Sharp Boundary Estimates and Harnack Inequalities for Fractional Porous Medium Type Equations

Abstract. This paper provides sharp quantitative and constructive estimates of nonnegative solutions [Formula: see text] to the nonlinear fractional diffusion equation, [Formula: see text] also known as filtration equation, posed in a smooth bounded domain [Formula: see text] with suitable homogeneous Dirichlet boundary conditions. Both the operator [Formula: see text] and the nonlinearity [Formula: see text] belong to a general class. The assumptions on the operator [Formula: see text] are set in terms of the kernel of [Formula: see text] and [Formula: see text], depending on the result, and allow for operators with degenerate kernel at the boundary of [Formula: see text]. The main examples of [Formula: see text] are the three different Dirichlet fractional Laplacians on bounded domains, and the nonlinearity can be nonhomogeneous, for instance, [Formula: see text]. Previous results were known in the porous medium case, i.e., [Formula: see text] with [Formula: see text], whose homogeneity allows us to considerably simplify the proofs. We take the opportunity of exposing a complete basic theory of existence, uniqueness, and boundedness for a quite general class of weak (dual) solutions, scattered in previous papers by Vazquez and the first author. Our aim here is to perform the next step: a delicate analysis of regularity through quantitative, constructive (all the constants can be explicitly computed), and sharp a priori estimates. Our first main results are global Harnack-type inequalities of the form [Formula: see text] where the expressions of [Formula: see text], and [Formula: see text] are explicit and change according to the choice of operator [Formula: see text] and nonlinearity [Formula: see text]. The sharpness of such estimates is proven by means of examples and counterexamples: on the one hand, we can match the powers (i.e., [Formula: see text]) when the operator has a nondegenerate kernel. On the other hand, when the operator [Formula: see text] has a kernel that degenerates at the boundary [Formula: see text], there appears an intriguing anomalous boundary behavior: the size of the initial data determines the sharp boundary behavior of the solution, which can be different for “small” and “large” initial data. We conclude the paper with higher regularity results: solutions are always Hölder continuous in the interior, and even classical when the operator allows it.