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Some examples of hyperbolic equations without local solvability. (English) Zbl 0702.35146
Three examples are presented, illustrating a feature of initial value problems for hyperbolic equations of the form $u_{tt}(x,t)- (A(x,t)u_ x(x,t))_ x=f(x),\quad u(x,0)=u_ 0(x),\quad u_ t(x,0)=u_ 1(x),$ with smooth right-hand sides but only Hölder continuous, strictly positive and bounded coefficient A. The examples show that in general we cannot expect even only a distribution valued local-in-time $$C_ 2$$-solution. This is demonstrated by showing that the solution lacks the continuity property of distribution-valued functions. The construction of the examples displaying this feature is based on two ODE Lemmas. The examples themselves are presented in the form of theorems. The ‘free space’ case is considered first. The ‘spatially’ periodic case is then obtained as a technical variant of the first example. The last example is a refinement of the first showing that in general not even a distribution valued local-in-time $$C_ 1$$-solution can be expected to exist.
Reviewer: R.Picard

MSC:
 35L15 Initial value problems for second-order hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35R05 PDEs with low regular coefficients and/or low regular data
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References:
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