], Theorem
2.2.
Proof of Theorem 2. We first assume that exists
in the classical sense on i.e.,
(The interval of existence of is unbounded, or is
defined in the whole interval ), and then
show that this leads to a contradiction. We select an
of the following form for
Then
(29)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
162
Volume 18, 2023
we distinguish two cases:
1. Case.1 , for all . Through (14)
we can choose as such
(30)
By adding , and
making us (21), (29), and (25) it yields
(31)
Let be an auxiliary function defined as
where is taken small enough such that
and large enough (if needed), so that
(32)
Therefore,
(33)
(34)
From (33), we obtain
(35)
where , then
(36)
Noting that
Therefore,
Using Holder and Young’s inequalities gives
(37)
From (34) and (36), we get
(38)
Now, from (38), (35), (31) and (37), the following
estimates ensured:
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
163
Volume 18, 2023
Recalling the values of and and taking into
account that , it result
Now, in this case we show that cannot be
infinite, and therefore there is no weak solution all
the time.
From Lemma 2, it follows that there exists a
such that as , where
Since is continuous with respect to , we
conclude that there exists a such that
.
Hence, discontinuing at some finite time
, that is to means, not exist for all time, i.e.
blows up at a time , which will lead to the
nonexistence result stated in the theorem, then
blows up at time in -norm, which
contradicts. Hence, for the data satisfies (14) any
solution possesses finite explosion time.
2. Case 2. Assume that there exists such that
, . We define
, so . By the
fact that is deceasing in , we can get:
(39)
Define , then we have as in
(26)
Then, it follows that:
(40)
For satisfy (H2), the following embedding
hold. Therefore, from , we obtain
that:
(41)
Using (12), and (41) from (40), we can get
(42)
where is a best embedding constant and
By , so . We can conclude
that
that is
(43)
Using (43) and (44), we have
(44)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
164
Volume 18, 2023
where
. Using (44) we
can derive the following result
The above inequality implies that blows up at
finite time
, which is a contraction.
3.2 Second Blow-Up Result
In this subsection, we will address problem (5) by
establishing a blow-up criterion and obtaining
bounds for the blow-up time of weak solutions
through the use of differential inequality techniques.
For our result, we need to consider the following
auxiliary functions.
(45)
and for (a small positive number) and (a precise
positive constant) to be picked later;
(46)
and
(47)
Let and be positive auxiliary
constants satisfying
(48)
The second result of the blow-up is as follows.
Theorem 3 Assuming that , , and satisfy
conditions with . Then the local
solution of problem (5) under boundary conditions
satisfying
blows up in
finite time , which provide the following estimates
where
(49)
and are defined in (80), (78), (70) and
(73), respectively.
Our desired result depends heavily on the following
lemma 9.
Lemma 9 Let be defined by
(50)
then has the following properties:
(i) is increasing for and
decreasing for ,
(ii)
and ,
(iii) ,
where is given in (46), and are given in
(49).
Proof. is continuous and differentiable in
which means that
(51)
Then (i) follows. Since , we have
. A simple computation yields to
. Then (ii) holds valid. By Lemma 3
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
165
Volume 18, 2023
Using , (15) and Lemma 3, we have
Then (iii) holds true.
Lemma 10 Assuming the conditions in Theorem 8
are fulfilled, there is a positive constant
such that
(52)
(53)
where , and are given in (49).
Proof. Since and is a continuous
function, there exist and with
such that which join with
Lemma 9 give:
(54)
From Lemma 9(i), we infer that:
(55)
so (51) holds for .
Now we prove (53), we proceed by contradiction
and assume there exist such that ,
then we distinguish two cases,
Case 1. If , we know
through Lemma 9 and (52) that
which contradicts Lemma 9(iii).
Case 2. If , then
. Setting
, then is a
continuous function, and by applying
(56), . Hence, there exists such
that , that means
, which
signifies
This contradicts to Lemma 9(iii), hence (51)
follows. By (15), we have
which give
then the second inequality in (54) holds. Let
(56)
The following lemma hold
Lemma 11 Under the assumptions of Theorem 3, if
the functional defined in (57)
satisfies the following estimates:
(57)
Proof. Lemma 3 ensure that is nondecreasing
in . Thus
(58)
By (49) and Lemma 10, we have
for all , which gives
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
166
Volume 18, 2023
(59)
(57) follows from (58) and (59).
Lemma 12 Assume that the conditions in Theorem 3
hold, then there exists a positive constant such
that
(60)
for all
Proof. By Lemma 10 and , we have
which combining with (49) imply
(61)
combining (57), (61) and the definition of , we
have
(62)
Then the desired result, with
The proof of Theorem 3 is shown below, based on
the lemmas presented above
Proof of Theorem 3 .
