Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains
In this interesting paper, the authors provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium- type of the form \[ \partial_t u +L u^m = 0, \] with \(m > 1\), where the operator \(L\) belongs to a general class of linear operators, and the equation acts in a bounded domain of \({\mathbb R}^N\). \(L\) can be a fractional power of a uniformly elliptic operator with \(C^1\). Here, the authors 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. These 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, it is proved that \begin{itemize} \item when \(2s > 1-1/m\), for large times all solutions behave as dist\(^{1/m}\) near the boundary; \item when \(2s\leq 1-1/m\), different solutions may exhibit different boundary behavior. \end{itemize} This unexpected phenomenon is a completely new and it is not present in the case of local semilinear elliptic equations.
scientific article