On the Cauchy problem for analytic semigroups with weak singularity (Q1192560)

Let \(A\) be the generator of an analytic one-parameter semigroup on the Banach space \(X\). Assume that \(A\) satisfies the following weak singularity condition for some \(\theta\) with \(1/2<\theta<1\): \[ |(A- \lambda)^{-1}|\leq C(\varepsilon)(1+|\lambda|)^{- \theta} \] for all \(\lambda\in \{z\): \(|\arg(z-z_ 0|<\pi/2+\varepsilon\}\) where \(z_ 0<0\) is a fixed constant and \(0<\varepsilon<\omega<\pi/2\) (here \(\omega\) is the constant characterizing the domain of analyticity of the semigroup). The author considers the following Cauchy problem: \[ {{du} \over {dt}}(t)= Au(t)+f(t), \qquad u(0)=u_ 0. \] The main result states that if the function \(f\) belongs to a certain Besov space and if \(u_ 0\) is in \(D((-A)^ \alpha)\) with \(1-\theta<\alpha<\theta\) then this Cauchy problem has a unique solution which is given by the variation of constant-formula.