On the analyticity of solutions of first-order nonlinear PDE

Abstract

Let (x, r) e Rm x R and u e C2(Rm x R).We discuss local and microlocal analyticity for solutions u to the nonlinear equation ut = f(x, t, u, ux).Here f(x, /, fo .0 's complex valued and analytic in all arguments.We also assume / to be holomorphic in (Co, C) € C x Cm .In particular we show that WF^ u c Char(/_")where WF^ denotes the analytic wave-front set and Char(L") is the characteristic set of the linearized operator L" = d/dt-J2df/di;j(x,t,u, ux)d/dXj.If we assume u 6 C3(Äm x R) then we show that the analyticity of u propagates along the elliptic submanifolds of Lu .