Initial value problems and extrapolation spaces

Abstract

Up until now we have dealt with evolutionary equations of the form (∂t,νM(∂t,ν)+A¯)U=F (∂ t,ν M(∂ t,ν )+A)U=F for some given F∈ L2,ν(ℝ; H) for some Hilbert space H, a skew-selfadjoint operator A in H and a material law M defined on a suitable half-plane satisfying an appropriate positive definiteness condition with ν∈ ℝ chosen suitably large. Under these conditions, we established that the solution operator,, is eventually independent of ν and causal; that is, if F = 0 on (− ∞, a] for some a∈ ℝ, then so too is U.