L 2 -stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s

Salvatore Rionero

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 4, page 227-238
  • ISSN: 1120-6330

Abstract

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The L 2 -stability (instability) of a binary nonlinear reaction diffusion system of P.D.E.s - either under Dirichlet or Neumann boundary data - is considered. Conditions allowing the reduction to a stability (instability) problem for a linear binary system of O.D.E.s are furnished. A peculiar Liapunov functional V linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.

How to cite

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Rionero, Salvatore. "$L^{2}$-stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.4 (2005): 227-238. <http://eudml.org/doc/252453>.

@article{Rionero2005,
abstract = {The $L^\{2\}$-stability (instability) of a binary nonlinear reaction diffusion system of P.D.E.s - either under Dirichlet or Neumann boundary data - is considered. Conditions allowing the reduction to a stability (instability) problem for a linear binary system of O.D.E.s are furnished. A peculiar Liapunov functional $V$ linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.},
author = {Rionero, Salvatore},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nonlinear stability; Lyapunov direct method; Reaction-diffusion system},
language = {eng},
month = {12},
number = {4},
pages = {227-238},
publisher = {Accademia Nazionale dei Lincei},
title = {$L^\{2\}$-stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s},
url = {http://eudml.org/doc/252453},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Rionero, Salvatore
TI - $L^{2}$-stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/12//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 4
SP - 227
EP - 238
AB - The $L^{2}$-stability (instability) of a binary nonlinear reaction diffusion system of P.D.E.s - either under Dirichlet or Neumann boundary data - is considered. Conditions allowing the reduction to a stability (instability) problem for a linear binary system of O.D.E.s are furnished. A peculiar Liapunov functional $V$ linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.
LA - eng
KW - Nonlinear stability; Lyapunov direct method; Reaction-diffusion system
UR - http://eudml.org/doc/252453
ER -

References

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  2. MURRAY, J.D., Mathematical Biology. I. An Introduction. 3rd éd., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York2002, 600 pp. Zbl1006.92001MR1908418
  3. MURRAY, J.D., Mathematical Biology. II. Spatial Models and Biomedical Applications. 3rd ed., Inter-disciplinary Applied Mathematics, vol. 18, Springer-Verlag, New York2003, 811 pp. Zbl1006.92002MR1952568
  4. STRAUGHAN, B., The energy method, stability, and nonlinear convection. 2nd ed., Appl. Math. Sci. Ser. vol. 91, Springer-Verlag, New York-London2004, 240 pp. Zbl1032.76001MR2003826
  5. CANTRELL, R.S. - COSNER, C., Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology, Wiley, Chichester2003, 411 pp. Zbl1059.92051MR2191264DOI10.1002/0470871296
  6. FLAVIN, J.N. - RIONERO, S., Qualitative estimates for partial differential equations: an introduction. CRC Press, Boca Raton, Florida1996, 360 pp. Zbl0862.35001MR1396085
  7. RIONERO, S., A nonlinear L 2 -stability analysis for two-species population dynamics with dispersal. Mathematical Biosciences and Engineering, vol. 3, n. 1, 2006, 189-204. Zbl1090.92039MR2192134DOI10.3934/mbe.2006.3.189
  8. RIONERO, S., A rigorous reduction of the L 2 -stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s, Journal of Mathematical Analysis and Applications, to appear. Zbl1099.35041MR2255006
  9. RIONERO, S., Asymptotic properties of solutions to nonlinear possibly degenerated parabolic equations in unbounded domains. Mathematics and Mechanics of Solids, vol. 10, 2005, 541-557. Zbl1085.35082MR2167055DOI10.1177/1081286505036418

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