Solutions to a One-Dimensional Combustion-Type Free Boundary Problem via Maximal Regularity

Abstract

We study a one-dimensional free boundary problem arising in combustion theory, where the motion of the interface is governed by a prescribed Neumann boundary flux and a zero Dirichlet boundary condition. We treat both the half-line case and the bounded interval case. For both settings, we employ maximal $L^p$-$L^q$ regularity as our main analytical tool. In the half-line case, the solutions need not decay at infinity, even though the spatial derivatives belong to $L^q(\mathbb{R}_+)$. To handle the evolution law of the free boundary, we introduce a derivative formulation that avoids second-order boundary traces. By combining maximal $L^p$-$L^q$ regularity and Schauder estimates, we establish the local-in-time existence, uniqueness, and regularity of solutions, as well as the evolution law of the free boundary.