CONCAVITY METHOD: A CONCISE SURVEY
Lakshmipriya Narayanan
Assistant Professor, Department of Mathematics, NSS College, Nemmara, Palakkad, Kerala , India
E-mail:lakshmipriya993@gmail.com (N.Lakshmipriya)
ORCID:
Gnanavel Soundararajan
Assistant Professor, Department of Mathematics,Central University of Kerala, Kerala - 671 320, India.
E-mail:gnanavel.math.bu@gmail.com
ORCID:
Reception: 30/08/2022 Acceptance: 14/09/2022 Publication: 29/12/2022
Suggested citation:
Lakshmipriya Narayanan and Gnanavel Soundararajan (2022). Concavity Method: A concise survey 3C Empresa.
Investigación y pensamiento crítico,11 (2), 94-102. https://doi.org/10.17993/3cemp.2022.110250.94-102
https://doi.org/10.17993/3cemp.2022.110250.94-102
ABSTRACT
This short review article discusses the concavity method, one of the most effective ways to deal with
parabolic equations with unbounded solutions in finite time. If the solution ceases to exist for some time,
we say it blows up. The solution or some of its derivatives become singular depending on the equation.
We focus on situations where the solution becomes unbounded in finite time, and our objective is to
review some of the key blowup theory papers utilising the concavity method.
KEYWORDS
Parabolic equations, Concavity method, weak solutions, blowup, variable exponent spaces,
p
(
x
)-Laplacian
operator.
https://doi.org/10.17993/3cemp.2022.110250.94-102
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
94
CONCAVITY METHOD: A CONCISE SURVEY
Lakshmipriya Narayanan
Assistant Professor, Department of Mathematics, NSS College, Nemmara, Palakkad, Kerala , India
E-mail:lakshmipriya993@gmail.com (N.Lakshmipriya)
ORCID:
Gnanavel Soundararajan
Assistant Professor, Department of Mathematics,Central University of Kerala, Kerala - 671 320, India.
E-mail:gnanavel.math.bu@gmail.com
ORCID:
Reception: 30/08/2022 Acceptance: 14/09/2022 Publication: 29/12/2022
Suggested citation:
Lakshmipriya Narayanan and Gnanavel Soundararajan (2022). Concavity Method: A concise survey 3C Empresa.
Investigación y pensamiento crítico,11 (2), 94-102. https://doi.org/10.17993/3cemp.2022.110250.94-102
https://doi.org/10.17993/3cemp.2022.110250.94-102
ABSTRACT
This short review article discusses the concavity method, one of the most effective ways to deal with
parabolic equations with unbounded solutions in finite time. If the solution ceases to exist for some time,
we say it blows up. The solution or some of its derivatives become singular depending on the equation.
We focus on situations where the solution becomes unbounded in finite time, and our objective is to
review some of the key blowup theory papers utilising the concavity method.
KEYWORDS
Parabolic equations, Concavity method, weak solutions, blowup, variable exponent spaces,
p
(
x
)-Laplacian
operator.
https://doi.org/10.17993/3cemp.2022.110250.94-102
95
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
1 INTRODUCTION
The idea of unbounded solutions, known as blowup theory, holds a specific place in the study of nonlinear
equations. Blow-up is a phenomenon where solutions of differential equations cease to exist because of
the infinite growth of the variables describing the evolution processes. Before the successful calculation of
mathematical methods to deal with the unboundedness of solutions, the physical significance of blowup
was understood because it occurs in processes such as heat conduction, combustion, volcanic eruption,
gas dynamics, etc. In addition, some of the current notable works in this field include the blowing up of
cancer cells, nuclear blowup, electrical blow up, laser fusion, blow up in pandemic simulations, etc.
The solutions to a problem can be unbounded at a finite or infinite time. In this work, we only deal
with papers studying finite time blowup of solutions using the concavity method. Finite time blowup is
a sufficient condition for the nonexistence of global solutions, since the solutions grow without bound
in finite time intervals. There are nonlinear PDEs, with local solutions for time
t<T
⋆
, which blowup
at a finite time T⋆. Thus, we can give a formal definition to finite time blowup as, if T⋆<∞and
lim sup
t→T⋆∥z(t)∥=∞,(1)
then we say the solution zof a given problem blows up at a finite time T⋆.
From 1960s onward, the following equations,
zt=∆z+|z|p−1z, (2)
and
zt=∆z+λez,λ>0(3)
have become fundamental models for blow-up study [4,20,38,52]. Fujita, Hayakawa [16,17,22], Kaplan [24]
and Friedman [18] studied these problems and obtained several critical results on blowup of solutions.
These fundamental results from the 1960s initiated deep research of blowup solutions for various
nonlinear evolution PDEs in the next decade [2,3,25,28
–
30,48,49]. Some of the fascinating problems in
this area are finding whether the solutions blow up at a finite time, obtaining lower and upper bounds
for blowup time and getting the blowup rate. Challenges in this theoretical study have attracted many
scientists, and useful techniques have been developed to deal with certain nonlinear parabolic problems.
They include eigenfunction method, explicit inequality method, logarithmic convexity method, Fourier
coefficient method, comparison method, concavity method and differential inequality techniques. We
solely concentrate on the concavity approach out of all these methods.
Concavity Method
The concavity method was introduced by H. A. Levine [28] and proved very successful with a wide
range of applications. The method uses the concavity of an auxiliary functional, say
M−η
, with
η>
0
.
