Parameter Differentiability of the Solution of a Nonlinear Abstract Cauchy Problem

In this article, an abstract nonlinear evolution equation with a parameter appearing in the nonlinear part is considered. By using the theory of analytic semigroups and a generalization of Gronwall’s lemma for singular kernels, sufficient conditions to ensure differentiability of the solution with respect to the parameter are derived in terms of the smoothness of the nonlinear part of the equation. The results are applied to a nonlinear system of partial differential equations modeling the dynamics of shape memory alloys