The Jump Conditions for Second Order Quasilinear Degenerate Parabolic Equations (Journal of Partial Differential Equations, 1990)
ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D&sup2_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x &#8712 R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} &#8747^u_Ak(s)ds, G(u)=^{def} &#8747^u_Ag(s)ds, and F(u) =^{def} &#8747^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, &#8230, u_0(x, t) must satisfy the following jump conditions, 1&#176. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, &#8230} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, &#8230, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2&#176. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0