  We consider the Cauchy problem for a $3$-evolution operator $P$ with
$(t,x)$-depending coefficients and complex valued lower order terms. We assume
the initial data to be Gevrey regular and to admit an exponential decay at
infinity, that is, the data belong to some Gelfand-Shilov spaces of type
$\mathscr{S}$. Under suitable assumptions on the decay at infinity of the
imaginary parts of the coefficients of $P$ we prove the existence of a solution
with the same Gevrey regularity of the data and we describe its behavior for
$|x| \to\infty$.
