Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-19T06:01:37.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Blow up for a diffusion-advection equation

Published online by Cambridge University Press:  14 November 2011

Nicholas D. Alikakos ,
Peter W. Bates  and
Christopher P. Grant
Nicholas D. Alikakos
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A
Peter W. Bates
Affiliation:
Applied Mathematics, National Science Foundation, Washington, D.C. 20550, U.S.A
Christopher P. Grant
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A
Get access Rights & Permissions [Opens in a new window]

Synopsis

These results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The “no flux” boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise “fast association” there is a criticalmass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 181 - 190
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amann, H.. Parabolic evolution equations and non-linear boundary conditions. J. Differential Equations 72 (1988), 201–269.CrossRefGoogle Scholar
2Benilan, P.. Lecture notes (University of Kentucky at Lexington, 1981).Google Scholar
3Bertsch, M., Gurtin, M., Hilhorst, D. and Peletier, L.. On interacting populations that disperse to avoid crowding; The effect of sedentary colony. J. Math. Biol. 19 (1984), 1–12.CrossRefGoogle Scholar
4Crandall, M. G. and Tartar, L.. Some relations between non-expansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980), 385–390.CrossRefGoogle Scholar
5Fujita, H.. Foundations of Ultracentrifugal Analysis (Chichester: John Wiley and Sons, 1975).Google Scholar
6Gilding, B. H.. Improved theory for a non-linear degenerate parabolic equation. Ann. Scuola Norm. Sup. Pisa Cl. Set. (4) (to appear).Google Scholar
7Grant, C. P.. Masters Thesis (Brigham Young University, 1988).Google Scholar
8Giga, Y. and Kohn, R.. Characterizing blow up using similarly variables. Indiana Univ.Math. J. 36 (1987), 1–40.CrossRefGoogle Scholar
9Gurtin, M. E. and Pipkin, A. C.. A note on interacting populations that disperse to avoidcrowding. Quart. Appl. Math. 42 (1984), 87–94.CrossRefGoogle Scholar
10Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
11Ladyzenskaja, O. A., Solonnikov, V. A. and Ural'ceva, N. N.. Linear and Quasilinear Equations of Parabolic Type (Providence, R.I.: American Mathematical Society, 1968).CrossRefGoogle Scholar
12Levine, H., Payne, L., Sacks, P. and Straughan, B.. (II). Analysis of a convective reaction—diffusion equation. SIAM J. Math. Anal. 20 (1989) 133–147.CrossRefGoogle Scholar
13Nagai, T.. Some non-linear degenerate diffusion equations with a non-locally convectiveterm in ecology. Hiroshima Math. J. 13 (1983), 449–464.CrossRefGoogle Scholar
14Saidi, F.. The Large Time Behaviour of Solutions of a Diffusion Equation Involving a Non-local Convective Term (Ph.D. Thesis, University of Maryland, Baltimore County, 1988).Google Scholar
15Shigesada, N., Kawasaki, K. and Teramoto, E.. Spatial segregation of interacting species. J. Theoret. Biol. 79 (1979), 89–99.CrossRefGoogle ScholarPubMed
16Yoshikawa, A.. On the behaviour of the solutions to the Lamm equation of the ultracentrifuge. SIAM J. Math. Anal. 15 (1984), 686–711.CrossRefGoogle Scholar
17Zelenyak, T. I.. Stabilisation of solutions of boundary value problems for a second-order equation with one space variable. Differential Equations 4 (1968), 17–22.Google Scholar