Geodesic
Stability of Traveling Waves in Partly Parabolic Systems
A. Ghazaryan1 ; Y. Latushkin2 ; S. Schecter3
1 Department of Mathematics, Miami University, Oxford, OH 45056 USA
2 Department of Mathematics, University of Missouri, Columbia, MO 65211 USA
3 Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695 USA
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 31-47

Voir la notice de l'article provenant de la source EDP Sciences

We review recent results on stability of traveling waves in partly parabolic reaction-diffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently mathematical technicalities has led to theorems that allow one to convert spectral calculations, which are used in the sciences and engineering to study stability of a wave, into detailed, theoretically-based information about the behavior of perturbations of the wave.
DOI : 10.1051/mmnp/20138503

A. Ghazaryan 1 ; Y. Latushkin 2 ; S. Schecter 3

1 Department of Mathematics, Miami University, Oxford, OH 45056 USA
2 Department of Mathematics, University of Missouri, Columbia, MO 65211 USA
3 Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695 USA 
@article{10_1051_mmnp_20138503,
     author = {A. Ghazaryan and Y. Latushkin and S. Schecter},
     title = {Stability of {Traveling} {Waves} in {Partly} {Parabolic} {Systems}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {31--47},
     publisher = {mathdoc},
     volume = {8},
     number = {5},
     year = {2013},
     doi = {10.1051/mmnp/20138503},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138503/}
}
TY  - JOUR
AU  - A. Ghazaryan
AU  - Y. Latushkin
AU  - S. Schecter
TI  - Stability of Traveling Waves in Partly Parabolic Systems
JO  - Mathematical modelling of natural phenomena
PY  - 2013
SP  - 31
EP  - 47
VL  - 8
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138503/
DO  - 10.1051/mmnp/20138503
LA  - en
ID  - 10_1051_mmnp_20138503
ER  - 
%0 Journal Article
%A A. Ghazaryan
%A Y. Latushkin
%A S. Schecter
%T Stability of Traveling Waves in Partly Parabolic Systems
%J Mathematical modelling of natural phenomena
%D 2013
%P 31-47
%V 8
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138503/
%R 10.1051/mmnp/20138503
%G en
%F 10_1051_mmnp_20138503
A. Ghazaryan; Y. Latushkin; S. Schecter. Stability of Traveling Waves in Partly Parabolic Systems. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 31-47. doi: 10.1051/mmnp/20138503

[1] I. Y. Akkutlu, Y. C. Yortsos Combustion and Flame 2003 229 247

[2] S. Balasuriya, G. Gottwald, J. Hornibrook, S. Lafortune SIAM J. Appl. Math. 2007 464 486

[3] P. W. Bates, C. K. R. T. Jones Dynamics Reported 2

[4] P. Bates, K. Lu, C. Zeng Trans. Amer. Math. Soc. 2000 4641 4676

[5] P. Bates, K. Lu, C. Zeng Mem. Amer. Math. Soc.

[6] A. Bayliss, B. Matkowsky SIAM J. Appl. Math. 1990 437 459

[7] V. Giovangigli SIAM J. Math. Anal. 1990 1305 1325

[8] M. Beck, A. Ghazaryan, B. Sandstede J. Differential Equations 2009 4371 4390

[9] T. Brand, M. Kunze, G. Schneider, T. Seelbach Arch. Ration. Mech. Anal. 2004 263 296

[10] R. J. Briggs. Electron-stream interaction with plasmas. MIT Press, Cambridge, MA, 1964.

[11] V. Capasso, L. Maddalena J. Math. Biology 1981 173 184

[12] X.-Y. Chen, J. K. Hale, B., Tan J. Differential Equations 1997 283 318

[13] C. Chicone, Y. Latushkin. Evolution semigroups in dynamical systems and differential equations. Math. Surv. Monogr., 70, AMS, Providence, 1999.

[14] C. Chicone, Y. Latushkin J. Differential Equations 1997 356 399

[15] B. Deng SIAM J. Math. Anal. 1991 653 679

[16] K. Engel, R. Nagel. One-parameter semigroups for linear evolution equations. Springer, New York, 2000.

[17] J. W. Evans Indiana Univ. Math. J. 1972 577 593

[18] E. Feireisl Diff. Int. Eqns. 1996 1147 1156

[19] A. Ghazaryan Indiana Univ. Math. J. 2009 181 212

[20] A. Ghazaryan, C. K. R. T. Jones Discrete Contin. Dyn. Syst. 2009 809 826

[21] A. Ghazaryan, Y. Latushkin, S. Schecter, A. J. De Souza Arch. Ration. Mech. Anal. 2010 981 1030

[22] A. Ghazaryan, Y. Latushkin, S. Schecter Indiana Univ. Math. J. 2011 443 472

[23] A. Ghazaryan, Y. Latushkin, S. Schecter SIAM J. Math. Anal. 2010 2434 2472

[24] A. Ghazaryan, B. Sandstede SIAM J. Appl. Dyn. Syst. 2007 319 347

[25] A. Ghazaryan, P. Simon, S. Schecter. Gasless combustion fronts with heat loss. To appear in SIAM J. Appl. Math.

