Abstract
For gasless combustion in a one-dimensional solid, we show a type of nonlinear stability of the physical combustion front: if a perturbation of the front is small in both a spatially uniform norm and an exponentially weighted norm, then the perturbation stays small in the spatially uniform norm and decays in the exponentially weighted norm, provided the linearized operator has no eigenvalues in the right half-plane other than zero. Using the Evans function, we show that the zero eigenvalue must be simple. Factors that complicate the analysis are: (1) the linearized operator is not sectorial, and (2) the linearized operator has good spectral properties only when the weighted norm is used, but then the nonlinear term is not Lipschitz. The result is nevertheless physically natural. To prove it, we first show that when the weighted norm is used, the semigroup generated by the linearized operator decays on a subspace complementary to the operator’s kernel, by showing that it is a compact perturbation of the semigroup generated by a more easily analyzed triangular operator. We then use this result to help establish that solutions stay small in the spatially uniform norm, which in turn helps establish nonlinear convergence in the weighted norm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balasuriya S., Gottwald G., Hornibrook J., Lafortune S.: High Lewis number combustion wavefronts: a perturbative Melnikov analysis. SIAM J. Appl. Math. 67, 464–486 (2007)
Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. Dynamics reported 2, 1–38, Dynam. Report. Ser. Dynam. Systems Appl., Vol. 2. Wiley, Chichester, 1989
Bayliss A., Matkowsky B.: Two routes to chaos in condensed phase combustion. SIAM J. Appl. Math. 50, 437–459 (1990)
Beck M., Ghazaryan A., Sandstede B.: Nonlinear convective stability of traveling fronts near Turing and Hopf instabilities. J. Differ. Equ. 246, 4371–4390 (2009)
Benzoni-Gavage S., Serre D., Zumbrun K.: Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32, 929–962 (2001)
Berestycki, H., Larrouturou, B., Roquejoffre, J.-M.: Mathematical investigation of the cold boundary difficulty in flame propagation theory. Dynamical Issues in Combustion Theory, Vol. 35 (Minneapolis, MN, 1989) IMA Vol. Math. Appl., Springer, New York, 37–61, 1991
Beyn W.J., Lorenz J.: Nonlinear stability of rotating patterns. Dyn. Partial Differ. Equ. 5, 349–400 (2008)
Billingham J.: Phase plane analysis of one-dimensional reaction diffusion waves with degenerate reaction terms. Dyn. Stab. Syst. 15, 23–33 (2000)
Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Math. Surv. Monogr. Vol. 70. AMS, Providence, 1999
da Mota J.C., Schecter S.: Combustion fronts in a porous medium with two layers. J. Dyn. Differ. Equ. 18, 615–665 (2006)
de Souza A.J., Akkutlu I.Y., Marchesin D.: Wave sequences for solid fuel adiabatic in-situ combustion in porous media. Comput. Appl. Math. 25, 27–54 (2006)
Doedel E., Friedman M.J.: Numerical computation of heteroclinic orbits. Continuation techniques and bifurcation problems. J. Comput. Appl. Math. 26, 155–170 (1989)
Edmunds D.E., Evans W.D.: Spectral Theory and Differential Operators. Clarendon Press, Oxford (1989)
Engel K., Nagel R.: One-parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Evans, J.W.: Nerve axon equations. III. Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577–593(1972/73)
Friesecke G., Pego R.L.: Solitary waves on FPU lattices: II. Linear implies nonlinear stability. Nonlinearity 15, 1343–1359 (2002)
Gallay T., Schneider G., Uecker H.: Stable transport of information near essentially unstable localized structures. Discret. Contin. Dyn. Syst. Ser. B 4, 349–390 (2004)
Ghazaryan G.: On the convective nature of the instability of a front undergoing a supercritical Turing bifurcation. Math. Comput. Simul. 80, 10–19 (2009)
Ghazaryan G.: Nonlinear stability of high Lewis number combustion fronts. Indiana Univ. Math. J. 58, 181–212 (2009)
Ghazaryan G., Sandstede B.: Nonlinear convective instability of Turing-unstable fronts near onset: a case study. SIAM J. Appl. Dyn. Syst. 6, 319–347 (2007)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. Springer, Berlin-New York, 1981
Hoffman A., Wayne C.E.: Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice. Nonlinearity 21, 2911–2947 (2008)
Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1966)
Logak E.: Mathematical analysis of a condensed phase combustion model without ignition temperature. Nonlinear Anal. Theory Methods Appl. 28, 1–38 (1997)
Luo Z.-H., Guo B.-Z., Morgul O.: Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, London (1999)
Matkowsky B., Sivashinsky G.: Propagation of a pulsating reaction front in solid fuel combustion. SIAM J. Appl. Math. 35, 465–478 (1978)
Miller J.R., Weinstein M.I.: Asymptotic stability of solitary waves for the regularized long-wave equation. Commun. Pure Appl. Math. 49, 399–441 (1996)
Palmer K.: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ. 55, 225–256 (1984)
Palmer K.: Exponential dichotomies and Fredholm operators. Proc. Am. Math. Soc. 104, 149–156 (1988)
Pazy P.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Pego R.L., Weinstein M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 (1994)
Sandstede, B.: Stability of travelling waves. Handbook of Dynamical Systems, Vol. 2. North-Holland, Amsterdam, 983–1055, 2002
Sandstede, B., Scheel, A. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145, 233–277
Sandstede B., Scheel A.: Relative Morse indices, Fredholm indices, and group velocities. Discret. Contin. Dyn. Syst. 20, 139–158 (2008)
Schecter S.: The saddle-node separatrix-loop bifurcation. SIAM J. Math. Anal. 18, 1142–1156 (1987)
Sell, G., You, Y.: Dynamics of evolutionary equations. Applied Mathematical Sciences, Vol. 143, Springer, New York, 2002
Varas F., Vega J.: Linear stability of a plane front in solid combustion at large heat of reaction. SIAM J. Appl. Math. 62, 1810–1822 (2002)
Webb G.F.: Theory of Nonlinear Age-dependent Population Dynamics. Decker, New York (1985)
Wu Y., Xing X., Ye Q.: Stability of travelling waves with algebraic decay for n-degree Fisher-type equations. Discret. Contin. Dyn. Syst. 16, 47–66 (2006)
Yamashita M.: Melnikov vector in higher dimensions. Nonlinear Anal. 18, 657–670 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C.M. Dafermos
We thank Dan Marchesin for discussions that led to this work. The work was supported in part by the National Science Foundation under grants DMS-0406016, 0338743, 0354339, and 0410267, and by CNPq-Brazil under grant 200403/05-2. Y. Latushkin. gratefully acknowledges the support of the Research Board and the Research Council of the University of Missouri and of the EU Marie Curie “Transfer of Knowledge” program. A. J. de Souza gratefully acknowledges the hospitality of North Carolina State University, and A. Ghazaryan., Y. Latushkin., and S. Schecter. gratefully acknowledge the hospitality of the Mathematical Sciences Research Institute in Berkeley, California, during part of this work.
Rights and permissions
About this article
Cite this article
Ghazaryan, A., Latushkin, Y., Schecter, S. et al. Stability of Gasless Combustion Fronts in One-Dimensional Solids. Arch Rational Mech Anal 198, 981–1030 (2010). https://doi.org/10.1007/s00205-010-0358-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-010-0358-y