Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Estimates for the principal spectrum point for certain time-dependent parabolic operators


Authors: V. Hutson, W. Shen and G. T. Vickers
Journal: Proc. Amer. Math. Soc. 129 (2001), 1669-1679
MSC (2000): Primary 35K20, 35P15; Secondary 92D25
DOI: https://doi.org/10.1090/S0002-9939-00-05808-1
Published electronically: November 2, 2000
MathSciNet review: 1814096
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Non-autonomous parabolic equations are discussed. The periodic case is considered first and an estimate for the principal periodic-parabolic eigenvalue is obtained by relating the original problem to the elliptic one obtained by time-averaging. It is then shown that an analogous bound may be obtained for the principal spectrum point in the almost periodic case. These results have applications to the stability of nonlinear systems and hence, for example, to permanence for biological systems.


References [Enhancements On Off] (What's this?)

  • 1. H. Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. S.I.A.M. Review, 18 (1976), 621-709. MR 54:3519; errata MR 57:7269
  • 2. E. J. Avila-Vales and R. S. Cantrell. Permanence in periodic-parabolic ecological systems with spatial heterogenity, in `Dynamical Systems and Applications', Ed. R.P.Agarwal. World Scientific Series in Applicable Analysis 4 63-76. World Scientific, Singapore, 1995. MR 97b:35173
  • 3. T. Burton and V. Hutson. Permanence for non-autonomous predator-prey systems. Diff. and Int. Eqns. 4 (1991), 1269-1280. MR 93f:34019
  • 4. S.N. Chow and H. Levia, Dynamical Spectrum for Time Dependent Linear Systems in Banach Spaces, Japan J. Indust. Appl. Math. 11 (1994), 379-415. MR 95i:34106
  • 5. S.N. Chow and K. Lu, Invariant Manifolds for Flows in Banach Spaces, J. Diff. Equ. 74 (1988), 285-317. MR 89h:58163
  • 6. A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics 377, Springer-Verlag, Berlin (1974). MR 57:792
  • 7. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ (1964). MR 31:6062
  • 8. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin (1981). MR 83j:35084
  • 9. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics 247 (1991). MR 92h:35001
  • 10. L.T. Magalhães, The Spectrum of Invariant Sets for Dissipative Semiflows, in Dynamics of Infinite-dimensional Systems (1988), NATO ASI Series, No. F-37, Springer Verlag, New York, 161-168. CMP 20:06
  • 11. P. Polácik and I. Terescák, Exponential Separation and Invariant Bundles for Maps in Ordered Banach Spaces with Applications to Parabolic Equations, J. Dyn. Diff. Equ. 5 (1993), 279-303. MR 94d:47064; erratum CMP 94:08
  • 12. R. J. Sacker and G. R. Sell, Dichotomies for Linear Evolutionary Equations in Banach Spaces, J. Diff. Equ. 113 (1994), 17-67. MR 96k:34136
  • 13. W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-product Semiflows, Part II. Skew-product Semiflows, Memoirs of A. M. S. 647 (1998).
  • 14. W. Shen and Y. Yi, Convergence in Almost Periodic Fisher and Kolmogorov Models, J. Math. Biol. 37 (1998), 84-102. MR 99k:92037

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35K20, 35P15, 92D25

Retrieve articles in all journals with MSC (2000): 35K20, 35P15, 92D25


Additional Information

V. Hutson
Affiliation: School of Mathematics and Statistics, The University of Sheffield, Sheffield S3 7RH, United Kingdom
Email: v.hutson@sheffield.ac.uk

W. Shen
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: ws@cam.auburn.edu

G. T. Vickers
Affiliation: School of Mathematics and Statistics, The University of Sheffield, Sheffield S3 7RH, United Kingdom
Email: g.vickers@sheffield.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-00-05808-1
Received by editor(s): September 7, 1999
Published electronically: November 2, 2000
Additional Notes: The second author was partially supported by NSF grant DMS-9704245.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society