On the Cauchy problem and initial traces for a degenerate parabolic equation

Abstract

Í u, -div(|Z>u|"-2Z>«) \ u(x,0) = u0(x), Abstract. We consider the Cauchy problem i) = 0 in R* x (0, oo), p > 2, x€RN, and discuss existence of solutions in some strip St a HN x (0, T), 0 < T < oo , in terms of the behavior of x -* uo(x) as |x| -► oo . The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of ¿^.(R-^).

On the long-time behaviour of solutions of the p-Laplacian parabolic system

“…The existence of solutions under our assumptions has been proved by Alt and Luckhaus [2]; similar questions have also been addressed by DiBenedetto and Herrero [6], Kuusi and Parviainen [11] and others.…”

A connection between a general class of superparabolic functions and supersolutions

“…DiBenedetto-Herrero [5], BoccardoDall'Aglio-Gallouët-Orsina [3]. Indeed, the convergence result also implies the useful weak convergence of the non-negative Radon measures generated by supersolutions.…”

“…Here p > 2 and λ = n(p − 2) + p. Another example is the equation 5) which was introduced by Lions in [14]. Observe that the equation is separable in the stationary case.…”

Existence for a degenerate Cauchy problem

“…Furthermore, the existence result is sharp: We show that the existence fails for more general Radon measures in the Euclidean setting. In [20], DiBenedetto and Herrero proved the existence and uniqueness for the evolutionary p-Laplace equation ∂u ∂t = div |∇u| p−2 ∇u , p > 2 in the Euclidean setting. The existence of this important special case is included in our results.…”