Case 1. If , then by
differentiating (47), we get
Integrating by parts on recalling Eq (5), we
obtain
P
utting (13) in (17), we get
(63)
Employing Young inequality, we can obtain:
(64)
By substituting (65) in (64) and then applying
(15), we can choose such that
, we
can deduce
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
167
Volume 18, 2023
(65)
By combining (11) and (66), we obtain:
(66)
Where
obviously
(67)
Using (67) in (66) and rewriting as
,with
produce
At this point, we choose that is sufficiently large
so that
After determining a fixed value for , we select a
small enough to meet the necessary conditions
(68)
Then there is a constant satisfying
(69)
and
(70)
which combining with (69) infer
Choosing such that
, recalling
Lemma 10 and then, we have
(71)
By utilizing Holder’s and Young’s inequalities,
and keeping in mind the embedding
, it can be observed that:
(72)
where:
Let be a positive constant such:
(73)
Using (47), (72), (73), and Cauchy-Schwarz’s
inequality,
(74)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
168
Volume 18, 2023
We join (72) and (73) with (71), it result
(75)
After integrating (65) over the interval , we
can deduce that:
(76)
Consequently, blows up in a finite time
Since , (77) shows that ,
where
This ends the proof.
Case 2.If , we can use Lemma 12
by setting to obtain a result similar
to Lemma 12. Before this, we had
and
. By taking
in (47) and using the same approach as in Case1, we
can reach our desired outcome.
We still need to determine an upper bound of
the blowing-up time, we can calculate it as follows;
Using (15), (16) and Lemma 3, the derivative of
(48) give:
(77)
To estimate the term on the right-hand side of
the inequality above, we need to analyze the
following three scenarios
Case.1. . The inequality embedding
has led us to
Case.2.
. Using Hölder’s and
embedding inequalities, we have
Case.3.
. Through the
simulation of Case 2, we have obtained the
following results.
Hence, we get :
(78)
Where equals
for the cases
mentioned above.
Using (48) and definition, we can see that:
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
169
Volume 18, 2023
(79)
where
. Joining (78)-(79),
taking into account that
which means ; we get
Where
C=
.
(80)
By the definition of ;
we obtain that
The proof is complete.
Acknowledgements:
The authors would like to thank the anonymous
referee(s) and the handling editor(s) for their kind
comments.
4 General Comments and Issues
This paper is devoted to studying new class of
mixed pseudo parabolic p(.)-Laplacian type
equation with viscoelastic term on a bounded and
regular domain (5), which that equations appear in
dynamics of viscoelastic structures, besides the most
common case is the evolution p(.)-Laplace equation,
where the exponent p(.) is dependent on the external
electromagnetic field.
We provide a blow-up threshold resulting in a
finite time of solutions, yielding a new blow-up
criterion. The upper bound estimate of the blow-up
time is also derived. We show that blow-up may
occur under appropriate smallness conditions on the
initial datum, in which case we also establish a
lower bound estimate.
The significance of this study is that it will
determine a new criterion and upper and lower
bounds estimate of the blow-up time, which have
not stayed vocalised in either case for the value of
q(.) (constant or variable) for this type of equation.
References:
[1] Benjamin T.B, Bona, J.L, Mahony J.J. Model
equations for long waves in nonlinear
dispersive systems. Philos. Trans. R. Soc.
Lond.Ser. A. Math. Phys. Sci., vol. 272,
pp.47-78, 1972.
[2] Vctor P. Effect of aggregation on population
recovery modeled by a forward-backward
pseudoparabolic equation. Trans. Am. Math.
Soc., vol.356(7), pp.2739-2756, 2004.
[3] Aripov M, Mukimov A, Mirzayev B. To
Asymptotic of the Solution of the Heat
Conduction Problem with Double
Nonlinearity with Absorption at a Critical
Parameter. Mathematics and Statistics,
vol.7(5), pp.205-217, 2019.
[4] Korpusov M.O, Sveshnikov A.G. Three-
dimensional nonlinear evolutionary
pseudoparabolic equations in mathematical
physics. Zh. Vych. Mat. Fiz., vol.43(12),
pp.1835-1869, 2003.
[5] Abita R. Logarithmic Wave Equation
Involving Variable-exponent Nonlinearities:
well posedness and Blow-up. WSEAS
Transactions on Mathematics, vol.21, pp.825-
837, 2022,
https://doi.org/10.37394/23206.2022.21.94.
[6] Soufiane B, Abita R. The Exponential Growth
of Solution, Upper and Lower Bounds for the
Blow-Up Time for a Viscoelastic Wave
Equation with Variable- Exponent
Nonlinearities. WSEAS Transactions on
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
170
Volume 18, 2023
Mathematics, vol.22, pp.451-465, 2023,
https://doi.org/10.37394/23206.2023.22.51.
