  We consider a class of elliptic and parabolic problems, featuring a specific
nonlocal operator of fractional-laplacian type, where integration is taken on
variable domains. Both elliptic and parabolic problems are proved to be
uniquely solvable in the viscosity sense. Moreover, some spectral properties of
the elliptic operator are investigated, proving existence and simplicity of the
first eigenvalue. Eventually, parabolic solutions are proven to converge to the
corresponding limiting elliptic solution in the long-time limit.
