On the analyticity of solutions of first-order nonlinear PDE

Hanges, Nicholas; Trèves, François

doi:10.1090/S0002-9947-1992-1061776-6

On the analyticity of solutions of first-order nonlinear PDE
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by Nicholas Hanges and François Trèves PDF

Trans. Amer. Math. Soc. 331 (1992), 627-638 Request permission

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 \[ {u_t}= f(x,t,u,{u_x})\] . 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 \[ {\text {WF}}_A u \subset \operatorname {Char}({L^u})\] where ${\text {WF}}_A$ denotes the analytic wave-front set and $\operatorname {Char}({L^u})$ is the characteristic set of the linearized operator \[ {L^u}= \partial /\partial t - \sum \partial f/\partial {\zeta _j}(x,t,u,{u_x})\;\partial /\partial {x_j}\] . 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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