On an estimate for the modulus of continuity of a nonlinear inverse problem

A reverse time problem is considered for a semilinear parabolic equation. Two-sided estimates are obtained for the norms of values of a nonlinear operator in terms of the norms of values of the corresponding linear operator. As a consequence, two-sided estimates are established for the modulus of continuity of a semilinear inverse problem in terms of the modulus of continuity of the corresponding linear problem.