  Variable order space-fractional diffusion equation derived as an important
model to describe complex anomalous diffusion phenomenon. In this article,
well-posedness theory has been constructed for equations with the "Dirichlet"
or the "Neumann" type volume-constrained conditions by using the technique of
the nonlocal vector calculus. Then some regularity properties have been
obtained under the variable-order Sobolev space framework. By choosing a
space-independent weight function and using the technique of the nonlocal
vector calculus, a Carleman type estimate has been obtained. At last, the
Carleman type estimate has been used both to a backward diffusion problem and
an inverse source problem to obtain some uniqueness and stability results.
