Perturbation of analytic semigroups in scales of Banach spaces and applications to linear parabolic equations with low regularity data

Abstract

We study linear perturbations of analytic semigroups defined on a scale of Banach spaces. Fitting the action of the linear perturbation between two spaces of the scale determines the spaces of existence and regularity of solutions for the perturbed semigroup, within the original scale. Also continuity of the resulting perturbed semigroup with respect to the perturbation is analyzed. As the main tools we exploit the smoothing of the original semigroup on the scale and the variation of constants formula. These general results are applied to several situations for linear partial differential equations of parabolic type. The main attention is set on low regularity perturbations of linear diffusion equations in either bounded or unbounded domains. Different scales of spaces are considered such as Lebesgue or Bessel spaces. However, the application of the abstract results are not limited to such examples and many other situatons can be considered.