Here M(t)is a positive energy functional of the solution to a PDE. Since M−ηis concave, we get
d2M(t)
dt2≤0.(4)
Hence by integrating the above inequality, one can arrive at
Mη(t)≥Mη+1(t)
M(0) −tηM′(0),(5)
this implies if
M′
(0)
>
0, then
Mη
(
t
)is bounded below by a function which becomes unbounded at a
finite time. To apply the concavity method, we need to show that
M
(
t
)obeys the inequality (4). Hence
we have the following inequality,
d2M−η(t)
dt2=−ηM−η−2(t)[M(t)M′′(t)−(1 + η)(M′(t))2].(6)
https://doi.org/10.17993/3cemp.2022.110250.94-102
Now, for M(t)>0we get
M(t)M′′(t)−(1 + η)(M′(t))2>0.(7)
Hence this inequality (7) is a sufficient condition for the existence of blowup. Moreover, the inequality
(5) helps to derive an upper bound for blowup time.
In [28], Levine studied an abstract parabolic equation
Pdz
dt =−A(t)z+f(z(t)),t∈[0,∞)
z(0) = z0,
(8)
where
P
and
A
are positive linear operators defined on a dense subdomain
D
of a real or complex
Hilbert space. In which he obtained blowup results under the conditions
2(α+ 1)F(x)≤(x, f(x)),F(z0(x)) >1
2(z0(x), Az0(x)),(9)
for every
x∈D
, where
F
(
x
)=
1
0
(
f
(
ρx
)
,x
)
dρ.
This study has been acknowledged as an innovative
and elegant way, known as "the concavity method,"for providing criteria for the blowup.
Later, Philippin and Proytcheva [39] transformed the procedure from its abstract form into a concrete
form and used it to solve the same equation with
P
=2. Since then, the concavity method has been
used for several variations of the equations (8) or other equations to get the blowup solutions. Levine,
together with Payne [29,30], established nonexistence theorems for the heat equation with nonlinear
boundary conditions and for the porous medium equation backwards in time using the concavity
argument. In [37] Payne, Philippin and Piro dealt with the blowup of the solutions to a semilinear
second-order parabolic equation with nonlinear boundary conditions. They demonstrated that blowup
would occur at some finite time under specific conditions on the nonlinearities and data. Junning [23]
investigated the following initial boundary value problem
zt=div(|∇z|(p−2)∇z)+f(∇z, z,x,t),(x, t)∈Ω×(0,t)
z(x, 0) = z0(x),x∈Ω
z(x, t)=0,x∈∂Ω
(10)
and obtained results on the existence and nonexistence of solutions under specific conditions. Erdem [15]
estabilished sufficient conditions for the global nonexistence of solutions of a second order quasilinear
parabolic equations.
In 2017, the finiteness of the time for blow up of a semilinear parabolic equation with Dirichlet
boundary conditions discussed by Chung and Choi [9]. They continued the study and obtained a new
condition for the concavity method of blowup solutions to
p
-Laplacian parabolic equations [10]. The
authors then developed a condition for blowup solutions to discrete
p
-Laplacian parabolic equations
under the mixed boundary conditions on networks with Hwang [11]. For a reaction-diffusion equation
with a generalised Lewis function and nonlinear exponential growth, Dai and Zhang established results
on nonexistence of solutions using concavity method [12]. In [47], the authors took into account the
results of finite time blowup for a parabolic equation coupled with a superlinear source term and a
local linear boundary dissipation. The adequate conditions for the solutions to blow up in a finite
time were deduced using the concavity argument. The existence of finite time blowup solutions with
arbitrarily high initial energy and the upper and lower bound of the blowup time were specifically
obtained. Galaktionov [19] provide the sufficient conditions for the unboundedness of the solutions of
boundary value problems for a class of quasilinear equations and systems of parabolic type, describing
the propagation of heat in media with nonlinear heat conduction and volume liberation of energy.
Blow up of solutions for a semilinear heat equation with a viscoelastic term was studied in [21] for
nonlinear flux on the boundary. The authors obtained blowup results for initial negative energy by
https://doi.org/10.17993/3cemp.2022.110250.94-102
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
96
1 INTRODUCTION
The idea of unbounded solutions, known as blowup theory, holds a specific place in the study of nonlinear
equations. Blow-up is a phenomenon where solutions of differential equations cease to exist because of
the infinite growth of the variables describing the evolution processes. Before the successful calculation of
mathematical methods to deal with the unboundedness of solutions, the physical significance of blowup
was understood because it occurs in processes such as heat conduction, combustion, volcanic eruption,
gas dynamics, etc. In addition, some of the current notable works in this field include the blowing up of
cancer cells, nuclear blowup, electrical blow up, laser fusion, blow up in pandemic simulations, etc.
The solutions to a problem can be unbounded at a finite or infinite time. In this work, we only deal
with papers studying finite time blowup of solutions using the concavity method. Finite time blowup is
a sufficient condition for the nonexistence of global solutions, since the solutions grow without bound
in finite time intervals. There are nonlinear PDEs, with local solutions for time
t<T
⋆
, which blowup
at a finite time T⋆. Thus, we can give a formal definition to finite time blowup as, if T⋆<∞and
lim sup
t→T⋆∥z(t)∥=∞,(1)
then we say the solution zof a given problem blows up at a finite time T⋆.
From 1960s onward, the following equations,
zt=∆z+|z|p−1z, (2)
and
zt=∆z+λez,λ>0(3)
have become fundamental models for blow-up study [4,20,38,52]. Fujita, Hayakawa [16,17,22], Kaplan [24]
and Friedman [18] studied these problems and obtained several critical results on blowup of solutions.
