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Regularized semigroups and systems of linear partial differential equations. (English) Zbl 0789.35075
This is a nice expository paper on the subject of the abstract Cauchy problem \[ y'(t)= A(y(t)), \qquad y(0)=x, \] where \(A\) is a closed operator on a subset of some Banach space \(E\) into \(E\) and \(x\in E\). The discussion concentrates on the case in which \(A\) is the generator of a regularized semigroup of operators on \(E\). An effort is made to unify the results on this subject scattered in an extensive bibliography. The concept of regularized semigroup is connected with that of a regularizing operator \(C\) for \(A\).
If \(\emptyset\neq \Omega\subset\mathbb{C}\) is disjoint from the point spectrum of \(A\) then a bounded one to one operator \(C\) on \(E\) is a regularizing operator for \(A\) on \(\Omega\) provided that the following are satisfied. First, \(CA=AC\) on the domain of \(A\) and the image of \(C\) is contained in \(\bigcap_{\lambda\in\Omega} \text{Im}(\lambda-A)\). Last, the function \(\psi:\lambda\to (\lambda-A)^{-1}C\), which maps \(\Omega\) into \(L(E)\) because of the first condition, is holomorphic on \(\Omega\).
If \(A\) generates a regularized semigroup then the Cauchy problems mentioned above have unique exponentially bounded solutions which are defined in terms of the semigroup. One of the theorems is that if the point spectrum of \(A\) is disjoint from the positive half plane \(\Omega= (\text{Re } \lambda>0)\) then \(A\) generates a regularized semigroup if and only if there is an operator \(C\) which regularizes \(A\) on \(\Omega\) so that the corresponding function \(\psi\) on \(\Omega\) satisfies \(\|\psi(\lambda)\|\leq p(| \lambda|)\) on \(\Omega\) for some polynomial \(p\).
In a concluding section, the abstract functional analytic results are shown to be useful in a number of specific settings.

MSC:
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
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