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We consider the equation $u_t\mathrm{-}i\mathrm{\Lambda }u=0$, where $\mathrm{\Lambda }=\lambda \left(D_x\right)$ is a first order pseudo-differential operator with real symbol $\lambda \left(\xi \right)$. Under a suitable convexity assumption on $\lambda$ we find the decay properties for $u\left(t,x\right)$. These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem $u_t\mathrm{-}i\mathrm{\Lambda }u=f\left(u\right)$, $u\left(0,x\right)=g\left(x\right)$. If $f\left(u\right)$ is a smooth function which satisfies $f\left(u\right)\simeq {\left|u\right|}^p$ near $u=0$, and $g$ is small in suitably Sobolev norm, we prove global existence theorems provided $p$ is greater than a critical exponent.

We study the Cauchy problem for utt − ∆u + V (x)u^5 = 0 in 3–dimensional case. The function V (x) is positive and regular, in particular we are interested in the case V (x) = 0 in some points. We look for the global classical solution of this equation under a suitable hypothesis on the initial energy.