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

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 \[ {u_t} = f(x,t,u,{u_x}),\] 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 \[ WF(u) \subset \operatorname {char}({L^u}),\] 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 \[ {L^u} = \frac {\partial }{{\partial t}} - \sum \limits _{j = 1}^m {\left ( {\frac {{\partial f}}{{\partial {\zeta _j}}}} \right )} (x,t,u,{u_x})\frac {\partial }{{\partial {x_j}}}.\]