Existence theorems for a free boundary problem in combustion theory

We deal with a parabolic problem arising from combustion theory: ut −(K(u)ux)x = f(u)H(u − 1), 0 < x < l, t > 0, subject to the boundary conditions: Au(i, t) + Bux(i, t) = gi(t) for i = 0, 1 and the initial value u(x, 0) = u0(x), 0 x < l, where H(s) is the Heaviside function and (A,B) is either (0, 1) or (1, 0). By smoothing the Heaviside function and using a standard compactness argument, we obtain a global strong solution, under certain conditions on K(u) and f(u). The main result of the paper is to establish that the set {(x, t): u(x, t) = 1} is comprised of a graph of a Lipschitz function x = s(t) (a free boundary) which separates the region {(x, t): u(x, t) > 1} and {(x, t): u(x, t) < 1}.