HYPERBOLIC PHENOMENA IN A STRONGLY DEGENERATE PARABOLIC EQUATION

We consider the equation u(t) = (phi(u) psi(u(x)))x, where phi > 0 and where psi is a strictly increasing function with lim(s-->infinity) psi(s) = psi(infinity) < infinity. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u(t) = psi(infinity) (phi(u))x. Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.