On a singular evolution equation in Banach spaces (Q1063771)

We study the singular evolution equation \(tu'(t)+Au(t)=f(t)\) where the unknown function u and the given function f are defined on \({\mathbb{R}}^+\) and take values in a complex Banach space E, while A is a closed operator in E, with the following property: there exist \(\alpha\),\(\beta\in {\mathbb{R}}\) such that if \(\lambda\in {\mathbb{C}}\) and Re \(\lambda\in \{\alpha,\beta \}\), then \(\lambda\in \rho (A)\) and \(\| (\lambda - A)^{-1}\|_{L(E)}\leq M(1+| Im \lambda |)^{-1}.\) Global properties on \({\mathbb{R}}^+\) of existence, uniqueness and regularity of the solutions are studied. More explicitly we assume that f belongs to a Banach space X of functions on \({\mathbb{R}}^+\) (where \(X=W^{k,p}({\mathbb{R}}^+,E)\) or \(X=C^ k({\mathbb{R}}^+,E)\) in this case possibly with limits at 0 and/or \(\infty)\) and prove results on existence and uniqueness of a strong solution (in the sense of Da Prato-Grisvard). For this \(\alpha\) and \(\beta\) are to be chosen in a suitable way depending on k and p. We prove also maximal regularity results in suitable interpolation spaces.