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Evolution d’une onde simple pour des équations non-linéaires générales. (Evolution of a simple wave for general nonlinear equations). (French) Zbl 0629.35088
Current topics in partial differential equations, Pap. dedic. S. Mizohata Occas. 60th Birthday, 63-90 (1986).
[For the entire collection see Zbl 0604.00006.]
Let u be a real-valued solution in $$\Omega \subset R^ n$$ (open set) of the equation $F(x,u(x),...,u^{(\alpha)}(x),...)\quad =0\quad | \alpha | <m-1,\quad u^{\alpha}(x)\equiv \partial^{| \alpha |}u(x)/\partial x^{\alpha},$ where F is $$C^{\infty}$$ of its arguments. Moreover $$P=\sum_{| \alpha | \leq m} \partial F/\partial u^{\alpha}\partial_ x^{\alpha}$$ is strictly hyperbolic with respect to the surfaces t $$=$$ constant, and there exists a surface $$S: x_ 1=\Phi (t,x')\in H^{\sigma}_{loc}$$ $$(\sigma >n/2+7/2)$$ that is a characteristic for P and $$\phi \in C^{\infty}$$ for $$t<0$$. The main result of the paper is the following: if for $$s>n/2+7/2$$ and for $$t<0$$, $$u\in H^{s+m,\infty}(S)$$, then S is of class $$C^{\infty}$$ in $$\Omega$$ and $$u\in H^{s+m,\infty}(S)$$.
Reviewer: R.Salvi

##### MSC:
 35L75 Higher-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35L25 Higher-order hyperbolic equations
##### Keywords:
singularity; Sobolev spaces; strictly hyperbolic; characteristic