Lemle, Ludovic Dan; Wu, Liming Uniqueness of \(C _{0}\)-semigroups on a general locally convex vector space and an application. (English) Zbl 1227.47026 Semigroup Forum 82, No. 3, 485-496 (2011). 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. Reviewer: Vincenzo Vespri (Firenze) Cited in 7 Documents 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 Keywords:\(C_{0}\)-semigroups; \(L^{1}\)-uniqueness; weak solution; mass transport equation Citations:Zbl 0585.47030 PDFBibTeX XMLCite \textit{L. D. Lemle} and \textit{L. Wu}, Semigroup Forum 82, No. 3, 485--496 (2011; Zbl 1227.47026) Full Text: DOI arXiv References: [1] Arendt, W.: The abstract Cauchy problem, special semigroups and perturbation. In: Nagel, R. (eds.) One Parameter Semigroups of Positive Operators. Lect. Notes in Math., vol. 1184. Springer, Berlin (1986) [2] Choe, Y.H.: C 0-semigroups on locally convex space. J. Math. Anal. Appl. 106, 293–320 (1985) · Zbl 0595.47031 · doi:10.1016/0022-247X(85)90115-5 [3] Eberle, A.: Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators. Lect. Notes Math., vol. 1718. Springer, Berlin (1999) · Zbl 0957.60002 [4] Fattorini, H.O.: Ordinary differential equations in linear topological spaces. J. Differ. Equ. 5, 72–105 (1968) · Zbl 0175.15101 [5] Lemle, L.D., Wu, L.M.: Unicité des pré-générateurs dans les espaces localement convexes. C. R. Math. 347, 1281–1284 (2009) · Zbl 1175.47037 · doi:10.1016/j.crma.2009.10.001 [6] Lemle, L.D., Bînzar, T., Pater, F.: L 1-uniqueness of weak solution for the one-dimensional mass transport equation In: A.I.P. Conference Proceedings, vol. 1168, pp. 160–163. Melville (2009) · Zbl 1182.80003 [7] Moore, R.T.: Banach algebras of operators in locally convex spaces. Bull. Am. Math. Soc. 75, 68–73 (1969) · Zbl 0189.13302 · doi:10.1090/S0002-9904-1969-12147-6 [8] Wu, L.: Uniqueness of Schrödinger operators restricted in a domain. J. Funct. Anal. 153(2) 276–319 (1998) · Zbl 0913.35035 · doi:10.1006/jfan.1997.3181 [9] Wu, L., Zhang, Y.: A new topological approach to the L uniqueness of operators and the L 1-uniqueness of Fokker-Planck equations. J. Funct. Anal. 241, 557–610 (2006) · Zbl 1111.47035 · doi:10.1016/j.jfa.2006.04.020 [10] Yosida, K.: Functional Analysis. Springer, New York (1971) · Zbl 0217.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.