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Nonlinear wave equations with data singular at one point. (English) Zbl 0552.35055
Microlocal analysis, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 27, 83-95 (1984).
[For the entire collection see Zbl 0527.00007.]
The propagation of singularities to semi-linear wave equations in n space dimensions is discussed. From the introduction: ”If $$n>1$$, there is a function $$\beta \in C_ 0^{\infty}({\mathbb{R}}^{n+1})$$ and a choice of initial data singular only at the origin for which the solution of $$\square u=\beta u^ 3$$ is singular on the entire interior of the light cone over the origin. On the other hand, if $$n=1$$ this spreading of singularities does not occur. In higher dimensions, additional conditions on the smoothness of the data with respect to angular differentiation $$(x_ j \partial /\partial x_ k-x_ k \partial /\partial x_ j)$$ will prevent the spreading of singularities.”
The paper is clearly written, but not all the proofs of the results stated are completely given.
Reviewer: H.D.Alber

##### MSC:
 35L67 Shocks and singularities for hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations