# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] M. S. Baouendi, C. H. Chang, and F. Trèves, Microlocal hypo-analyticity and extension of CR functions, J. Differential Geom. 18 (1983), no. 3, 331 – 391. · Zbl 0575.32019 [2] M. S. Baouendi and F. Trèves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), no. 2, 387 – 421. · Zbl 0491.35036 [3] J.-Y. Chemin, Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires, Duke Math. J. 56 (1988), no. 3, 431 – 469 (French). · Zbl 0676.35009 [4] Nicholas Hanges and François Trèves, Propagation of holomorphic extendability of CR functions, Math. Ann. 263 (1983), no. 2, 157 – 177. · Zbl 0494.32004 [5] J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), no. 4, 403 – 450 (French, with English summary). · Zbl 0742.58053 [6] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.