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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