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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the analyticity of solutions of first-order nonlinear PDE


Authors: Nicholas Hanges and François Trèves
Journal: Trans. Amer. Math. Soc. 331 (1992), 627-638
MSC: Primary 35A20; Secondary 35A30, 35F20
MathSciNet review: 1061776
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Abstract: Let $ (x,t) \in {R^m} \times R$ and $ u \in {C^2}\,({R^m} \times R)$. We discuss local and microlocal analyticity for solutions $ u$ to the nonlinear equation

$\displaystyle {u_t}= f(x,t,u,{u_x})$

. 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

$\displaystyle {\text{WF}}_A\,u \subset \operatorname{Char}({L^u})$

where $ {\text{WF}}_A$ denotes the analytic wave-front set and $ \operatorname{Char}({L^u})$ is the characteristic set of the linearized operator

$\displaystyle {L^u}= \partial /\partial t - \sum \partial \,f/\partial \,{\zeta _j}(x,t,u,{u_x})\;\partial /\partial \,{x_j}$

.

If we assume $ u \in {C^3}\;({R^m} \times R)$ then we show that the analyticity of $ u$ propagates along the elliptic submanifolds of $ {L^u}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1061776-6
PII: S 0002-9947(1992)1061776-6
Keywords: Characteristic set, analytic wave-front set, Hamiltonian
Article copyright: © Copyright 1992 American Mathematical Society