On the Analyticity of Solutions of First-Order Nonlinear PDE

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 ${L^u}$.

Locations

• Transactions of the American Mathematical Society - View - PDF