A free boundary value problem related to the combustion of a solid: flux boundary conditions
We demonstrate the existence, uniqueness, and continuous dependence upon the data for the solution (u, v, s) of the free boundary value problem: ut = auxx, vt = (3vxx, 0 < x < s(t), 0 < t < T, u(x, 0) = (x), v(x, 0) = 0 < x < s(0) = b. -aux(0, t) = f(t), -0vx(O, t) g(t), 0 < t < T, aux(s(t),t) = -(7 + u(s(t),t))s(t), (3vx(s(t),t) = (n v(s(t),t))s(t), 0 < t < T, s(t) = u(v(s(t),t)) exp(—8/v(s(t),t))F(u(s(t), t)), 0 < t < T, where a,/3,7,6, and /x are positive constants related to the physical constants.