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Counterexamples to local existence for semi-linear wave equations. (English) Zbl 0855.35080
The author studies how much regularity of initial data is needed to ensure existence of a local solution to a semilinear wave equation. He gives counterexamples to local existence involving the Cauchy problem for semilinear wave equation \(\partial^2_t u- \sum^3_{i= 1} \partial^2_{x_l} u= (D^l u) D^{k- l} u\), where \(D= (\partial_{x_1}- \partial_t)\) and \(0\leq l\leq k- l\leq 1\). The given counterexamples (which improve some previous ones due to the author and C. Sogge) are obtained by concentrating solutions in one direction, close to a characteristic.
Reviewer: C.Popa (Iaşi)

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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