The purpose of this thesis is to ascertain whether linear differential operators with vanishing coefficients make suitable operators in the Cauchy problem (CP) utx,t -Aux,t=0on U×0,∞ u=gonU× t=0, CP where U ⊂ R is a bounded and open set. Well-posedness for linear Cauchy problems - characterized by existence, uniqueness, and continuous dependence on the initial data - depends on a ray being in the spectrum of the operator and an estimate for the resolvent operator along this ray. This was originally shown by Hille and Yosida [34] for operators when (0, ∞) ⊂ ρ( A) and later generalized by Feller, Miyadera, and Phillips in [12, 20, 23] when (a, ∞) ⊂ ρ(A) (for some a ∈ R ).;We establish ill-posedness of (CP) by analyzing the spectral properties of A and showing σ(A) = C for a wide variety of differential operators with vanishing coefficients. If the differential operator A is of the form p n(x, ∂x) f(x) where pn( x, ∂x) is an n-th order differentiable operator and f has roots of finite multiplicity, we develop simple criteria for establishing ill-posedness of (CP). For the operator A = f(x) pn(∂x), we establish point spectral results when f is real-valued with only simple roots and n ≥ 2, in particular we show σp( Ā) = C in this case.;Much less is known when the coefficients of the differential operator A depend on time. For these non-autonomous Cauchy problems (NCPs) only sufficient conditions for well-posedness are known, with necessary conditions still lacking. In this thesis we make strides with establishing necessary spectral conditions for well-posed NCPs in the case where the family of operators is continuous.
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