ON THE ANALYTICITY OF SOLUTIONS OF 1ST-ORDER NONLINEAR PDE

Let (x, t) is-an-element-of R(m) x R and u is-na-element-of C2(R(m) x R). We discuss local and microlocal analyticity for solutions u to the nonlinear equation u(t) = f(x, t, u, u(x)). Here f(x, t, zeta-0, zeta) is complex valued and analytic in all arguments. We also assume f to be holomorphic in (zeta-0, zeta) is-an-element-of C x C(m). In particular we show that WF(A) u subset-of Char(L(u)) where WF(A) denotes the analytic wave-front set and Char(L(u)) is the characteristic set of the linearized operator L(u) = partial derivative/partial derivative t - SIGMA partial derivative f/partial derivative zeta(j)(x, t, u, u(x))partial derivative/partial derivative x(j) If we assume u is-an-element-of E C3(R(m) x R) then we show that the analyticity of u propagates along the elliptic submanifolds of L(u).