Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains (original) (raw)

Abstract

This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form ∂ t u + Lu m = 0, m > 1, where the operator L belongs to a general class of linear operators, and the equation is posed in a bounded domain Ω ⊂ R N . As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, L can be a power of a uniformly elliptic operator with C 1 coefficients. Since the nonlinearity is given by u m with m > 1, the equation is degenerate parabolic. The basic well-posedness theory for this class of equations has been recently developed in . Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when L is a uniformly elliptic operator, and provide new estimates even in this setting. A surprising aspect discovered in this paper is the possible presence of non-matching powers for the long-time boundary behavior. More precisely, when L = (-∆) s is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that: -when 2s > 1 -1/m, for large times all solutions behave as dist 1/m near the boundary; -when 2s ≤ 1 -1/m, different solutions may exhibit different boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation Lu m = u.

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