Global classical solutions for a free-boundary problem modeling combustion of solid propellants

Abstract

We study a free-boundary problem for the heat equation in one space dimension, describing the burning of a semi-infinite adiabatic solid propellant subjected to external thermal radiation (typically, a laser). The model includes the presence on the moving solid-gas interface (the free boundary) of heat release, due both to propellant degradation and conductive heat feedback from the gas phase reactions. The pyrolysis law and the flame submodel, relating burning rate to the boundary temperature and the heat feedback, respectively, satisfy general and physically significant conditions. We prove existence and uniqueness of a classical solution, local in time, for continuous initial thermal profiles. In addition, if the initial datum is exponentially bounded at infinity, we derive the main result of existence in the large and some uniform bounds for the solution.

A nonlinear parabolic problem from combustion theory: attractors and stability

“…Indeed, considering the estimates in more detail, we obtain as a consequence of Lemma 4.1 in [9] that there exists a constant K = K (t) > 0 depending on t but independent of u 0 such that…”

“…Then a function u = u (x, t) is a (classical global) solution of problem (1) if (1) hold. Existence and uniqueness was established in [9]: Theorem 2.1. We assume that the hypothesis H1 is satisfied.…”

“…It has been established in [9] that, for data u 0 in X 0 , there exist ε > 0 and > 0 such that K possesses a unique fixed point in the space Y = C 0 0, ⊕ C 0 0, , equipped with the supremum norm. In addition, if u 0 belongs to a given bounded set of X 0 , is bounded away from zero and K(v, u 0 ) is a contraction mapping with respect to v, uniformly with respect to u 0 .…”

“…Concerning well-posedeness, it is shown in [9] that for exponentially decaying, continuous data u 0 problem (1) has a unique global-in-time classical solution u (x, t); in addition, the same exponential behavior holds for u and its spatial derivative for all positive t. Our aim in this paper is to address some questions of long-time asymptotics. This study is motivated on the basis of experimental as well as numerical evidences [4].…”