Semilinear wave equations with angularly smooth data. (English) Zbl 0614.35058

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1984, Conf. No. 10, 5 p. (1984).
Consider the problem \[ (P)\quad \square u=(\partial^ 2/\partial t^ 2-\sum^{n}_{1}\partial^ 2/\partial x^ 2_ i)u=f(t,x,u,Du),\quad u=u_ 0,\quad u_ t=u_ 1\quad for\quad t=0, \] where \(u_ i\in H^{s-i}({\mathbb{R}}^ n)\), \(i=0,1\), \(S=2+n/2\), and f is a \(C^{\infty}\) function of its arguments. J. M. Bony [Sémin. Goulaouic-Meyer- Schwartz, Equations Deriv. Partielles 1981-82, Exp. No.2, 11 p. (1982; Zbl 0498.35017) and ibid. 1983-1984, Exp. No.10, 27 p. (1984; Zbl 0555.35118)] has formulated ”conormal” hypotheses which are very restrictive, forcing the singularities of the data to be contained in submanifolds. J. Rauch and M. Reed [Indiana Univ. Math. J. 34, 337-353 (1985; Zbl 0537.35057)] considered conditions placed only on derivatives in directions parallel to a family of smooth hypersurfaces. Here the case of ”angularly” smooth data is considered; the hypersurfaces (spheres) flow out into surfaces (cones) which form caustics at \(x=0\). The basic idea is that by using arguments from the \(n=1\) case, one can control derivatives in one direction, allowing the ”conormal” hypotheses to be relaxed.
For \(p=(t_ 0,x_ 0)\in (0,\infty)\times {\mathbb{R}}^ n\setminus 0\) let \(L_ p\) be the union of the two backward characteristics through p which project onto the line from \(x_ 0\) to p. For p inside the light cone over 0 (so that \(L_ p\cap \{x=0\}\neq \emptyset)\) let \(N_ p\) be the backward light cone from \(L_ p\cap \{x=0\}\). Otherwise let \(N_ p=\emptyset\). Finally, set \(C_ p=L_ p\cup N_ p\). The outlines of proofs [for details see the author, Microlocal analysis, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 27, 83-95 (1984; Zbl 0552.35055)] of Theorem 1. If u is a solution of (P) with angularly smooth data, and if the data are \(C^{\infty}\) near \(C_ p\cap \{t=0\}\), then u is \(C^{\infty}\) near p; and Theorem 2. There exists a solution u of (P) with angularly smooth data, though the data are restricted to a relatively small set.
Reviewer: N.Kazarinoff


35L70 Second-order nonlinear hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
35L67 Shocks and singularities for hyperbolic equations
Full Text: Numdam EuDML