Hyperbolic Phenomena in a Degenerate Parabolic Equation

Abstract

M. Bertsch and R. Dal Passo [1] considered the equation u_t =  (φ(u)ψ(u_z))x., where φ > 0 and ψ is a strictly increasing function with lim_{s → ∞} ψ(s) = ψ_∞ < ∞. They have solved the associated Cauchy problem for an increasing initial function. Furthermore, they discussed to what extent the solution behaves like the solution of the first order conservation law u_t = ψ_∞(φ(u))_x. The condition φ > 0 is essential in their paper. In the present paper, we study the above equation under the degenerate condition φ(0) = 0. The solution also possesses some hyperbolic phenomena like those pointed out in [1].