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Uniqueness of \(C _{0}\)-semigroups on a general locally convex vector space and an application. (English) Zbl 1227.47026
In this paper, the authors generalize a well known result due to W. Arendt [in: R. Nagel (ed.), “One-parameter semigroups of positive operators”, Lecture Notes in Mathematics 1184. Berlin etc.: Springer-Verlag (1986; Zbl 0585.47030), doi:10.1007/BFb0074924, Theorem 1.33, p. 46]. Their result concerns the uniqueness of \(C_0\)-semigroups in the setting of general locally convex vector spaces. More precisely, they prove that the cores are the only domains of uniqueness for \(C_0\)-semigroups on locally convex spaces. They apply their result to the mass transport equation and they find a necessary and sufficient condition for the uniqueness of an \(L^1\) weak solution of such an equation.

MSC:
47D06 One-parameter semigroups and linear evolution equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
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