Haraux, Alain On a uniqueness theorem of L. Amerio and G. Prouse. (English) Zbl 0555.35090 Proc. R. Soc. Edinb., Sect. A 96, 221-230 (1984). The author considers the boundary value problem \(f-u_{tt}+\Delta u\in \beta (u_ t)\) on \(J\times \Omega\); \(u=0\) on \(J\times \partial \Omega\), where J is a closed real interval, \(\Omega\) is a bounded open set in \({\mathbb{R}}^ n\) with regular boundary, \(t\to f(t,x)\) is locally integrable, \(x\to f(t,x)\) is square integrable on \(\Omega\) and \(\beta\) is a monotone operator with real domain whose values are subsets of reals and which satisfies \(\beta (0)=\{0\}\). Generalizing previous results by Amerio and Prouse, uniqueness theorems for weak solutions are established under certain additional hypotheses on the solutions u, on the forcing term f and on the damping term \(\beta\) such as integrability of \(u_{tt}-\Delta u\), absolute continuity of \(u_ t\), almost periodicity of u, periodicity of f, strong (or strict) monotonicity of \(\beta\), etc. Examples are included to indicate the necessity of the various hypotheses. Reviewer: P.Ramankutty Cited in 1 Document MSC: 35L70 Second-order nonlinear hyperbolic equations 47H05 Monotone operators and generalizations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:monotone operator; uniqueness; weak solutions; integrability; absolute continuity; almost periodicity; monotonicity PDFBibTeX XMLCite \textit{A. Haraux}, Proc. R. Soc. Edinb., Sect. A, Math. 96, 221--230 (1984; Zbl 0555.35090) Full Text: DOI References: [1] DOI: 10.1016/0022-0396(80)90017-0 · Zbl 0413.35011 · doi:10.1016/0022-0396(80)90017-0 [2] Biroli, Ricerche Mat. 22 pp 190– (1973) [3] Amerio, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 46 pp 1– (1969) [4] Haraux, Proc. Roy. Soc. Edinburgh Sect. A 94 pp 195– (1983) · Zbl 0589.35076 · doi:10.1017/S0308210500015584 [5] Brezis, Opérateurs maximaux monotones et semi-groups de contractions dans les espaces de Hilbert (1973) [6] DOI: 10.1016/0022-0396(82)90006-7 · Zbl 0458.35063 · doi:10.1016/0022-0396(82)90006-7 [7] Grun-Rehomme, J. Math. Pures Appl. 56 pp 149– (1977) [8] DOI: 10.1007/BF02760227 · doi:10.1007/BF02760227 [9] Brezis, J. Math. Pures Appl. 51 pp 1– (1972) [10] Haraux, Proc. Roy. Soc. Edinburgh Sect. A 84 pp 213– (1979) · Zbl 0429.35013 · doi:10.1017/S0308210500017091 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.