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On a uniqueness theorem of L. Amerio and G. Prouse. (English) Zbl 0555.35090

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

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)
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[1] DOI: 10.1016/0022-0396(80)90017-0 · Zbl 0413.35011 · doi:10.1016/0022-0396(80)90017-0
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