Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T01:51:33.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Gradient-like parabolic semiflows on BUC(ℝN)

Published online by Cambridge University Press:  14 November 2011

Daniel Daners  and
Sandro Merino
Daniel Daners
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia e-mail: [email protected]
Sandro Merino
Affiliation:
Mathematisches Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

We prove that a class of weighted semilinear reaction diffusion equations on RN generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on RN. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigroup.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 6 , 1998 , pp. 1281 - 1291
Copyright
Copyright © Royal Society of Edinburgh 1998

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

1Arendt, W. and Batty, C. J. K.. Exponential stability of a diffusion equation with absorption. Differential Integral Equations 6 (1993), 427–48.CrossRefGoogle Scholar
2Batty, C. J. K.. Asymptotic stability of Schrodinger semigroups: path integral methods. Math. Ann. 292(1992), 457–92.CrossRefGoogle Scholar
3Brunovský, P., Poláčik, P. and Sandstede, B.. Convergence in general periodic parabolic equations in one space dimension. Nonlinear Anal. 18 (1992), 209–15.CrossRefGoogle Scholar
4Daners, D. and Medina, P. Koch. Abstract evolution equations, periodic problems and applications, Pitman Research Notes in Mathematics Series 279 (Harlow: Longman Scientific & Technical, 1992).Google Scholar
5Daners, D. and Medina, P. Koch. Exponential stability, change of stability and eigenvalue problems for linear time-periodic parabolic equations on RN. Differential Integral Equations 7 (1994), 1265–84.CrossRefGoogle Scholar
6Davies, E. B.. Heat kernels and spectral theory (Cambridge: Cambridge University Press, 1989).CrossRefGoogle Scholar
7Edmunds, D. E. and Evans, W. D.. Spectral theory and differential operators (Oxford: Clarendon Press, 1987).Google Scholar
8Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.. Higher transcendental functions, Vol. II (New York: McGraw-Hill).Google Scholar
9Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn (Berlin: Springer, 1983).Google Scholar
10Hale, J. K. and Raugel, G.. Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43 (1992), 63–124.CrossRefGoogle Scholar
11Ladyženskaja, O. A., Solonnikov, V. A. and Ural'ceva, N. N.. Linear and quasilinear equations of parabolic type (Providence, RI: American Mathematical Society, 1968).CrossRefGoogle Scholar
12Ladyzhenskaya, O.. Attractors for semigroups and evolution equations (Cambridge: Cambridge University Press, 1991).CrossRefGoogle Scholar
13Merino, S.. On the existence of the compact global attractor for semilinear reaction diffusion systems on RN. J. Differential Equations 132 (1996), 87–106.CrossRefGoogle Scholar
14Zelenyak, T. I.. Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Differentsial'nye Uravneniya 4 (1968), 35–45.Google Scholar