Initial Value Problems and Extrapolation Spaces

Abstract

AbstractUp until now we have dealt with evolutionary equations of the form $$\displaystyle \big (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\big )U=F $$ ( ∂ t , ν M ( ∂ t , ν ) + A ¯ ) U = F for some given $$F\in L_{2,\nu }(\mathbb {R};H)$$ F ∈ L 2 , 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 $$\nu \in \mathbb {R}$$ ν ∈ ℝ chosen suitably large. Under these conditions, we established that the solution operator, "Equation missing", is eventually independent of ν and causal; that is, if F = 0 on $$\left (-\infty ,a\right ]$$ − ∞ , a for some $$a\in \mathbb {R}$$ a ∈ ℝ , then so too is U.