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
tTz(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|p1z, (2)
and
zt=∆z+λez>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 a