A note on the norm-continuity for evolution families arising from non-autonomous forms

Abstract

We consider evolution equations of the forṁwhere A(t), t ∈ [0, T ], are associated with a non-autonomous sesquilinear form a(t, ·, ·) on a Hilbert space H with constant domain V ⊂ H. In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces V, H and on the dual space V ′ of V. The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrödinger operator with time dependent potentials.

Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities

“…The thesis of previous lemma remains true if E m (t, s) is replaced with the adjoint E m (t, s) * and E(t, s) is replaced by E(t, s) * . Indeed, the following formula holds for the adjoint (see also [19,22]):…”

Solutions to Nonlocal Evolution Equations Governed by Non-autonomous Forms and Demicontinuous Nonlinearities

“…The thesis of previous lemma remains true if Em(t, s) is replaced with the adjoint Em(t, s) * and E(t, s) is replaced by E(t, s) * . Indeed, the following formula holds for the adjoint (see also [19,22]):…”

Solutions of nonlocal semilinear non-autonomous evolution equations with $L^2-$maximal regularity

“…The thesis of the previous lemma remains true if E m (t, s) is replaced with the adjoint E m (t, s) * and E(t, s) is replaced by E(t, s) * . Indeed, the following formula holds for the adjoint (see [15,16])…”