L2-stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s
Rionero, Salvatore
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 227-238 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

The L2-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.

Publié le : 2005-12-01
@article{RLIN_2005_9_16_4_227_0,
     author = {Salvatore Rionero},
     title = {$L^{2}$-stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {16},
     year = {2005},
     pages = {227-238},
     zbl = {1150.35012},
     mrnumber = {2255006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2005_9_16_4_227_0}
}

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, Tome 16 (2005) pp. 227-238. http://gdmltest.u-ga.fr/item/RLIN_2005_9_16_4_227_0/

[1] Okubo, A. - Levin, S.A., Diffusion and ecological problems: modern prospectives. 2nd ed., Interdisciplinary Applied Mathematics, vol. 14, Springer-Verlag, New York 2001, 488 pp. | MR 1895041 | Zbl 1027.92022

[2] Murray, J.D., Mathematical Biology. I. An Introduction. 3rd éd., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York 2002, 600 pp. | MR 1908418 | Zbl 1006.92001

[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. | MR 1952568 | Zbl 1006.92002

[4] Straughan, B., The energy method, stability, and nonlinear convection. 2nd ed., Appl. Math. Sci. Ser. vol. 91, Springer-Verlag, New York-London 2004, 240 pp. | MR 2003826 | Zbl 1032.76001

[5] Cantrell, R.S. - Cosner, C., Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology, Wiley, Chichester 2003, 411 pp. | MR 2191264 | Zbl 1059.92051

[6] Flavin, J.N. - Rionero, S., Qualitative estimates for partial differential equations: an introduction. CRC Press, Boca Raton, Florida 1996, 360 pp. | MR 1396085 | Zbl 0862.35001

[7] Rionero, S., A nonlinear L2-stability analysis for two-species population dynamics with dispersal. Mathematical Biosciences and Engineering, vol. 3, n. 1, 2006, 189-204. | MR 2192134 | Zbl 1090.92039

[8] Rionero, S., A rigorous reduction of the L2-stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s, Journal of Mathematical Analysis and Applications, to appear. | MR 2255006 | Zbl 1099.35041

[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. | MR 2167055 | Zbl 1085.35082