Stability of Gasless Combustion Fronts in One-Dimensional Solids

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. 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.deSouzagratefully 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. 982 Anna Ghazaryan et al.