These fundamental results from the 1960s initiated deep research of blowup solutions for various
nonlinear evolution PDEs in the next decade [2,3,25,28
–
30,48,49]. Some of the fascinating problems in
this area are finding whether the solutions blow up at a finite time, obtaining lower and upper bounds
for blowup time and getting the blowup rate. Challenges in this theoretical study have attracted many
scientists, and useful techniques have been developed to deal with certain nonlinear parabolic problems.
They include eigenfunction method, explicit inequality method, logarithmic convexity method, Fourier
coefficient method, comparison method, concavity method and differential inequality techniques. We
solely concentrate on the concavity approach out of all these methods.
Concavity Method
The concavity method was introduced by H. A. Levine [28] and proved very successful with a wide
range of applications. The method uses the concavity of an auxiliary functional, say
M−η
, with
η>
0
.
Here M(t)is a positive energy functional of the solution to a PDE. Since M−ηis concave, we get
d2M(t)
dt2≤0.(4)
Hence by integrating the above inequality, one can arrive at
Mη(t)≥Mη+1(t)
M(0) −tηM′(0),(5)
this implies if
M′
(0)
>
0, then
Mη
(
t
)is bounded below by a function which becomes unbounded at a
finite time. To apply the concavity method, we need to show that
M
(
t
)obeys the inequality (4). Hence
we have the following inequality,
d2M−η(t)
dt2=−ηM−η−2(t)[M(t)M′′(t)−(1 + η)(M′(t))2].(6)
https://doi.org/10.17993/3cemp.2022.110250.94-102
Now, for M(t)>0we get
M(t)M′′(t)−(1 + η)(M′(t))2>0.(7)
Hence this inequality (7) is a sufficient condition for the existence of blowup. Moreover, the inequality
(5) helps to derive an upper bound for blowup time.
In [28], Levine studied an abstract parabolic equation
Pdz
dt =−A(t)z+f(z(t)),t∈[0,∞)
z(0) = z0,
(8)
where
P
and
A
are positive linear operators defined on a dense subdomain
D
of a real or complex
Hilbert space. In which he obtained blowup results under the conditions
2(α+ 1)F(x)≤(x, f(x)),F(z0(x)) >1
2(z0(x), Az0(x)),(9)
for every
x∈D
, where
F
(
x
)=
1
0
(
f
(
ρx
)
,x
)
dρ.
This study has been acknowledged as an innovative
and elegant way, known as "the concavity method,"for providing criteria for the blowup.
Later, Philippin and Proytcheva [39] transformed the procedure from its abstract form into a concrete
form and used it to solve the same equation with
P
=2. Since then, the concavity method has been
used for several variations of the equations (8) or other equations to get the blowup solutions. Levine,
together with Payne [29,30], established nonexistence theorems for the heat equation with nonlinear
boundary conditions and for the porous medium equation backwards in time using the concavity
argument. In [37] Payne, Philippin and Piro dealt with the blowup of the solutions to a semilinear
second-order parabolic equation with nonlinear boundary conditions. They demonstrated that blowup
would occur at some finite time under specific conditions on the nonlinearities and data. Junning [23]
investigated the following initial boundary value problem
zt=div(|∇z|(p−2)∇z)+f(∇z, z,x,t),(x, t)∈Ω×(0,t)
z(x, 0) = z0(x),x∈Ω
z(x, t)=0,x∈∂Ω
(10)
and obtained results on the existence and nonexistence of solutions under specific conditions. Erdem [15]
estabilished sufficient conditions for the global nonexistence of solutions of a second order quasilinear
parabolic equations.
In 2017, the finiteness of the time for blow up of a semilinear parabolic equation with Dirichlet
boundary conditions discussed by Chung and Choi [9]. They continued the study and obtained a new
condition for the concavity method of blowup solutions to
p
-Laplacian parabolic equations [10]. The
authors then developed a condition for blowup solutions to discrete
p
-Laplacian parabolic equations
under the mixed boundary conditions on networks with Hwang [11]. For a reaction-diffusion equation
with a generalised Lewis function and nonlinear exponential growth, Dai and Zhang established results
on nonexistence of solutions using concavity method [12]. In [47], the authors took into account the
results of finite time blowup for a parabolic equation coupled with a superlinear source term and a
local linear boundary dissipation. The adequate conditions for the solutions to blow up in a finite
time were deduced using the concavity argument. The existence of finite time blowup solutions with
arbitrarily high initial energy and the upper and lower bound of the blowup time were specifically
obtained. Galaktionov [19] provide the sufficient conditions for the unboundedness of the solutions of
boundary value problems for a class of quasilinear equations and systems of parabolic type, describing
the propagation of heat in media with nonlinear heat conduction and volume liberation of energy.
Blow up of solutions for a semilinear heat equation with a viscoelastic term was studied in [21] for
nonlinear flux on the boundary. The authors obtained blowup results for initial negative energy by
https://doi.org/10.17993/3cemp.2022.110250.94-102
97
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
employing the concavity method. Li and Han [32] then improved these results for positive initial energy
with the support of the potential well method. Sun et. al. [46] investigated an initial boundary value
problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear
source term and obtained results on finite time blowup of solutions under suitable assumptions on the
initial data and the relaxation function.
