Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T07:00:35.779Z Has data issue: false hasContentIssue false
  • English
  • Français

The persistence of logconcavity for positive solutions of the one dimensional heat equation

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

G. Keady
G. Keady
Affiliation:
Mathematics Department, University of Western AustraliaNedlands, W. A. 6009, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

MSC classification

Secondary: 35K05: Heat equation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 2 , April 1990 , pp. 246 - 263
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Acker, A., Payne, L. E. and Philippin, G., ‘On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem’, Z. Angew. Math. Phys. 32 (1981), 683–694.CrossRefGoogle Scholar
[2]Benilan, P. and Vazquez, J. L., ‘Concavity of solutions of the porous medium equation’, Trans. Amer. Math. Soc. 299 (1987), 81–93.CrossRefGoogle Scholar
[3]Brascamp, H. J. and Lieb, E. H., ‘On extensions of the Brunn Minkowski and Prekopa Liendler inequalities, including inequalities for logconcave functions, and with an application to the diffusion equation’, J. Functional Anal. 22 (1976), 366–389.CrossRefGoogle Scholar
[4]Caffarelli, L. A. and Friedman, A., ‘Convexity of solutions of semilinear elliptic equations’, Duke Math. J. 52 (1985), 431–456.CrossRefGoogle Scholar
[5]Ellis, R. S. and Newman, C. M., ‘Extensions of the Maximum Principle: exponential preservation by the heat equation’, J. Differential Equations 30 (1978), 365–379.CrossRefGoogle Scholar
[6]Engler, H., ‘Contractive properties for the heat equation in Sobolev spaces’, J. Funct. Anal. 64 (1985), 412–435.CrossRefGoogle Scholar
[7]Friedman, A., Partial differential equations of parabolic type (Prentice-Hall, 1964).Google Scholar
[8]Gidas, B., Ni, W. M. and Nirenberg, L., ‘Symmetry and related problems via the Maximum Principle’, Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
[9]Karlin, S., Total positivity (Stanford University Press, 1968).Google Scholar
[10]Kawohl, B., Rearrangements and convexity of level sets in P.D.E., Lecture Notes in Math., vol. 1150 (Springer, Berlin, 1985).CrossRefGoogle Scholar
[11]Kawohl, B., ‘Qualitative properties of solutions to semilinear heat equations’, Expositiones Math. 4 (1986), 257–270.Google Scholar
[12]Keady, O., ‘The power concavity of solutions of some semilinear elliptic boundary-value problems’, Bull. Austral. Math. Soc. 31 (1985), 181–184.CrossRefGoogle Scholar
[13]Keady, G., The persistence of log-concavity for positive solutions of the one-dimensional heat equation, Res. Rep. Math. Dept. Univ. of Western Australia (02 1987). With corrections, January 1988.Google Scholar
[14]Keady, G. and Stakgold, I., Combustion of convex solids, Res. Rep. Math. Dept. Univ. of Western Australia (05 1987).Google Scholar
[15]Kennington, A. U., ‘Concavity of level curves for an initial value problem’, J. Math. Anal. Appl. 133 (1988), 324–330.CrossRefGoogle Scholar
[16]Korevaar, N., ‘Convex solutions to nonlinear elliptic and parabolic boundary value problems’, Indiana Univ. Math. J. 32 (1983), 603–614.CrossRefGoogle Scholar
[17]Makar-Limanov, L. G., ‘Solution of Dirichlet's problem for the torsion equation in a convex region’, Math. Notes Acad. Sci. U.S.S.R. 9 (1971), 52–53.Google Scholar
[18]Matano, H., ‘Nonincrease in the lap number for a one-dimensional semilinear parabolic equation’, J. Fac. Sci. Univ. Tokyo IA 29 (1982), 401–441.Google Scholar
[19]Nickel, K., ‘Gestaltaussagen über Losungen parabolischer Differentialgleichungen’, J. Reine Angew. Math. 211 (1962), 78–94.CrossRefGoogle Scholar
[20]Polya, G., ‘Qualitatives der warmeausgleich’, Z. Angew. Math. Mech. 13 (1933), 125–128.CrossRefGoogle Scholar
[21]Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations (Prentice-Hall, Englewood Cliffs, N.J., 1967).Google Scholar
[22]Sperb, R., Maximum principles and their applications (Academic Press, New York, 1981).Google Scholar