C-regularized solutions of ill-posed problems defined by strong strip-type operators

Abstract

Abstract Cauchy problems of the form d ⁢ u d ⁢ t = q ⁢ ( A ) ⁢ u + h ⁢ ( t ) {\frac{du}{dt}=q(A)u+h(t)} , 0 < t < T {0<t<T} , u ⁢ ( 0 ) = φ {u(0)=\varphi} , are studied in a Banach space X where A is a strong strip-type operator and q ⁢ ( A ) {q(A)} is a complex polynomial in A. In this case, the spectrum of A lies within a horizontal strip of height θ, and so potentially neither A nor - A {-A} generates a strongly continuous semigroup on X. Therefore, depending on the definition of q ⁢ ( A ) {q(A)} , the original problem may be severely ill-posed. We utilize a functional calculus for strip-type operators in order to define an approximate operator f β ⁢ ( A ) {f_{\beta}(A)} such that f β ⁢ ( A ) {f_{\beta}(A)} is bounded for each β > 0 {\beta>0} and f β ⁢ ( A ) ⁢ χ → q ⁢ ( A ) ⁢ χ {f_{\beta}(A)\chi\rightarrow q(A)\chi} as β → 0 {\beta\rightarrow 0} for χ in a suitable domain. We show that this approximation gives rise to regularization for the original problem with respect to a graph norm related to C-regularized semigroups. We also fit the theory of the paper into a special case where iA generates a bounded, strongly continuous group on X. Under this assumption, which implies that A is a strong strip-type operator of height 0, results follow for a wide variety of ill-posed PDEs in L p {L^{p}} spaces.