The existence of solutions for a pseudo-parabolic equation with memory was deduced by Di and
Shang [13] using Galerkin method and potential well theory. Then using the concavity method, derived
finite time blowup results for both negative and non-negative initial energy. Sun et. al. [45] studied the
problem and came up with existence and finite time blow-up results using Galerkin method, concavity
argument and potential well theory by making a slight change in the source term. They derived an upper
bound for the blowup time and obtained the existence of solutions which blow up in finite time with
arbitrary initial energy conditions. Di and Shang [14] worked on a class of nonlinear pseudo-parabolic
equations with a memory term under Dirichlet boundary condition. They proved a finite time blowup
result for specific initial energy and relaxation function. In 2019, Messaoudi and Talahmeh [33] studied
a semilinear viscoelastic pseudo-parabolic problem with variable exponent and demonstrated any weak
solution with initial data at arbitrary energy level blows up in finite time. Furthermore, they obtained
an upper bound for the blowup time using the concavity method. Chen and Xu [5] studied a finitely
degenerate semilinear pseudo-parabolic problem and showed the global existence and blowup in finite
time of solutions with sub-critical and critical initial energy. The asymptotic behaviour of the global
solutions and a lower bound for blowup time of the local solution are also obtained. A pseudo-parabolic
equation with variable exponents under initial and Dirichlet boundary value conditions is the subject of
study in [53]. In [53], Zhou et. al. established the global existence and blowup results of weak solutions
with arbitrarily high initial energy.
Existence and blow-up studies of the following
p
(
x
)-Laplacian parabolic equation with memory was
studied by Lakshmipriya and Gnanavel [34],
zt−∆z−µ∇(|∇z|p(x)−2∇z)+t
0
h(t−τ)∆z(x, τ)dτ =β|z|b(x)−2z, x ∈Ω,t≥0
z(x, t)=0
,x
∈∂Ω,t≥0
z(x, 0) = z0(x)
,x
∈Ω
(11)
where Ω
⊂RN,
(
N≥
1
,N
= 2) is a bounded domain with smooth boundary
∂
Ω.
β>
0
,µ ≥
0are
constants. The authors established the existence and finite time blow up of weak solutions of the
problem. Further, obtained upper and lower bounds for the blowup time of solutions, by employing the
concavity method and differential inequality technique, respectively. In [35], the authors analysed and
interpreted unbounded solutions of a viscoelastic
p
(
x
)
−
Laplacian parabolic equation with logarithmic
nonlinearity. Here the problem was considered for initial data corresponding to the sub-critical initial
energy. In this attempt, Lakshmipriya and Gnanavel obtained the local existence of solutions on an
interval [0
,T
). Moreover, it extracted an upper bound for the blowup time by applying the concavity
method.
The paper [27] deals with the existence and blowup of weak solutions of the following pseudo-parabolic
equation with logarithmnonlinearityity
wt−∆wt−div(|∇w|p(x)−2∇w)=|w|s(x)−2w+|w|h−2wlog|w|,(x, t)∈Ω×(0,∞)
w(x, t)=0,(x, t)∈∂Ω×[0,∞)
w(x, 0) = w0(x)
,x
∈Ω
(12)
where Ω
⊂Rn
(
n≥
1) is a bounded domain with smooth boundary
∂
Ω. The model consider is used to
describe the non-stationary process in semiconductors in the presence of sources; the first two terms
represent the free electron density rate and logarithmic and polynominonlinearityity stands for the
source of free electron current [26]. Lakshmipriya and Gnanavel [36] analysed the blowup of solutions
https://doi.org/10.17993/3cemp.2022.110250.94-102
to the following problem
zt(x, t)=∆
p(x)z(x, t)+g(z(x, t)),(x, t)∈Ω×(0,∞)
z(x, t)=0,(x, t)∈∂Ω×[0,∞)
z(x, 0) = z0(x)≥0,x∈Ω
(13)
where Ω
⊂RN
(
N≥
1) is a bounded domain with smooth boundary
∂
Ω. The model is involved in
image processing, elastic mechanics and electro-rheological fluids [1,40, 42]. The authors considered a
condition on the nonlinear function g(z)given by,
ςz
0
g(s)ds ≤zg(z)+ηzb(x)+µ, z > 0.
Obtained results on blowup and established an upper bound for the blowup time with the help of the
concavity method.
Now, we consider the most recent works involving the Concavity method. Ruzhansky et.al. [41]
proved a global existence and blowup of the positive solutions to the initial-boundary value problem of
the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified
Lie groups based on the concavity argument and the Poincare inequality. A nonlinear porous medium
equation under a new nonlinearity condition is considered in a bounded domain by Sabitbek and
Torebek [43]. They presented the blowup of the positive solution to the considered problem for the
negative initial energy. For the subcritical and critical initial energy cases, obtained a global existence,
asymptotic behaviour and blowup phenomena in a finite time of the positive solution to the nonlinear
porous medium equation. In [31], Li and Fang are concerned with the blowup phenomena for a semilinear
pseudo-parabolic equation with general nonlinearity under the null Dirichlet boundary condition. When
the nonlinearity satisfies a new structural condition, they obtain some new blowup criteria with different
initial energy levels. They derived the growth estimations and life span of blowup solutions.
The concavity method is also used to understand the blowup behaviour of system of nonlinear
parabolic equations [8]. Apart from parabolic and pseudo-parabolic equations, the method plays a
significant role in studying unbounded solutions of hyperbolic equations and systems. Some of the latest
works are as follows [6,7,44,50,51]. Hence the Concavity method is a simple and powerful tool to use
in the blow up studies of solutions to differential equation problems.
REFERENCES
[1]
Antontsev,S. N. and Shmarev, S. I.(2005). A model porous medium equation with variable
exponent nonlinearityity: Existence uniqueness and localisation properties of solutions, Nonlinear
Anal., 515-545.
[2]
Aronson, D. G. and Weinberger, H. F. Multidimensional nonlinear diffusion arising in
population genetics.Adv. Math. 30(1978), 33-76.
