Non-Autonomous Evolutionary Equations

Abstract

AbstractPreviously, we focussed on evolutionary equations of the form $$\displaystyle \left (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\right )U=F. $$ ∂ t , ν M ( ∂ t , ν ) + A ¯ U = F . In this chapter, where we turn back to well-posedness issues, we replace the material law operator M(∂t,ν), which is invariant under translations in time, by an operator of the form $$\displaystyle \mathcal {M}+\partial _{t,\nu }^{-1}\mathcal {N}, $$ ℳ + ∂ t , ν − 1 N , where both $$\mathcal {M}$$ ℳ and $$\mathcal {N}$$ N are bounded linear operators in $$L_{2,\nu }(\mathbb {R};H)$$ L 2 , ν ( ℝ ; H ) . Thus, it is the aim in the following to provide criteria on $$\mathcal {M}$$ ℳ and $$\mathcal {N}$$ N under which the operator $$\displaystyle \partial _{t,\nu }\mathcal {M}+\mathcal {N}+A $$ ∂ t , ν ℳ + N + A is closable with continuous invertible closure in $$L_{2,\nu }(\mathbb {R};H)$$ L 2 , ν ( ℝ ; H ) . In passing, we shall also replace the skew-selfadjointness of A by a suitable real part condition. Under additional conditions on $$\mathcal {M}$$ ℳ and $$\mathcal {N}$$ N , we will also see that the solution operator is causal. Finally, we will put the autonomous version of Picard’s theorem into perspective of the non-autonomous variant developed here.