an:03903126
Zbl 0566.35017
Alinhac, S.; Metivier, G.
Propagation de l'analyticit?? des solutions d'??quations non- lin??aires de type principal
FR
Commun. Partial Differ. Equations 9, No. 6, 523-537 (1984).
00149365
1984
j
35F25 35A20
propagation of analyticity; nonlinear equation; linearization; real principal type
Let u be a \(C^{\infty}\) solution in \(\Omega\) for the nonlinear equation \(F(y,u(y),...,u^{(\alpha)}(y),...)=0,\) where F is an analytic function of the variables \((y,u^{(\alpha)}\), \(| \alpha | \leq m)\). Denote by P the linearization of F at the solution u and let \(\phi\) be some \(C^ 2\) function on \(\Omega\) with real values such that \(d\phi\) \(\neq 0\) on \(\Omega\). Further consider \(y^ 0\) with \(\phi (y^ 0)=0\), assume that P is of real principal type, that \(\{\phi =0\}\) is noncharacteristic for P and that all real characteristics for P are transverse to \(\{\phi =0\}\) at \(y^ 0\). The main result of the article is then the following: If u is analytic for \(\phi <0\), then it is also analytic in a full neighborhood of \(y^ 0\). The result extends a previous result of the same authors for nonlinear hyperbolic systems [Invent. Math. 75, 189-204 (1984; Zbl 0545.35063)].
O.Liess
Zbl 0545.35063