We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form , , where the operator belongs to a general class of linear operators, and the equation is posed in a bounded domain . 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, can be a fractional power of a uniformly elliptic operator with coefficients. Since the nonlinearity is given by with , the equation is degenerate parabolic.
The basic well-posedness theory for this class of equations was recently developed by Bonforte and Vázquez (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 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 is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that
when , for large times all solutions behave as near the boundary;
when , 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 .
"Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains." Anal. PDE 11 (4) 945 - 982, 2018. https://doi.org/10.2140/apde.2018.11.945