Stability of Gasless Combustion Fronts in One-Dimensional Solids

Stability of Traveling Waves in Partly Parabolic Systems

“…For larger β the traveling wave apparently loses stability when a pair of complex eigenvalues crosses the imaginary axis (see numerical evidence in [6]). In [21] we verified analytically that the 0 eigenvalue is simple for all β.…”

“…The nonlinear operator N is [69], Nii [44], and Sandstede [52] show that L is spectrally stable. With the aid of [21], it follows that the same is true for E 0 = H 1 (R). Theorem 2.3 and the result of Bates and Jones are used in [22] to obtain the following result.…”

“…In the paper [68] on a scalar nonlocal reaction-diffusion equation, properties of the bifurcating periodic pattern are used. In [19,21,23] convective stability is derived using an assumption that the left state has stability for some components of the solution (see Hypotheses 3.1 and 3.2 below). We shall discuss [23] in more detail, since it applies to a class of partly parabolic systems.…”

Gasless Combustion Fronts with Heat Loss

“…For example, they vanish in the presence of heat loss to the environment, and they are only seen in simulations when carefully prepared, physically implausible initial conditions are used. There is a theoretical explanation for this observation; see [8].…”

“…When κ = 0 there is the additional difficulty that the essential spectrum includes a vertical line, so L is not sectorial. We shall make use of machinery appropriate to dealing with these issues that was developed in [9] and [10], generalizing the methods of [8] (for gasless combustion with δ = 0 and κ = 0) and [6] (for gasless combustion with δ = 0 and κ > 0).…”