Top

Journal of Engineering Mathematics

Published in:

01-06-2015

On the stability of initial conditions for the parabolic Gelfand problem

Author: Alejandro Omón Arancibia

Published in: Journal of Engineering Mathematics | Issue 1/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This work investigates the initial conditions for a classical quasilinear parabolic equation with exponential nonlinearity such that blowup is developed. This is a very old question in parabolic partial differential equations because it corresponds, for example, to the question of blowup that develops in a thermal runaway. Very definite criteria are given for the initial conditions under which blowup develops.

previous article A parametric study of lycopodium dust flame

next article Approximation of limit cycles in two-dimensional nonlinear systems near a Hopf bifurcation by canonical transformations

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Gelfand IM (1963) Some problems in the theory of quasilinear equations. Amer Math Soc Trans 29:295–381

Frank-Kamenetskii DA (1955) Diffusion and heat exchange in chemical kinetics. Princeton University Press, Princeton

Fujita H (1969) On the asymptotic stability of solutions of the equation \(v_{t}=\Delta v+{\rm e}^{v}\). In: Proceedings of the international conference on functional analysis and related topics,Tokyo

Kassoy DR (1976) Extremely rapid transient phenomena in combustion, ignition and explosion. In: O’Malley RE (ed) Asymptotic methods and singular perturbations, SIAM-AMS Proceedings, vol 10

Bebernes J, Kassoy DR (1981) Analysis of blowup for thermal reactions-the spacially nonhomogeneous case. SIAM J Appl Math 40:476–484CrossRefMATHMathSciNet

Boddington T, Gray P, Wake G (1977) Criteria for thermal explosions with and without reactant consumption. Proc R Soc Lond A357:403–433CrossRefADS

Boddington T, Feng C-G, Gray P (1983) Thermal explosions, critically and the disappearance of critically in systems with distributed temperatures I. Arbitrary Biot numbers and general reactions rates. Proc R Soc Lond A 390:247–264

Keller HB (1969) Some positone problems suggested by nonlinear heat generation. In: Keller HB, Altman S (eds) Bifurcation theory and nonlinear eigenvalue problems. Benjamin, New York, pp 217–256

Keener JP, Keller HB (1974) Positive solutions of convex nonlinear eigenvalue problems. J Diff Equ 16:103–125CrossRefADSMATHMathSciNet

10.

Bandle C (1975) Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems. Arch Ration Mech Anal 58:219–238CrossRefMATHMathSciNet

11.

Lacey A (1983) Mathematical analysis of thermal runaway for spacially inhomogeneous reactions. SIAM J Appl Math 43:1350–1366CrossRefMATHMathSciNet

12.

Lacey A, Tzanetis D (1988) Complete blow-up for a semilinear diffusion equation with a sufficient large initial condition. IMA J Appl Math 41:207–215CrossRefMATHMathSciNet

13.

Lacey A, Wake GC (1992) Critical initial conditions for spatially-distributed thermal explosions. J Aust Math Soc Ser B Appl Math 33:350–362CrossRefMATHMathSciNet

14.

Lacey A, Tzanetis D (1993) Global unbounded solutions to a parabolic equation. J Diff Equ 101:80–102CrossRefADSMATHMathSciNet

15.

Dold J (1985) Analysis of the early stage of thermal runaway. Quart J Mech Appl Math 38:361–387CrossRefMATH

16.

Bebernes J, Eberly D (1989) Mathematical problems from combustion theory, Applied mathematical sciences, vol 83. Springer, New York

17.

Cazenave T, Haraux A (1998) An introduction to semilinear evolution equations, Oxford lectures in mathematics and its applications, vol 13. Oxford University Press, Oxford

18.

Tello JI (2006) Stability of steady states of the Cauchy problem for the exponential reaction–diffusion equation. J Math Anal Appl 324:381–396CrossRefMATHMathSciNet

19.

Fila M, Pulkkinen A (2008) Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation. Tohoku Math J 60:303–328CrossRefMATHMathSciNet

20.

Shi J (2002) Semilinear Neumann boundary value problems on a rectangle. Trans Am Math Soc 354:3117–3154CrossRefMATH

21.

Berchio E, Gazzola F, Pierotti D (2010) Gelfand type elliptic problems under Steklov boundary conditions. Ann Inst H Poincare Ann Non Linear 27:315–335CrossRefADSMATHMathSciNet

22.

Lions PL (1982) Asymptotic behaviour of some nonlinear heat equations. Physica D 53:293–306CrossRefADS

23.

Lions PL (1984) Structure of the set of steady-state solutions and asymptotic behaviour of semilinear equations. J Diff Equ 53:362–386CrossRefADSMATH

24.

Ni WM (1984) On the asymptotic behaviour of solutions of certain quasilinear parabolic equations. J Diff Equ 54:97–120CrossRefADSMATH

25.

Lions PL (1996) Mathematical topics in fluid dynamics, vol 1: Incompressible models, Oxford lectures in mathematics and its applications, vol 13. Oxford University Press, Oxford

26.

Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics, vol 68, Appied mathematical scienceSpringer, New York

27.

Chipot M (2002) l goes to plus infinity. Birkhauser Advanced Text, Birkhauser

28.

Sperb R (1981) Maximum principles and their applications, Mathematics in science and engineering, vol 157. Academic Press, New York

29.

Ni WM (1987) Lane–Emden equations and related topics in nonlinear elliptic and parabolic problems. Aspects Math 10:135–152

30.

Sattinger D (1972) Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ Math J 21:979–1000CrossRefMATHMathSciNet

31.

Ball J (1977) Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart J Math Oxford 2 28:473–786CrossRefMATH

32.

Liñán A, Williams F (1993) Fundamental aspects of combustion, Oxford engineering science series, vol 34. Oxford University Press, New York

33.

Matano H (1979) Asymptotic behaviour and stability of solution of semilinear diffusion equations. Publ RIMS, Kyoto Univ 15:401–454

Title: On the stability of initial conditions for the parabolic Gelfand problem
Author: Alejandro Omón Arancibia
Publication date: 01-06-2015
Publisher: Springer Netherlands
Published in: Journal of Engineering Mathematics / Issue 1/2015
Print ISSN: 0022-0833
Electronic ISSN: 1573-2703
DOI: https://doi.org/10.1007/s10665-014-9759-5

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 1/2015

Slip flow and heat transfer over a specific wedge: an exactly solvable Falkner–Skan equation

Asymptotic analysis of Rayleigh–Taylor flow for Newtonian miscible fluids

A time spectral method for solving the nonlinear dynamic equations of a rectangular elastic plate

Approximation of limit cycles in two-dimensional nonlinear systems near a Hopf bifurcation by canonical transformations

A study of the effects of electric field on two-dimensional inviscid nonlinear free surface flows generated by moving disturbances

A parametric study of lycopodium dust flame

Premium Partners