On initial inverse problem for Caputo fractional elliptic equation

Abstract

In this work, we are interested to study the initial inverse problem for elliptic equation with Caputo derivative. We  obtained some  different results regarding the well-posedness  of the solution corresponding to the given  input data.   In addition, we also derive the upper bound of the derivative of the mild solution. The principal analysis is based on the estimation of  the Mittag-Leffler function, combined with main analysis in Hilbert scale authorized, and we also use fixed point techniques to prove the inequalities in this paper.