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

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.