On the existence of the weak solution and the regularity of the free boundary to a one-dimensional two-phase degenerate Stefan problem

AbstractExistence and regularity of the free boundary s(t) are demonstrated for the weak solution u = u(x, t) and s = s(t) of the degenerate Stefan problem α1(u) ut = uxx, 0 < x < s(t), 0 < t < T; α2(u) ut = uxx, s(t) < x < 1, 0 < t < T; u(0, t) = ƒ1(t) >0, 0 ⩽ t ⩽ T; u(1, t) < ƒ2(t) < 0, 0 ⩽ t ⩽ T; u(x, 0) = u0(x), 0 ⩽ x ⩽ 1, u(s(t), t) = 0, 0 ⩽ t ⩽ T; (t) = − ux(s(t) − 0, t) + ux(s(t) + 0, t), 0 < t ⩽ T; s(0) = s0, where αi(u) ⩾ 0 and αi(0) = 0, i = 1, 2