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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)

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
Full Text: DOI
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