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

Nicholas Braun Rodrigues and Antonio Victor da Silva

Mathematische Nachrichten, 2021, vol. 294, issue 8, 1452-1471

Abstract: In this paper we study microlocal regularity of a C2‐solution u of the equation ut=f(x,t,u,ux),where f(x,t,ζ0,ζ) is ultradifferentiable in the variables (x,t)∈RN×R and holomorphic in the variables (ζ0,ζ)∈C×CN. We proved that if CM is a regular Denjoy–Carleman class (including the quasianalytic case) then: WFM(u)⊂Char(Lu),where WFM(u) is the Denjoy–Carleman wave‐front set of u and Char(Lu) is the characteristic set of the linearized operator Lu: Lu=∂∂t−∑j=1N∂f∂ζj(x,t,u,ux)∂∂xj.

Date: 2021
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https://doi.org/10.1002/mana.201800516

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