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 a