[3]
Ball, J. M. (1977), Remarks on blow-up and nonexistence theorems for nonlinear evolution
equations, Q. J. Math.28, 473-486.
[4]
Bebernes, J., and Eberly, D. (2013). Mathematical problems from combustion theory, Vol. 83,
Springer Science & Business Media,
[5]
Chen, H., and Xu, H. Y. (2019). Global existence and blowup in finite time for a class of finitely
degenerate semilinear pseudo-parabolic equations. Acta Mathematica Sinica, English Series, 35(7),
1143-1162.
[6]
Choi, M. J. (2022). A condition for blowup solutions to discrete semilinear wave equations on
networks. Applicable Analysis, 101(6), 2008-2018.
https://doi.org/10.17993/3cemp.2022.110250.94-102
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
98
employing the concavity method. Li and Han [32] then improved these results for positive initial energy
with the support of the potential well method. Sun et. al. [46] investigated an initial boundary value
problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear
source term and obtained results on finite time blowup of solutions under suitable assumptions on the
initial data and the relaxation function.
The existence of solutions for a pseudo-parabolic equation with memory was deduced by Di and
Shang [13] using Galerkin method and potential well theory. Then using the concavity method, derived
finite time blowup results for both negative and non-negative initial energy. Sun et. al. [45] studied the
problem and came up with existence and finite time blow-up results using Galerkin method, concavity
argument and potential well theory by making a slight change in the source term. They derived an upper
bound for the blowup time and obtained the existence of solutions which blow up in finite time with
arbitrary initial energy conditions. Di and Shang [14] worked on a class of nonlinear pseudo-parabolic
equations with a memory term under Dirichlet boundary condition. They proved a finite time blowup
result for specific initial energy and relaxation function. In 2019, Messaoudi and Talahmeh [33] studied
a semilinear viscoelastic pseudo-parabolic problem with variable exponent and demonstrated any weak
solution with initial data at arbitrary energy level blows up in finite time. Furthermore, they obtained
an upper bound for the blowup time using the concavity method. Chen and Xu [5] studied a finitely
degenerate semilinear pseudo-parabolic problem and showed the global existence and blowup in finite
time of solutions with sub-critical and critical initial energy. The asymptotic behaviour of the global
solutions and a lower bound for blowup time of the local solution are also obtained. A pseudo-parabolic
equation with variable exponents under initial and Dirichlet boundary value conditions is the subject of
study in [53]. In [53], Zhou et. al. established the global existence and blowup results of weak solutions
with arbitrarily high initial energy.
Existence and blow-up studies of the following
p
(
x
)-Laplacian parabolic equation with memory was
studied by Lakshmipriya and Gnanavel [34],
zt−∆z−µ∇(|∇z|p(x)−2∇z)+t
0
h(t−τ)∆z(x, τ)dτ =β|z|b(x)−2z, x ∈Ω,t≥0
z(x, t)=0,x∈∂Ω,t≥0
z(x, 0) = z0(x),x∈Ω
(11)
where Ω
⊂RN,
(
N≥
1
,N
= 2) is a bounded domain with smooth boundary
∂
Ω.
β>
0
,µ ≥
0are
constants. The authors established the existence and finite time blow up of weak solutions of the
problem. Further, obtained upper and lower bounds for the blowup time of solutions, by employing the
concavity method and differential inequality technique, respectively. In [35], the authors analysed and
interpreted unbounded solutions of a viscoelastic
p
(
x
)
−
Laplacian parabolic equation with logarithmic
nonlinearity. Here the problem was considered for initial data corresponding to the sub-critical initial
energy. In this attempt, Lakshmipriya and Gnanavel obtained the local existence of solutions on an
interval [0
,T
). Moreover, it extracted an upper bound for the blowup time by applying the concavity
method.
The paper [27] deals with the existence and blowup of weak solutions of the following pseudo-parabolic
equation with logarithmnonlinearityity
wt−∆wt−div(|∇w|p(x)−2∇w)=|w|s(x)−2w+|w|h−2wlog|w|,(x, t)∈Ω×(0,∞)
w(x, t)=0,(x, t)∈∂Ω×[0,∞)
w(x, 0) = w0(x),x∈Ω
(12)
where Ω
⊂Rn
(
n≥
1) is a bounded domain with smooth boundary
∂
Ω. The model consider is used to
describe the non-stationary process in semiconductors in the presence of sources; the first two terms
represent the free electron density rate and logarithmic and polynominonlinearityity stands for the
source of free electron current [26]. Lakshmipriya and Gnanavel [36] analysed the blowup of solutions
https://doi.org/10.17993/3cemp.2022.110250.94-102
to the following problem
zt(x, t)=∆
p(x)z(x, t)+g(z(x, t)),(x, t)∈Ω×(0,∞)
z(x, t)=0,(x, t)∈∂Ω×[0,∞)
z(x, 0) = z0(x)≥0,x∈Ω
(13)
where Ω
⊂RN
(
N≥
1) is a bounded domain with smooth boundary
∂
Ω. The model is involved in
image processing, elastic mechanics and electro-rheological fluids [1,40, 42]. The authors considered a
condition on the nonlinear function g(z)given by,
ςz
0
g(s)ds ≤zg(z)+ηzb(x)+µ, z > 0.
Obtained results on blowup and established an upper bound for the blowup time with the help of the
concavity method.
