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Non-uniqueness in hyperbolic Cauchy problems. (English) Zbl 0649.35051
The authors give some examples of Cauchy problems for hyperbolic equations of order two for which non-uniqueness holds. A hyperbolic equation is said to possess the uniqueness property at a point \((t_ 0,x_ 0)\) if, for any solution \(u\in C^{\infty}(V)\), \(V\) being a neighborhood of \((t_ 0,x_ 0)\) such that \(u=0\) on \(V\cap (t<t_ 0)\), \(u=0\) identically on a neighborhood of \((t_ 0,x_ 0)\). In particular, for the second order strictly hyperbolic equation \[ (\partial /\partial t)\quad 2u=a(t)(\partial /\partial x)\quad 2u+b(t,x)u \] where \(a(t)\) is non-negative and \(b(t,x)\) complex-valued, the authors show the existence of \(a(t)\) and \(b(t,x)\) satisfying certain conditions with \(b(t,x)=0\) for \(t\leq 0\) such that the above equation does not have the uniqueness property at any point of the hyperplane \((t=0)\). Analogous results are also obtained for weakly hyperbolic equations.
Reviewer: E.C.Young

35L15 Initial value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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