Abstract. We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called "focusing problem" in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive t, if the data are smooth.
We study travelling wave solutions of a one-dimensional two-phase Free Boundary Problem, which models premixed flames propagating in a gaseous mixture with dust. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative effects due to the presence of dust account for the divergence of the radiative flux entering the equation for temperature. This flux is modelled by the Eddington equation. In an appropriate limit the divergence of the flux takes the form of a nonlinear heat loss term. The resulting reduced model is able to capture a hysteresis effect that appears if the amount of fuel in front of the flame, or equivalently, the adiabatic temperature is taken as a control parameter.
Abstract. We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface.Its planar travelling-wave solutions are orbitally stable provided the bifurcation parameter u * does not exceed a critical value u c * . The latter is the limit of a decreasing sequence (u k * ) of bifurcation points. The paper deals with the study of the 2D bifurcated branches from the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we can apply the Crandall-Rabinowitz bifurcation theorem for a local study. We point out that the fully nonlinear reformulation of the FBP can also serve to develop efficient numerical schemes in view of global information, such as techniques based on arc length continuation. Résumé.On s'intéresseà un problèmeà frontière libre bidimensionnel, avec conditions de sautà l'interface. Le problème parabolique admet comme solutions des ondes progressives planes, qui sont orbitalement stables si le paramètre u * ne dépasse pas la valeur u c * . Ce point critique est la limite d'une suite décroissante de points de bifurcation (u k * ). Dans cet article, onétudie la structure des branches bifurquées 2Dà partir de la branche triviale formée des ondes planes, pour k petit. Notre technique consisteàéliminer le front inconnu, pour se ramenerà un problème totalement non linéaireéquivalent, auquel on applique le théorème de bifurcation de Crandall-Rabinowitz pour uneétude locale. La reformulation totalement non linéaire du problème s'avèreégalement bien adaptéeà la mise en oeuvre de méthodes numériques pour le suivi global des branches bifurquées, en particulier par continuation.Mathematics Subject Classification. 35R35, 35B32, 35K55, 65M06, 65F99.
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