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Variable order space-fractional diffusion equation derived as an importantmodel 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 ofthe nonlocal vector calculus. Then some regularity properties have beenobtained under the variable-order Sobolev space framework. By choosing aspace-independent weight function and using the technique of the nonlocalvector calculus, a Carleman type estimate has been obtained. At last, theCarleman type estimate has been used both to a backward diffusion problem andan inverse source problem to obtain some uniqueness and stability results.

A Carleman type estimate of variable order space-fractional diffusion equations and applications to some inverse problems