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Zbl 0878.35080
Zworski, Maciej
(Melrose, R.B.; Sá Barreto, A.)
Semilinear diffraction of conormal waves. (Joint work with Melrose and Sá Barreto). (English)
[J] Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau Sémin. 1992-1993, Exp. No.2, 21 p. (1993).

This exposé deals with the conormal regularity for a class of mixed hyperbolic problems with a diffractive boundary. So assume that $P$ is a strictly hyperbolic operator, $X$ is a $C^\infty$ manifold with the boundary $\partial X$, $X_-= \{x\in X: \varphi(x) <-T\}$ and $\varphi\in C^\infty$ is a time function for $P$. Consider the mixed problem: $$Pu= f(x,u),\ f\in C^\infty,\ u|_{\partial X} =0,\ u|_{X_-} =u_0. \tag 1$$ The initial data is supposed to be conormal to the incident front $F$. Let $\nu(X_-,F)$ be the Lie algebra of $C^\infty$ vector fields in $X_-$ tangent to $F$. Then the space of distributions of finite $L_2$-based conormal regularity with respect to $F$ is denoted by $I_kL^2 (X_-,F)$. This is the main result of this exposé: Let $u\in L^\infty_{\text {loc}} (X)$ be the solution of (1) and $u_0\in I_kL^2_{\text {loc}}(X_-)$. Then $u\in J_kL^2_{\text{loc}}(X)$, where $J_kL_{\text{loc}}^2$ is a pseudoconormal space for the manifold with boundary $X$.
[P.Popivanov (Sofia)]

MSC 2000:: *35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35A20 Analytic methods (PDE)
35L35 Higher order hyperbolic equations, boundary value problems

Keywords: propagation of singularities; conormal regularity

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