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A class of exponentially bounded distribution semigroups. (English) Zbl 1004.47024

The authors study a class of distribution semigroups (0-exponentially bounded distribution semigroup, 0-EDSG) for which an infinitesimal generator \(A\) is not necessarily densely defined. The composition law for a 0-EDSG is: \(\langle S(t+s,x),\beta(t,s)\rangle=\langle S(t,S( ,x)),\beta(t,s)\rangle\), \(\beta\in K_1(R^2)\), \(\operatorname{supp}\beta\subset[0,\infty)\times[0,\infty)\). Their class is in fact identical to Wang-Kunstmann’s, what follows from the results of the paper. The authors use a quite different approach which is in a sense more direct. They apply these results to an equation of the form \(\partial u/\partial t = Au + f\), where \(A\) is not necessarily densely defined and \(f\) is an exponential vector valued distribution supported by \([0,\infty)\).

MSC:

47D06 One-parameter semigroups and linear evolution equations
47D62 Integrated semigroups
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