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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
Cauchy problem
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