The authors study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation \[ u_t = |x|^{\gamma}\nabla \cdot(|x|^{-\beta}\nabla u^m), \] with \(0 < m < 1\) on cylinders of the type \((0, T) \times\mathbb{R}^N\). The weights \(|x|^{\gamma}\) and \(|x|^{-\beta}\), with \(\gamma< N\) and \(\gamma-2 < \beta \leq \gamma (N-2)/N\) can be both degenerate and singular and need not belong to the class \(\mathcal{A}_2\), a typical assumption for this kind of problems. The authors have chosen this range of parameters because it is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting.
The weights considered are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. They are compelled therefore to prove quantitative – with computable constants – upper and lower estimates for local weak solutions. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, they obtain Hölder continuity of the solutions, with a quantitative exponent. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Worth to be noted is that these techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, they also prove quantitative estimates, extending known results to a wider class of weights.