Nonlinear wave equations with data singular at one point. (English) Zbl 0552.35055

Beals, Michael

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

Keywords:

propagation of singularities; semi-linear wave equations; smoothness of the data

Citations:

Zbl 0527.00007