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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 $k\in \bbfN$, $0< \tau\leq \infty$. 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 $E(\alpha, \beta):= \{\lambda\in \bbfC$: $\text{Re} (\lambda) \geq \beta$, $|\text{Im } \lambda|\leq \exp(\alpha \text{ Re} (\lambda))\}$ 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 $\alpha> 0$, $\beta>0$, $M\geq 0$, $-1< k\in \bbfR$ 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 {\it H. Kellermann} and {\it M. Hieber} [Integrated semigroups, J. Funct. Anal. 84, No. 1, 160- 180 (1989; Zbl 0689.47014)] as well as those of {\it G. Lumer} [C. R. Acad. Sci., Paris, Sér. I 310, No. 7, 577-580 (1990; Zbl 0693.47033)].

MSC:
47D06One-parameter semigroups and linear evolution equations
34G10Linear ODE in abstract spaces
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