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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

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