Approximate solutions of vector fields and an application to Denjoy-Carleman regularity of solutions of a nonlinear PDE

In this paper we study microlocal regularity of a $\mathcal{C}^2$ solution $u$ of the equation \begin{equation*} u_t = f(x,t,u,u_x), \end{equation*} where $f(x,t,\zeta_0, \zeta)$ is ultradifferentiable in the variables $(x,t)\in \mathbb{R}^{N} \times \mathbb{R}$ and holomorphic in the variables $(\zeta_0,\zeta) \in \mathbb{C} \times \mathbb{C}^{N}$. We proved that if $\mathcal{C}^{\mathcal{M}}$ is a regular Denjoy-Carleman class (including the quasianalytic case) then: \begin{equation*} \mathrm{WF}_\mathcal{M} (u)\subset \mathrm{Char}(L^u), \end{equation*} where $\mathrm{WF}_\mathcal{M}(u)$ is the Denjoy-Carleman wave-front set of $u$ and $\mathrm{Char}(L^u)$ is the characteristic set of the linearized operator $L^u$: \begin{equation*} L^u = \dfrac{\partial}{\partial t} - \sum_{j=1}^{N}\frac{\partial f}{\partial\zeta_j}(x,t,u,u_x)\dfrac{\partial}{\partial x_j}. \end{equation*}