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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the $ C\sp \infty$ wave-front set of solutions of first-order nonlinear PDEs


Author: Claudio Hirofume Asano
Journal: Proc. Amer. Math. Soc. 123 (1995), 3009-3019
MSC: Primary 35S05; Secondary 35A99, 35F20
MathSciNet review: 1264801
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Abstract: Let $ \Omega \subset {{\mathbf{R}}^{m + 1}}$ be a neighborhood of the origin and assume $ u \in {C^2}(\Omega )$ is a solution of the nonlinear PDE

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

where $ f(x,t,{\zeta _0},\zeta )$ is $ {C^\infty }$ in the variables $ (x,t) \in {{\mathbf{R}}^m} \times {\mathbf{R}}$ and holomorphic in the variables $ ({\zeta _0},\zeta ) \in {\mathbf{C}} \times {{\mathbf{C}}^m}$. We present a proof that

$\displaystyle WF(u) \subset \operatorname{char}({L^u}),$

where $ WF(u)$ denotes the $ {C^\infty }$ wave-front set of u and $ \operatorname{char}({L^u})$ is the characteristic set of the linearized operator

$\displaystyle {L^u} = \frac{\partial }{{\partial t}} - \sum\limits_{j = 1}^m {\... ...partial {\zeta _j}}}} \right)} (x,t,u,{u_x})\frac{\partial }{{\partial {x_j}}}.$


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1264801-0
PII: S 0002-9939(1995)1264801-0
Keywords: $ {C^\infty }$ wave-front set, characteristic set, Hamiltonian
Article copyright: © Copyright 1995 American Mathematical Society