Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Abstract

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.