## Evolution d’une onde simple pour des équations non-linéaires générales. (Evolution of a simple wave for general nonlinear equations).(French)Zbl 0586.35005

Journ. Équ. Dériv. Partielles, Saint-Jean-De-Monts 1985, No. 1, Exp. No. 10, 7 p. (1985).
Let u be a solution of the nonlinear (evolution) equation $F(t,x,...,D^{\alpha}u,...)=0,\quad | \alpha | \leq m,\quad (t,x)\in \Omega \times {\mathbb{R}}^ n$ such that $$u\in H^{s+m}_{loc}(\Omega)$$, $$s>(n+7)/2$$ and $$0\in \Omega$$. Assume that the linearized operator P associated to F is strictly hyperbolic with respect to the surface $$t=Const$$. in $$\Omega$$ and that $$\Omega_+=\{(t,x)\in \Omega;\quad t\geq 0\}$$ is an influence domain of $$\Omega_-=\{(t,x)\in \Omega;\quad t\leq 0\}.$$ In addition, assume that ($$\exists)$$ a surface S, characteristic for P, $$x_ 1=\phi (t,x_ 2,...,x_ n)$$ in $$\Omega$$ with $$\phi \in H^{\sigma}_{loc}$$, $$\sigma >(n+3)/2,$$ $$\phi (0)=0$$, such that for $$t<0$$, $$S\in C^{\infty}$$ and $$u\in H^{s+m,\infty}(S)$$. Then, S is $$C^{\infty}$$ in $$\Omega$$, and $$u\in H^{s+m,\infty}(S)$$ in $$\Omega$$.
Reviewer: G.Moroşanu

### MSC:

 35B65 Smoothness and regularity of solutions to PDEs 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 58J47 Propagation of singularities; initial value problems on manifolds 35G20 Nonlinear higher-order PDEs
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