Uniqueness for fractional parabolic and elliptic equations with drift

Abstract

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of linear, nonlocal parabolic problems with a drift. More precisely, the problem is nonlocal due to the presence of the fractional Laplacian as diffusion operator. The drift term is driven by a smooth enough, possibly unbounded vector field $ b $ which satisfies a suitable growth condition in the set $ \{x\in {\mathbb{R}}^N:\langle b(x), x\rangle> 0\} $. In general, our uniqueness class includes unbounded solutions; in particular, we get uniqueness of bounded solutions. Furthermore, we show sharpness of the hypothesis on the drift term $ b $; in fact we show that, if the drift term $ b $ violates, in an appropriate sense, the mentioned growth condition (see (2.5)), then infinitely many bounded solutions to the problem exist. Finally, we also investigate uniqueness of a linear, nonlocal elliptic equation with a drift term obtaining similar results.