Regularity Results for a Strongly Degenerate Parabolic Equation

Abstract

M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.