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On the analyticity of solutions of first-order nonlinear PDE. (English) Zbl 0758.35018
(From the author’s abstract:) Let \(x\in R^ m\), \(t\in R^ 1\) and \(u\in C^ 2\). We discuss local and microlocal analyticity for solutions \(u\) to the nonlinear equation \(u_ t=f(x,t,u,u_ x)\). Here \(f(x,t,\zeta_ 0,\zeta)\) is complex valued and analytic in all arguments. We also assume \(f\) to be holomorphic in \((\zeta_ 0,\zeta)\in C\times C^ m\). In particular we show that \(WF_ Au\subseteq\text{Char}(L^ u)\), where \(WF_ A\) denotes the analytic wave front set and \(\text{Char}(L^ u)\) is the characteristic of the linearized operator \(L^ u=\partial/\partial t-\sum\partial f/\partial\zeta_ j(x,t,u,u_ x)\partial/\partial x_ j\). If we assume that \(u\in C^ 3(R^ m\times R)\), then we show the analyticity of \(u\) propagates along the elliptic submanifolds of \(L^ u\).
Reviewer: E.Barron (Chicago)

MSC:
35F20 Nonlinear first-order PDEs
35A20 Analyticity in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35S30 Fourier integral operators applied to PDEs
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