Juan Luis Vázquez

Universidad Autónoma de Madrid

On the theory of fractional \(p\)-Laplacian equations

We consider the time-dependent fractional \(p\)-Laplacian equation with parameter \(p>1\) and fractional exponent \(0<s<1\). It is the gradient flow corresponding to the Gagliardo–Slobodeckii fractional energy. Our main interest is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on delicate analysis. We will concentrate on the sublinear or “fast” regime, \(1<p<2\), since it offers a richer theory. Fine bounds in the form of global Harnack inequalities are obtained as well as solutions having strong point singularities (Very Singular Solutions) that exist for a very special parameter interval. They are related to fractional elliptic problems of nonlinear eigenvalue form.