Optimal estimates for fractional fast diffusion equations
Using parabolic comparison results from a previous work [ibid. (9) 101, No. 5, 553--582 (2014; Zbl 1290.35317)], the authors obtain precise a priori estimates with best constants for the solutions of the fractional diffusion equation \(\partial_t u+(-\Delta)^{\sigma/2}A(u)=f\), \(0<\sigma <2\), \(0<m\leq 1\). Some extra regularity conditions are accepted as needed, like \(A\) being smooth with \(A'(u) >0\) for all \(u >0\). The problem is posed in the whole space \(\mathbb{R}^N\). Special attention is paid to cases of the form \(A(u) =u^m\) with \(m >0\); the equation is then called the fractional heat equation when \(m =1\), the fractional porous medium equation if \(m >1\), and the fractional fast diffusion equation if \(m <1\). The linear equation \(\partial_t+(-\Delta)^{\sigma/2}A(u)=0\) is a model of the so-called anomalous diffusion, a much studied topic in physics. Classical results for \(\sigma=2\) are recovered in the limit.
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