# Lisbon WADE — Webinar in Analysis and Differential Equations

## Next session

### 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 fractional energy. Our main result 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. Extinction phenomena are discussed.

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Organizers: Hugo Tavares, James Kennedy and Nicolas Van Goethem

Joint iniciative of the research centers CAMGSD, CMAFcIO and GFM.