LOCAL SMOOTH SOLUTIONS OF THE NONLINEAR KLEIN-GORDON EQUATION

Given any mu(1), mu(2) is an element of C and alpha > 0, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation partial derivative(tt)u - Delta u + mu(1)u = mu(2)vertical bar u vertical bar(alpha)u on R-N, N >= 1, that do not vanish, i.e. vertical bar u(t,x)vertical bar > 0 for all x is an element of R-N and all sufficiently small t. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.