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Local integrated semigroups: Evolution with jumps of regularity. (English) Zbl 0833.47034
Let \(A\) be a closed operator in a Banach space \(S\), and let \(0< \tau\leq \infty\). The authors consider the Cauchy problem \(C_{k+1} (\tau)\) \[ v\in C([0, \tau), D(A))\cap C^1 ([0, \tau), X) \] \[ v'(t)= Av (t)+ t^k/ k! x,\;t\in [0, \tau), \qquad v(0)= 0, \] where \(x\in D(A)\), and \(D(A)\) is equipped with the graph norm.
The basic theorem reads as follows: Let \(k\in \mathbb{N}\), \(0< \tau\leq \infty\). Assume that \(C_{k+1} (\tau)\) is uniquely solvable for all \(x\in D(A)\) (i.e. \(C_{k+1} (\tau)\) is well-posed). Then for all \(0< \alpha< \tau/k\) there exists \(\beta >0\), \(M\geq 0\) such that \(E(\alpha, \beta):= \{\lambda\in \mathbb{C}\): \(\text{Re} (\lambda) \geq \beta\), \(|\text{Im } \lambda|\leq \exp(\alpha \text{ Re} (\lambda))\}\) is contained in the resolvent set \(\rho(A)\) and \(|(\lambda- A)^{- 1} |\leq M|\lambda|^k\) for all \(\lambda\in E(\alpha, \beta)\).
The converse theorem holds in the following manner: Let \(\alpha> 0\), \(\beta>0\), \(M\geq 0\), \(-1< k\in \mathbb{R}\) and assume \(E(\alpha, \beta) \subset \rho(A)\) and \(|(\lambda- A)^{-1} |\leq M|\lambda|^k\) \((\lambda\in E(\alpha, \beta))\). If \(N\ni p> k+1\), \(\tau= \alpha (p- (k+1))\) then \(C_{p+1} (\tau)\) is well-posed.
The results given here generalize those gained by H. Kellermann and M. Hieber [Integrated semigroups, J. Funct. Anal. 84, No. 1, 160- 180 (1989; Zbl 0689.47014)] as well as those of G. Lumer [C. R. Acad. Sci., Paris, Sér. I 310, No. 7, 577-580 (1990; Zbl 0693.47033)].

MSC:
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
Keywords:
Cauchy problem
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