Now, we consider the most recent works involving the Concavity method. Ruzhansky et.al. [41]
proved a global existence and blowup of the positive solutions to the initial-boundary value problem of
the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified
Lie groups based on the concavity argument and the Poincare inequality. A nonlinear porous medium
equation under a new nonlinearity condition is considered in a bounded domain by Sabitbek and
Torebek [43]. They presented the blowup of the positive solution to the considered problem for the
negative initial energy. For the subcritical and critical initial energy cases, obtained a global existence,
asymptotic behaviour and blowup phenomena in a finite time of the positive solution to the nonlinear
porous medium equation. In [31], Li and Fang are concerned with the blowup phenomena for a semilinear
pseudo-parabolic equation with general nonlinearity under the null Dirichlet boundary condition. When
the nonlinearity satisfies a new structural condition, they obtain some new blowup criteria with different
initial energy levels. They derived the growth estimations and life span of blowup solutions.
The concavity method is also used to understand the blowup behaviour of system of nonlinear
parabolic equations [8]. Apart from parabolic and pseudo-parabolic equations, the method plays a
significant role in studying unbounded solutions of hyperbolic equations and systems. Some of the latest
works are as follows [6,7,44,50,51]. Hence the Concavity method is a simple and powerful tool to use
in the blow up studies of solutions to differential equation problems.
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https://doi.org/10.17993/3cemp.2022.110250.94-102
99
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
[7]
Chu, Y., Wu, Y., and Cheng, L. (2022). Blow up and Decay for a Class of
p
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Hyperbolic Equation with Logarithmic Nonlinearity. Taiwanese Journal of Mathematics, 1(1),
1-23.
[8]
Chung, S. Y., and Hwang, J.. (2022). Blowup conditions of nonlinear parabolic equations and
systems under mixed nonlinear boundary conditions. Bound Value Probl 2022, 46.
[9]
Chung, S. Y., and Choi, M. J., (2017). A New Condition for the Concavity Method of Blowup
Solutions to Semilinear Heat Equations. arXiv preprint arXiv:1705.05629.
[10]
Chung, S. Y., and Choi, M. J., (2018). A new condition for the concavity method of blowup
solutions to p-Laplacian parabolic equations. Journal of Differential Equations, 265(12), 6384-6399.
[11]
Chung, S. Y., Choi, M. J., and Hwang, J. (2019). A condition for blowup solutions to discrete
p-Laplacian parabolic equations under the mixed boundary conditions on networks. Boundary Value
Problems, 2019(1), 1-21.
[12]
Dai, H., and Zhang, H. (2014). Energy decay and nonexistence of solution for a reaction-diffusion
equation with exponentinonlinearityity. Boundary Value Problems, 2014(1), 1-9.
[13]
Di, H. and Shang, Y. (2014), Global existence and nonexistence of solutions for the nonlinear
pseudo-parabolic equation with a memory term, Math. Meth. Appl. Sci., 38: 3923– 3936.
[14]
Di, H., and Shang, Y. (2014), Blow-up of solutions for a class of nonlinear pseudoparabolic
equations with a memory term. Abstract and Applied Analysis (Vol. 2014). Hindawi.
[15]
Erdem, D. , (1999). Blowup of solutions to quasilinear parabolic equations. Applied mathematics
letters, 12(3), 65-69.
[16]
Fujita, H. (1966). On the blowing up of solutions fo the Cauchy problem for
ut
=∆
u
+
u1+α
,J.
Fac. Sci. Univ. Tokyo. 13 , 109-124.
[17]
Fujita, H.(1968). On some nonexistence and non-uniqueness theorems for nonlinear parabolic
equations, In: Proc. Symp. Math., 18, 105–113.
[18]
Friedman, A. (1965). Remarks on nonlinear parabolic equations. Proc. Symp. in Appl. Math.
AMS. 13, 3-23.
[19]
Galaktionov, V. A. (1982). The conditions for there to be no global solutions of a class of
quasilinear parabolic equations. USSR Computational Mathematics and Mathematical Physics, 22(2),
73-90.
[20]
Gelfand, I. M. (1959) Some problems in the theory of quasi-linear equations, Uspekhi Mat. Nauk.
14, 87-158.
[21]
Han, Y., Gao, W. and Li, H. (2015), Blow-up of solutions to a semilinear heat equation with a
viscoelastic term and a nonlinear boundary flux, C.R.Acad.Sci.Paris. Ser.I, 353, 825-830.
[22]
Hayakawa, K.(1973), On nonexistence of global solutions of some semilinear parabolic differential
equations, Proc. Jpn. Acad. 49 503-505.
[23]
Junning, Z. (1993). Existence and nonexistence of solutions for
ut
=
÷
(
|∇u|p−2∇u
)+
f
(
∇u, u, x, t
).
J. Math. Anal. Appl., 172(1), 130-146.
[24]
Kaplan, S. (1963). On the growth of solutions of quasi-linear parabolic equations, Comm. Pure
Appl. Math. 16, 305-330.
[25]
Kobayashi, K. Sirao, T. and Tanaka, H. (1977). On the growing up problem for semilinear
heat equations, J. Math. Soc. Japan. 29, 407-424.
https://doi.org/10.17993/3cemp.2022.110250.94-102
[26]
Korpusov, M.O. and Sveshnikov, A.G. (2003). Three dimensional non-linear evolutionary
pseudo - parabolic equations in mathematical physics. Zhurnal Vychislitel’no
˘
i Matematiki i Matema-
tichesko˘
i Fiziki 43(12) 1835–1869.
[27]
Lakshmipriya, N., Gnanavel, S., Balachandran, K., and Ma, Y. K. (2022). Existence and
blowup of weak solutions of a pseudo-parabolic equation with logarithmnonlinearityity. Boundary
Value Problems, 2022(1), 1-17.
