A class of inverse problems for evolution equations with the Riemann‐Liouville derivative in the sectorial case

Abstract

We investigate the unique solvability of a class of linear inverse problems with a time‐independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Riemann‐Liouville derivative. We assume that the operator in the right‐hand side of the equation generates a family of resolving operators for the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that the inverse problem is well‐posed with respect to the graph norm of the generating operator only. A well‐posedness criterion is found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time‐fractional partial differential equations. An example, in particular, shows that in the case of an unbounded generating operator, the inverse problem can be ill‐posed with respect to the norm of the whole space.