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

top
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

top

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

top
  1. CANTRELL, R. S. - COSNER, C., Spatial ecology via reaction diffusion equations. Wiley, 2003. Zbl1059.92051MR2191264DOI10.1002/0470871296
  2. FLAVIN, J. N. - RIONERO, S., Qualitative estimates for partial differential equations: an introduction. Boca Raton (FL): CRC Press, 1996. Zbl0862.35001MR1396085
  3. RIONERO, S., Stability of ternary reaction-diffusion dynamical system. To appear on ``Rendiconti di Matematica'' of ``Accademia dei Lincei'' issue 3, 2011. MR2847472DOI10.4171/RLM/599
  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
  5. RIONERO, S., Long time behaviour of three competing species and mututalistic communitites. Asymptotic Methods in Nonlinear Wave phenomenon. World Sci., 2007, 171-185. Zbl1311.92167MR2370502DOI10.1142/9789812708908_0015
  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
  7. MERKIN, D. R., Introduction to the theory of stability. Springer texts in Applied Mathematics, vol. 24 (1997). MR1418401
  8. GANTMACHER, F. R., The theory of matrices. Vol I, AMS, 2000. MR1657129
  9. GANTMACHER, F. R., Lezioni di Meccanica analitica, Editori Riuniti, Roma1980, (Lektsii po analiticeskoj mechanike, Mir, Moscow). 
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.