[26] P. Gordon Math. Model. Nat. Phenom. 2007 56 76

[27] K. P. Hadeler, M. A. Lewis Canadian Appl. Math. Quart 2002 473 499

[28] S. Heinze, B. Schweizer Nonlinearity 2005 2455 2476

[29] D. Henry. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics vol. 840. Springer, New York, 1981.

[30] T. Kapitula, K. Promislow. An introduction to spectral and dynamical stability. Springer, New York, to appear.

[31] B. Kazmierczak, V. Volpert Math. Model. Nat. Phenom. 2007 106 125

[32] B. Kazmierczak, V. Volpert Nonlinearity 2008 71 96

[33] B. Kazmierczak, V. Volpert Arch. Mech. 2008 3 22

[34] B. Kazmierczak, V. Volpert Math. Methods and Models in Appl. Sci. 2008 883 912

[35] G. Kreiss, H.-O. Kreiss, N. A. Petersson SIAM J. Numer. Anal. 1994 157 1604

[36] M. Kunze, G. Schneider Z. Angew. Math. Phys. 2004 383 399

[37] Y. Latushkin, B. Layton Discrete Contin. Dynam. Systems 1999 233 268

[38] Y. Latushkin, J. Prüss, R. Schnaubelt Discrete Contin. Dyn. Syst. Ser. B 2008 595 633

[39] Y. Li, Y. Wu SIAM J. Math. Anal. 2012 1474 1521

[40] S. Luckhaus, L. Triolo Rend. Mat. Acc. Lincei„ 2004 215 223

[41] A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Progr. Nonlin. Diff. Eqns. Appl. vol. 16, Birkhäuser, Basel, 1995.

[42] A. Mielke, G. Schneider Nonlinearity 1995 743 768

[43] J. D. Murray. Mathematical biology. I. An introduction and II. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics vols. 17, 18. Springer, New York, 2002, 2003.

[44] S. Nii SIAM J. Math. Anal. 1997 1094 1112

[45] R. L. Pego, M. I. Weinstein Comm. Math. Phys. 1994 305 349

[46] L. Roques European J. Appl. Math. 2005 741 765

[47] J. Rottmann-Matthes. Computation and stability of patterns in hyperbolic-parabolic systems. Shaker Verlag, Aachen, 2010.

[48] J. Rottman-Matthes J. Dynam. Differential Equations 2011 365 393

[49] J. Rottmann-Matthes Dynamics of Part. Diff. Eqns. 2012 29 62

[50] J. Rottmann-Matthes J. Dynam. Differential Equations 2012 341 367

[51] J. Rottmann-Matthes IMA J. Appl. Math. 2012 420 429

[52] B. Sandstede SIAM J. Math. Anal. 1998 183 207

[53] B. Sandstede. Stability of traveling waves. In Handbook of dynamical systems, vol. 2, 983–1055. North-Holland, Amsterdam, 2002.

[54] B. Sandstede, A. Scheel Phys. D 2000 233 277

[55] D. H. Sattinger Adv. Math. 1976 312 355

[56] G. Sell, Y. You. Dynamics of evolutionary equations. Applied Mathematical Sciences vol. 143, Springer, 2002.

[57] P. Simon, J. Merkin, S. Scott Focus on Combustion Research (2006)

[58] P. Simon, S. Kalliadasis, J.H. Merkin, S.K. Scott IMA J. Appl. Math. 2003 537 562

[59] P. Simon, S. Kalliadasis, J.H. Merkin, S.K. Scott IMA J. Appl. Math. 2004 175 203

[60] P. Simon, S. Kalliadasis, J.H. Merkin, S.K. Scott J. Math. Chem. 2004 309 328

[61] P. Simon, J. Merkin, S. Scott Focus on Combustion Research (2006)

[62] J.-C. Tsai Discrete Contin. Dyn. Syst. 2008 601 623

[63] J.-C. Tsai, J. Sneyd SIAM J. Appl. Math. 2005 237 265

[64] J.-C. Tsai, W. Zhang, V. Kirk, J. Sneyd SIAM J. Appl. Dyn. Systems 2012 1149 1199

[65] J. M. A. M. van Neerven. The asymptotic behavior of semigroups of linear operators. Operator Theory: Advances and Applications vol. 88. Birkhäuser, Basel, 1996.

[66] W. Van Saarloos Phys. Rep. 2003 29 222

[67] F. Varas, J. Vega SIAM J. Appl. Math. 2002 1810 1822

[68] V. Volpert, V. Vougalter. Emergence and propagation of patterns in nonlocal reaction-diffusion equations arising in the theory of speciation. In Dispersal, individual movement and spatial ecology: a mathematical perspective. Lecture Notes in Mathematics vol. 2071, 331–354. Springer, New York, 2013.

[69] E. Yanagida Math. Comput. Modelling 1989 289 301

[70] K. Zumbrun, P. Howard Indiana Univ. Math. J. 1998 741 871

Cité par Sources :