[7] Abita R. Blow-up phenomenon for a semi
linear pseudo-parabolic equation involving
variable source. Applicable Analysis, 2021.
[8] Abu Zaytoon M.S, Hamdan M.H. Fluid
Mechanics at the Interface between a Variable
Viscosity Fluid Layer and a Variable
Permeability Porous Medium, WSEAS
Transactions on Heat and Mass Transfer,
vol.16, pp.159-169, 2021,
https://doi.org/10.37394/232012.2021.16.19.
[9] Xu R, Su J. Global existence and finite time
blow-up for a class of semilinear pseudo-
parabolic equations, J. Funct. Anal.,
vol.264(12), pp.2732-2763, 2013.
[10] Aboulaich R, Meskine D, Souissi A. New
diffusion models in image processing.
Comput. Math. Appl., vol.56(4), pp.874-882,
2008.
[11] Lian S, Gao W, Cao C, Yuan H. Study of the
solutions to a model porous medium equation
with variable exponent of nonlinearity. J.
Math. Anal. Appl., vol.342(1), pp.27-38, 2008.
[12] Antontsev S, Shmarev S. Blow-up of
solutions to parabolic equations with
nonstandard growth conditions. J. Comput.
Appl. Math., vol.234, pp.2633-2645, 2010.
[13] Pinasco J.P, Blow-up for parabolic and
hyperbolic problems with variable exponents.
Nonlinear Anal. TMA. vol.71, pp.1049–1058,
2009.
[14] S. Lian, W. Gao, C. Cao, H. Yuan, Study of
the solutions to a model porous medium
equation with variable exponent of
nonlinearity, J. Math. Anal. Appl., vol.342 (1)
, pp.27–38, 2008.
[15] Y. Chen, S. Levine, M. Rao, Variable
exponent, linear growth functionals in image
restoration, SIAM J. Appl. Math., vol.66,
pp.1383–1406, 2006.
[16] Tarek G. Emam, Boundary Layer Flow over a
Vertical Cylinder Embedded in a Porous
Medium Moving with non Linear Velocity,
WSEAS Transactions on Fluid Mechanics,
vol. 16, pp. 32-36, 2021.
[17] Songzhe L, Gao W, Cao C. Study of the
solutions to a model porousmedium equation
with variable exponent of nonlinearity. J Math
Anal Appl., vol.2008, 342, pp.27–38
[18] Diening L, Růžička, M. Calderón-Zygmund
operators on generalized Lebesgue spaces
and problems related to fluid
dynamics, J. Reine Angew. Math., vol.563,
pp.197-220, 2003.
[19] Gawade S.S, Jadhav A.A. A Review On
Electrorheological (ER) Fluids And Its
Applications. International Journal of
Engineering Research & Technology (IJERT),
Vol. 1, Issue 10, December 2012.
[20] Acerbi E, Mingione G. Regularity results for
electrorheological fluids, the stationary case,
C. R. Acad. Sci. Paris, vol.334, pp.817–822,
2002.
[21] Růžička M. Electrorheological Fluids,
Modeling and Mathematical Theory, Lecture
Notes in Mathematics, vol.1748, Springer,
2000.
[22] Diening L, Hästo P, Harjulehto P, Růžička M.
Lebesgue and Sobolev Spaces with Variable
Exponents, Springer Lecture Notes, vol. 2017,
Springer-Verlag, Berlin, 2011.
[23] Acerbi E, Mingione G, Seregin G.A.
Regularity results for parabolic systems
related to a class of non Newtonian fluids,
Ann. Inst. H. Poincaré Anal. Non Linéaire
vol.21(1), pp.25-60, 2004.
[24] Yin H.M. Weak and classical solutions of
some Volterra integro-differential equations.
Comm. Partial Differ. Equ., vol.17(7-8),
pp.1369-1385, 2019.
[25] Wu X, Yang X, Zhao Y. The Blow-Up of
Solutions for a Class of Semi-linear Equations
with -Laplacian Viscoelastic Term Under
Positive Initial Energy. Mediterr. J. Math.
vol.20, 272, 2023.
[26] Tian S.Y. Bounds for blow-up time in a
semilinear parabolic problem with viscoelastic
term. Computers and Mathematics with
Applications, vol.74(4), pp.736 743, 2017.
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
171
Volume 18, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Touil Nadji and Abita Rahmoune wrote the main
manuscript text. All authors reviewed the
manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.16
Touil Nadji, Abita Rahmoune
E-ISSN: 2224-347X
172
Volume 18, 2023