[28]
Levine, H. A.(1973). Some nonexistence and instability theorems for solutions of formally
parabolic equations of the form Pu
t=−Au +F(u),Arch. Ration. Mech. Anal. 51, 371-386.
[29]
Levine, H. A. and Payne, L. E., (1974).Nonexistence theorems for the heat equation with
nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential
Equations. 16, 319-334.
[30]
Levine, H. A. and Payne, L. E., (1974)Some nonexistence theorems for initial-boundary value
problems with nonlinear boundary constraints,Proc. Amer. Math. Soc. 46, 277-284.
[31]
Li, X., and Fang, Z. B. (2022). New blowup criteria for a semilinear pseudo-parabolic equation
with genernonlinearityity. Mathematical Methods in the Applied Sciences.
[32]
Li, H. and Han, Y. (2017), Blow-up of solutions to a viscoelastic parabolic equation with positive
initial energy. Bound. Value Probl., 1:1-9.
[33]
Messaoudi, S. A., and Talahmeh, A. A. (2019), Blow up in a semilinear pseudo-parabolic
equation with variable exponents, Annalli Dell Universita Di Ferrara, 65(2), 311-326.
[34]
Narayanan, L., and Soundararajan, G. (2022). Existence and blowup studies of a
p
(
x
)-
Laplacian parabolic equation with memory. Mathematical Methods in the Applied Sciences, 45(14),
8412-8429.
[35]
Narayanan, L., and Soundararajan, G. (2022). Nonexistence of global solutions of a viscoelastic
p
(
x
)-Laplacian equation with logarithmnonlinearityity. In AIP Conference Proceedings (Vol. 2451,
No. 1, p. 020024). AIP Publishing LLC.
[36]
Narayanan, L., and Soundararajan, G.(2021). Quasilinear p (x)-Laplacian parabolic problem:
upper bound for blowup time. In Journal of Physics: Conference Series (Vol. 1850, No. 1, p. 012007).
IOP Publishing.
[37]
Payne, L.E., Philippin, G.A. and Piro, S.V. (2010), Blowup phenomena for a semilinear
heat equation with nonlinear boundary condition, II, Nonlinear Anal. 73 971–978.
[38]
Peral, I. and Vázquez, J. L.(1995), On the stability or instability of the singular solution of the
semilinear heat equation with exponential reaction term, Arch. Ration. Mech. Anal. 129, 201-224.
[39]
Philippin, G.A. and Proytcheva, V. (2006). Some remarks on the asymptotic behaviour of
the solutions of a class of parabolic problems, Math. Methods Appl. Sci. 29 297–307.
[40]
Prusa, V. and Rajagopal, K. R.(2018). A New Class of Models to Describe the Response
of Electrorheological and other Field Dependent Fluids, Generalised Models and Non-classical
Approaches in Complex Materials.
[41]
Ruzhansky, M., Sabitbek, B., and Torebek, B. (2022). Global existence and blowup of
solutions to porous medium equation and pseudo-parabolic equation, I. Stratified groups. manuscripta
mathematica, 1-19.
[42]
Ruzicka, M. (2000). Electrorheological fluids: Modeling and mathematical theory, Lecture Notes
in Math., Vol.1748, Springer-Verlag, Berlin.
https://doi.org/10.17993/3cemp.2022.110250.94-102
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
100
[7]
Chu, Y., Wu, Y., and Cheng, L. (2022). Blow up and Decay for a Class of
p
-Laplacian
Hyperbolic Equation with Logarithmic Nonlinearity. Taiwanese Journal of Mathematics, 1(1),
1-23.
[8]
Chung, S. Y., and Hwang, J.. (2022). Blowup conditions of nonlinear parabolic equations and
systems under mixed nonlinear boundary conditions. Bound Value Probl 2022, 46.
[9]
Chung, S. Y., and Choi, M. J., (2017). A New Condition for the Concavity Method of Blowup
Solutions to Semilinear Heat Equations. arXiv preprint arXiv:1705.05629.
[10]
Chung, S. Y., and Choi, M. J., (2018). A new condition for the concavity method of blowup
solutions to p-Laplacian parabolic equations. Journal of Differential Equations, 265(12), 6384-6399.
[11]
Chung, S. Y., Choi, M. J., and Hwang, J. (2019). A condition for blowup solutions to discrete
p-Laplacian parabolic equations under the mixed boundary conditions on networks. Boundary Value
Problems, 2019(1), 1-21.
[12]
Dai, H., and Zhang, H. (2014). Energy decay and nonexistence of solution for a reaction-diffusion
equation with exponentinonlinearityity. Boundary Value Problems, 2014(1), 1-9.
[13]
Di, H. and Shang, Y. (2014), Global existence and nonexistence of solutions for the nonlinear
pseudo-parabolic equation with a memory term, Math. Meth. Appl. Sci., 38: 3923– 3936.
[14]
Di, H., and Shang, Y. (2014), Blow-up of solutions for a class of nonlinear pseudoparabolic
equations with a memory term. Abstract and Applied Analysis (Vol. 2014). Hindawi.
[15]
Erdem, D. , (1999). Blowup of solutions to quasilinear parabolic equations. Applied mathematics
letters, 12(3), 65-69.
[16]
Fujita, H. (1966). On the blowing up of solutions fo the Cauchy problem for
ut
=∆
u
+
u1+α
,J.
Fac. Sci. Univ. Tokyo. 13 , 109-124.
