  In this paper, we consider a Lewy-Stampacchia-type inequality for the
fractional Laplacian on a bounded domain in Euclidean space. Using this
inequality, we can show the well-posedness of fractional-type anomalous
unidirectional diffusion equations. This study is an extension of the work by
Akagi-Kimura (2019) for the standard Laplacian. However, there exist several
difficulties due to the nonlocal feature of the fractional Laplacian. We
overcome those difficulties employing the Caffarelli-Silvestre extension of the
fractional Laplacian.
