×

zbMATH — the first resource for mathematics

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
Full Text: Numdam EuDML