[17]
Fujita, H.(1968). On some nonexistence and non-uniqueness theorems for nonlinear parabolic
equations, In: Proc. Symp. Math., 18, 105–113.
[18]
Friedman, A. (1965). Remarks on nonlinear parabolic equations. Proc. Symp. in Appl. Math.
AMS. 13, 3-23.
[19]
Galaktionov, V. A. (1982). The conditions for there to be no global solutions of a class of
quasilinear parabolic equations. USSR Computational Mathematics and Mathematical Physics, 22(2),
73-90.
[20]
Gelfand, I. M. (1959) Some problems in the theory of quasi-linear equations, Uspekhi Mat. Nauk.
14, 87-158.
[21]
Han, Y., Gao, W. and Li, H. (2015), Blow-up of solutions to a semilinear heat equation with a
viscoelastic term and a nonlinear boundary flux, C.R.Acad.Sci.Paris. Ser.I, 353, 825-830.
[22]
Hayakawa, K.(1973), On nonexistence of global solutions of some semilinear parabolic differential
equations, Proc. Jpn. Acad. 49 503-505.
[23]
Junning, Z. (1993). Existence and nonexistence of solutions for
ut
=
÷
(
|∇u|p−2∇u
)+
f
(
∇u, u, x, t
).
J. Math. Anal. Appl., 172(1), 130-146.
[24]
Kaplan, S. (1963). On the growth of solutions of quasi-linear parabolic equations, Comm. Pure
Appl. Math. 16, 305-330.
[25]
Kobayashi, K. Sirao, T. and Tanaka, H. (1977). On the growing up problem for semilinear
heat equations, J. Math. Soc. Japan. 29, 407-424.
https://doi.org/10.17993/3cemp.2022.110250.94-102
[26]
Korpusov, M.O. and Sveshnikov, A.G. (2003). Three dimensional non-linear evolutionary
pseudo - parabolic equations in mathematical physics. Zhurnal Vychislitel’no
˘
i Matematiki i Matema-
tichesko˘
i Fiziki 43(12) 1835–1869.
[27]
Lakshmipriya, N., Gnanavel, S., Balachandran, K., and Ma, Y. K. (2022). Existence and
blowup of weak solutions of a pseudo-parabolic equation with logarithmnonlinearityity. Boundary
Value Problems, 2022(1), 1-17.
[28]
Levine, H. A.(1973). Some nonexistence and instability theorems for solutions of formally
parabolic equations of the form Pu
t=−Au +F(u),Arch. Ration. Mech. Anal. 51, 371-386.
[29]
Levine, H. A. and Payne, L. E., (1974).Nonexistence theorems for the heat equation with
nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential
Equations. 16, 319-334.
[30]
Levine, H. A. and Payne, L. E., (1974)Some nonexistence theorems for initial-boundary value
problems with nonlinear boundary constraints,Proc. Amer. Math. Soc. 46, 277-284.
[31]
Li, X., and Fang, Z. B. (2022). New blowup criteria for a semilinear pseudo-parabolic equation
with genernonlinearityity. Mathematical Methods in the Applied Sciences.
[32]
Li, H. and Han, Y. (2017), Blow-up of solutions to a viscoelastic parabolic equation with positive
initial energy. Bound. Value Probl., 1:1-9.
[33]
Messaoudi, S. A., and Talahmeh, A. A. (2019), Blow up in a semilinear pseudo-parabolic
equation with variable exponents, Annalli Dell Universita Di Ferrara, 65(2), 311-326.
[34]
Narayanan, L., and Soundararajan, G. (2022). Existence and blowup studies of a
p
(
x
)-
Laplacian parabolic equation with memory. Mathematical Methods in the Applied Sciences, 45(14),
8412-8429.
[35]
Narayanan, L., and Soundararajan, G. (2022). Nonexistence of global solutions of a viscoelastic
p
(
x
)-Laplacian equation with logarithmnonlinearityity. In AIP Conference Proceedings (Vol. 2451,
No. 1, p. 020024). AIP Publishing LLC.
[36]
Narayanan, L., and Soundararajan, G.(2021). Quasilinear p (x)-Laplacian parabolic problem:
upper bound for blowup time. In Journal of Physics: Conference Series (Vol. 1850, No. 1, p. 012007).
IOP Publishing.
[37]
Payne, L.E., Philippin, G.A. and Piro, S.V. (2010), Blowup phenomena for a semilinear
heat equation with nonlinear boundary condition, II, Nonlinear Anal. 73 971–978.
[38]
Peral, I. and Vázquez, J. L.(1995), On the stability or instability of the singular solution of the
semilinear heat equation with exponential reaction term, Arch. Ration. Mech. Anal. 129, 201-224.
[39]
Philippin, G.A. and Proytcheva, V. (2006). Some remarks on the asymptotic behaviour of
the solutions of a class of parabolic problems, Math. Methods Appl. Sci. 29 297–307.
[40]
Prusa, V. and Rajagopal, K. R.(2018). A New Class of Models to Describe the Response
of Electrorheological and other Field Dependent Fluids, Generalised Models and Non-classical
Approaches in Complex Materials.
[41]
Ruzhansky, M., Sabitbek, B., and Torebek, B. (2022). Global existence and blowup of
solutions to porous medium equation and pseudo-parabolic equation, I. Stratified groups. manuscripta
mathematica, 1-19.
[42]
Ruzicka, M. (2000). Electrorheological fluids: Modeling and mathematical theory, Lecture Notes
in Math., Vol.1748, Springer-Verlag, Berlin.
https://doi.org/10.17993/3cemp.2022.110250.94-102
101
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
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