On a Class of Quasilinear Parabolic Equations of Second Order with Double-degeneracy

Abstract

In this paper we study the first boundary value problem for nonlinear diffusion equation \frac{∂u}{∂t} + \frac{∂}{∂x}f(u) = \frac{∂}{∂x}A(\frac{∂}{∂x}B(u)) whereA(s) = ∫&sup1_0a(σ)dσ, B(s) = ∫&sup1_0b(σ)dσ with a(s) ≥ 0, b(s) ≥ 0. We prove the existence of BV solutions under the much general structural conditions lim_{s → + ∞} A(s) = +∞, lim_{s → - ∞} A(s) = -∞ Moreover, we show the uniqueness without any structural conditions.