Global UCP For Parabolic Fractional $p$-Laplace Equation With Very Rough Potentials

TL;DR

This paper proves a global unique continuation principle for a parabolic fractional p-Laplace equation with very rough potentials, advancing understanding in a challenging nonlocal PDE context.

Contribution

It establishes the first known global unique continuation result for this class of equations with rough potentials, without relying on extension techniques or Carleman estimates.

Findings

Unique continuation holds for the specified equation with rough potentials.

The proof avoids traditional extension and Carleman estimate methods.

Result is new even for the fractional p-Laplace operator, with local case still open.

Abstract

We show that the global unique continuation principle holds for the parabolic fractional $p-$Laplace equation with very rough potentials $V(x,t) \in L^{p'}_tW^{-s,p'}_x$. Whereas the result is new even for the fractional $p-$Laplace operator, the corresponding local problem remains open even with zero potential. The short proof eschews extension techniques and Carleman estimates.