Continuity method with movable singularities for classical complex Monge-Ampere equations

On a compact Kähler manifold (X, ω), we study the strong continuity of solutions with prescribed singularities of complex Monge-Ampère equations with integrable Lebesgue densities.Moreover, we give sufficient conditions for the strong continuity of solutions when the right-hand sides are modified to include all (log) Kähler-Einstein metrics with prescribed singularities.Our findings can be interpreted as closedness of new continuity methods in which the densities vary together with the prescribed singularities.For Monge-Ampère equations of Fano type, we also prove an openness result when the singularities decrease.As an application, we deduce a strong stability result for (log-)Kähler Einstein metrics on semi-Kähler classes given as modifications of {ω}.