October  2017, 22(8): 3145-3165. doi: 10.3934/dcdsb.2017168

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  June 2017

Fund Project: 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

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.

Citation: Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
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. Google Scholar

[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. Google Scholar

[3]

P. BatesK. 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. Google Scholar

[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. Google Scholar

[5]

X.-Y. ChenJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. GhazaryanY. 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. Google Scholar

[10]

A. GhazaryanY. LatushkinS. 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. Google Scholar

[11]

A. GhazaryanY. 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. Google Scholar

[12]

A. GhazaryanY. 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. Google Scholar

[13]

V. V. GubernovH. 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. Google Scholar

[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. Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators Springer-Verlag, New York, 1966. Google Scholar

[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. Google Scholar

[17]

Y. LatushkinJ. 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. Google Scholar

[18]

Y. LatushkinJ. 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. Google Scholar

[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. Google Scholar

[20]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995. Google Scholar

[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. Google Scholar

[22]

K. Palmer, Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. Google Scholar

[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. Google Scholar

[24]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. doi: 10.1007/BF02101705. Google Scholar

[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. Google Scholar

[26]

J. Rottmann-Matthes, Computation and Stability of Patterns in Hyperbolic-Parabolic Systems Shaker Verlag, Aachen, 2010.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

P. SimonJ. Merkin and S. Scott, Bifurcations in non-adiabatic flame propagation models, Focus on Combustion Research, (2006), 315-357. Google Scholar

[33]

P. SimonS. KalliadasisJ. 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. Google Scholar

[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. Google Scholar

[35]

J.-C. TsaiW. ZhangV. 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. Google Scholar

[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. Google Scholar

[37]

A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence (RI), 1994. Google Scholar

[38]

J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

P. BatesK. 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. Google Scholar

[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. Google Scholar

[5]

X.-Y. ChenJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. GhazaryanY. 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. Google Scholar

[10]

A. GhazaryanY. LatushkinS. 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. Google Scholar

[11]

A. GhazaryanY. 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. Google Scholar

[12]

A. GhazaryanY. 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. Google Scholar

[13]

V. V. GubernovH. 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. Google Scholar

[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. Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators Springer-Verlag, New York, 1966. Google Scholar

[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. Google Scholar

[17]

Y. LatushkinJ. 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. Google Scholar

[18]

Y. LatushkinJ. 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. Google Scholar

[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. Google Scholar

[20]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995. Google Scholar

[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. Google Scholar

[22]

K. Palmer, Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. Google Scholar

[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. Google Scholar

[24]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. doi: 10.1007/BF02101705. Google Scholar

[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. Google Scholar

[26]

J. Rottmann-Matthes, Computation and Stability of Patterns in Hyperbolic-Parabolic Systems Shaker Verlag, Aachen, 2010.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

P. SimonJ. Merkin and S. Scott, Bifurcations in non-adiabatic flame propagation models, Focus on Combustion Research, (2006), 315-357. Google Scholar

[33]

P. SimonS. KalliadasisJ. 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. Google Scholar

[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. Google Scholar

[35]

J.-C. TsaiW. ZhangV. 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. Google Scholar

[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. Google Scholar

[37]

A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence (RI), 1994. Google Scholar

[38]

J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296. Google Scholar

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