Archive for Rational Mechanics and Analysis

, Volume 198, Issue 3, pp 981–1030

Stability of Gasless Combustion Fronts in One-Dimensional Solids

  • Anna Ghazaryan
  • Yuri Latushkin
  • Stephen Schecter
  • Aparecido J. de Souza
Article

DOI: 10.1007/s00205-010-0358-y

Cite this article as:
Ghazaryan, A., Latushkin, Y., Schecter, S. et al. Arch Rational Mech Anal (2010) 198: 981. doi:10.1007/s00205-010-0358-y

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.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Anna Ghazaryan
    • 1
    • 2
  • Yuri Latushkin
    • 3
  • Stephen Schecter
    • 4
  • Aparecido J. de Souza
    • 5
  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Department of MathematicsMiami UniversityOxfordUSA
  3. 3.Department of MathematicsUniversity of MissouriColumbiaUSA
  4. 4.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  5. 5.Departamento de Matemática e EstatísticaUniversidade Federal de Campina GrandeCampina GrandeBrazil