A Peculiar Liapunov Functional for Ternary Reaction-Diffusion Dynamical Systems

Salvatore Rionero

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 3, page 393-407
  • ISSN: 0392-4041

Abstract

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A Liapunov functional W , depending - together with the temporal derivative W ˙ along the solutions - on the eigenvalues via the system coefficients, is found. This functional is ``peculiar'' in the sense that W is positive definite and simultaneously W ˙ is negative definite, if and only if all the eigenvalues have negative real part. An application to a general type of ternary system often encountered in the literature, is furnished.

How to cite

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Rionero, Salvatore. "A Peculiar Liapunov Functional for Ternary Reaction-Diffusion Dynamical Systems." Bollettino dell'Unione Matematica Italiana 4.3 (2011): 393-407. <http://eudml.org/doc/290749>.

@article{Rionero2011,
abstract = {A Liapunov functional $W$, depending - together with the temporal derivative $\dot\{W\}$ along the solutions - on the eigenvalues via the system coefficients, is found. This functional is ``peculiar'' in the sense that $W$ is positive definite and simultaneously $\dot\{W\}$ is negative definite, if and only if all the eigenvalues have negative real part. An application to a general type of ternary system often encountered in the literature, is furnished.},
author = {Rionero, Salvatore},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {393-407},
publisher = {Unione Matematica Italiana},
title = {A Peculiar Liapunov Functional for Ternary Reaction-Diffusion Dynamical Systems},
url = {http://eudml.org/doc/290749},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Rionero, Salvatore
TI - A Peculiar Liapunov Functional for Ternary Reaction-Diffusion Dynamical Systems
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/10//
PB - Unione Matematica Italiana
VL - 4
IS - 3
SP - 393
EP - 407
AB - A Liapunov functional $W$, depending - together with the temporal derivative $\dot{W}$ along the solutions - on the eigenvalues via the system coefficients, is found. This functional is ``peculiar'' in the sense that $W$ is positive definite and simultaneously $\dot{W}$ is negative definite, if and only if all the eigenvalues have negative real part. An application to a general type of ternary system often encountered in the literature, is furnished.
LA - eng
UR - http://eudml.org/doc/290749
ER -

References

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  4. RIONERO, S., A rigorous reduction of the stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.Es. to the stability of the solutions to a linear binary system of O.D.Es. J.M. A. A., vol. 319 (2006), 372-392. MR2227911DOI10.1016/j.jmaa.2005.05.059
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  6. RIONERO, S., On the reducibility of the L 2 -stability of ternary reaction-diffusion systems of P.D.Es. Proceedings Wascom 2009, World Sci., 2010, 321-331. Zbl1242.35043MR2762033DOI10.1142/9789814317429_0044
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  10. PRODI, G., Teoremi di tipo locale per il sistema di Navier-Stokes e stabilità delle soluzioni stazionarie. Rend. Sem. Mat. Univ. Padova, 32 (1962), 374-397. Zbl0108.28602MR189354
  11. FLAVIN, J. N. - RIONERO, S., Cross-diffusion influence on the nonlinear L 2 -stability analysis for the Lotka-Volterra reaction-diffusion system of P.D.Es. IMA J.A.M., vol. 72, n. 5 (2007), 540-557. Zbl1160.35032MR2361568DOI10.1093/imamat/hxm026
  12. RIONERO, S., L 2 -energy stability via new dependent variables for circumventing strongly nonlinear reaction terms. Nonlinear Analysis, vol. 70 (2009), 2530-2541. Zbl1175.34036MR2499720DOI10.1016/j.na.2008.03.039
  13. STRAUGHAN, B., The energy method, stability and nonlinear convection. Springer, 2004 (2nd Edit.). Zbl1032.76001MR2003826DOI10.1007/978-0-387-21740-6
  14. STRAUGHAN, B., Stability and wave motion in porous media. Springer, 2008. Zbl1149.76002MR2433781

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