SHARP GLOBAL ESTIMATES FOR LOCAL AND NONLOCAL POROUS MEDIUM-TYPE EQUATIONS IN BOUNDED DOMAINS

We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous mediumtype of the form partial derivative(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 Omega subset of 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 fractional 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 was recently developed by Bonforte and Vazquez (2015, 2016). 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 nonmatching powers for the long-time boundary behavior. More precisely, when L = (-Delta)(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.