Stable foliations near a traveling front for reaction diffusion systems

Yuri Latushkin; Roland Schnaubelt; Xinyao Yang

doi:10.3934/dcdsb.2017168

Article Contents

2017, Volume 22, Issue 8: 3145-3165. Doi: 10.3934/dcdsb.2017168

This issue Previous Article Dichotomy and periodic solutions to partial functional differential equations Next Article Exponential stability for nonautonomous functional differential equations with state-dependent delay

Stable foliations near a traveling front for reaction diffusion systems

1.
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
2.
Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

Received: July 2016

Revised: November 2016

Published: October 2017

Partially supported by the US National Science Foundation under Grants NSF DMS-1067929, DMS-1311313, by the Research Board and Research Council of the University of Missouri, and by the Simons Foundation. This research was funded by the IRSES program of the European Commission (PIRSES-GA-2012-318910). RS gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173
We thank S. Walsh for pointing out the important paper [24], and the anonymous referee for many valuable comments.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front.

Keywords:

Mathematics Subject Classification: 35B35, 35C07, 34C45.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	I. Y. Akkutlu and Y. C. Yortsos, The dynamics of in-situ combustion fronts in porous media, Combustion and Flame, 134 (2003), 229-247. doi: 10.1016/S0010-2180(03)00095-6.
[2]	P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, In: Dynamics Reported, Vieweg-Teubner Verlag, 2 (1989), 1-38.
[3]	P. Bates, K. Lu and C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc., 352 (2000), 4641-4676. doi: 10.1090/S0002-9947-00-02503-4.
[4]	H. Berestycki, B. Larrouturou and J. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory, In: Dynamical Issues in Combustion Theory, Vol. 35 (Minneapolis, MN, 1989) IMA Vol. Math. Appl. , Springer, New York, pp. 37-61,1991. doi: 10.1007/978-1-4612-0947-8_2.
[5]	X.-Y. Chen, J. Hale and B. Tan, Invariant foliations for $C^1$ semigroups in Banach spaces, J. Differential Equations, 139 (1997), 283-318. doi: 10.1006/jdeq.1997.3255.
[6]	M. Das and Y. Latushkin, Derivatives of the Evans function and (modified) Fredholm determinants for first order systems, Math. Nachr., 284 (2011), 1592-1638. doi: 10.1002/mana.201000074.
[7]	P. Fife, Pattern formation in gradient systems, In: Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 677-722. doi: 10.1016/S1874-575X(02)80034-0.
[8]	A. Ghazaryan, Nonlinear stability of high Lewis number combustion fronts, Indiana Univ. Math. J., 58 (2009), 181-212. doi: 10.1512/iumj.2009.58.3497.
[9]	A. Ghazaryan, Y. Latushkin and S. Schecter, Stability of traveling waves for degenerate systems of reaction diffusion equations, Indiana Univ. Math. J., 60 (2011), 443-472. doi: 10.1512/iumj.2011.60.4069.
[10]	A. Ghazaryan, Y. Latushkin, S. Shechter and A. de Souza, Stability of gasless combustion fronts in one-dimensional solids, Archive Rational Mech. Anal., 198 (2010), 981-1030. doi: 10.1007/s00205-010-0358-y.
[11]	A. Ghazaryan, Y. Latushkin and S. Shechter, Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models, SIAM J. Math. Anal., 42 (2010), 2434-2472. doi: 10.1137/100786204.
[12]	A. Ghazaryan, Y. Latushkin and S. Shechter, Stability of traveling waves in partly hyperbolic systems, Math. Model. Nat. Phenom., 8 (2013), 31-47. doi: 10.1051/mmnp/20138503.
[13]	V. V. Gubernov, H. S. Sidhu and G. N. Mercer, Combustion waves in a model with chain branching reaction and their stability, Combust. Theory Model., 12 (2008), 407-431. doi: 10.1080/13647830701716948.
[14]	T. Kapitula and K. Promislow, Spectral and dynamical stability of nonlinear waves Appl. Math. Sci. , 185 Springer, New York, (2013), xiv+361 pp. doi: 10.1007/978-1-4614-6995-7.
[15]	T. Kato, Perturbation Theory for Linear Operators Springer-Verlag, New York, 1966.
[16]	Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete Contin. Dyn. Syst., 5 (1999), 233-268. doi: 10.3934/dcds.1999.5.233.
[17]	Y. Latushkin, J. Prüss and R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576. doi: 10.1007/s00028-006-0272-9.
[18]	Y. Latushkin, J. Prüss and R. Schnaubelt, Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions, Discrete Cont. Dyn. Syst. Ser. B, 9 (2008), 595-633. doi: 10.3934/dcdsb.2008.9.595.
[19]	Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521. doi: 10.1137/100814974.
[20]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995.
[21]	B. Matkowsky and G. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion, SIAM J. Appl. Math., 35 (1978), 465-478. doi: 10.1137/0135038.
[22]	K. Palmer, Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1.
[23]	P. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[24]	R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. doi: 10.1007/BF02101705.
[25]	P. J. Rabier, Fredholm operators, semigroups and the asymptotic and boundary behavior of solutions of PDEs, J. Diff. Eqns., 193 (2003), 460-480. doi: 10.1016/S0022-0396(03)00094-9.
[26]	J. Rottmann-Matthes, Computation and Stability of Patterns in Hyperbolic-Parabolic Systems Shaker Verlag, Aachen, 2010.
[27]	J. Rottmann-Matthes, Linear stability of travelling waves in first-order hyperbolic PDEs, J. Dynam. Differential Equations, 23 (2011), 365-393. doi: 10.1007/s10884-011-9216-3.
[28]	J. Rottmann-Matthes, Stability of parabolic-hyperbolic traveling waves, Dynamics of Part. Diff. Eqns., 9 (2012), 29-62. doi: 10.4310/DPDE.2012.v9.n1.a2.
[29]	B. Sandstede, Stability of travelling waves, In: Handbook of Dynamical Systems, North-Holland, Elsevier, Amsterdam, 2 (2002), 983-1055. doi: 10.1016/S1874-575X(02)80039-X.
[30]	B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7.
[31]	D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.
[32]	P. Simon, J. Merkin and S. Scott, Bifurcations in non-adiabatic flame propagation models, Focus on Combustion Research, (2006), 315-357.
[33]	P. Simon, S. Kalliadasis, J. H. Merkin and S. K. Scott, On the structure of the spectra for a class of combustion waves, J. Math. Chem., 35 (2004), 309-328. doi: 10.1023/B:JOMC.0000034249.20215.1f.
[34]	D. Terman, Traveling wave solutions arising from a two-step combustion model, SIAM J. Math. Anal., 19 (1988), 1057-1080. doi: 10.1137/0519071.
[35]	J.-C. Tsai, W. Zhang, V. Kirk and J. Sneyd, Traveling waves in a simplified model of calcium dynamics, SIAM J. Appl. Dyn. Systems, 11 (2012), 1149-1199. doi: 10.1137/120867949.
[36]	F. Varas and J. Vega, Linear stability of a plane front in solid combustion at large heat of reaction, SIAM J. Appl. Math., 62 (2002), 1810-1822. doi: 10.1137/S0036139901386417.
[37]	A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence (RI), 1994.
[38]	J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296.