Methods and Applications of Analysis

Volume 24 (2017)

Number 3

On Microlocal Smoothness of Solutions of First Order Nonlinear PDE

Pages: 383 – 406



Abraham Hailu (Department of Mathematics, Arba Minch University, Arba Minch, Ethiopia)


We study the microlocal smoothness of $C^2$ solutions $u$ of the first-order nonlinear partial differential equation\[u_t = f(x, t, u, u_x)\]where $f(x, t, \zeta_0, \zeta)$ is a complex-valued function which is $C^{\infty}$ in all the variables $(x, t, \zeta_0, \zeta)$ and holomorphic in the variables $(\zeta_0, \zeta)$. If the solution $u$ is $C^2 , \sigma \in \mathrm{Char}(L^u)$ and $\frac{1}{\sqrt{-1}} \sigma ([L^u , L^{\overline{u}}]) \lt 0$, then we show that $\sigma \notin WF(u)$. Here $WF(u)$ denotes the $C^{\infty}$ wave front set of $u$ and $\mathrm{Char}(L^u)$ denotes the characteristic set of the linearized operator\[L^u = \frac{\partial}{\partial t} - \sum^{m}_{j=1} \frac{\partial f}{\partial \zeta_j} (x, t, u, u_x) \frac{\partial}{\partial x_j} \textrm{ .}\]


$C^{\infty}$ wave front set, linearized operator

2010 Mathematics Subject Classification

35A18, 35A21, 35A22, 42B10

Full Text (PDF format)

The author’s work was supported in part by ISP of Sweden.

Received 24 May 2016

Accepted 9 March 2017

Published 17 January 2018