  We consider the Cauchy problem for a class of non-linear evolution equations
in the form \[L(\partial_t,\partial_x) u=F(\partial_t^\ell u), \quad (t,x)\in
[0,\infty)\times \mathbb{R}^n;\] here, $L(\partial_t,\partial_x)$ is a linear
partial differential operator with constant coefficients, of order $m\geq 1$
with respect to the time variable $t$, and $\ell$ is a natural number
satisfying $0\leq \ell\leq m-1$. For several different choices of $L$, many
authors have investigated the existence of global (in time) solutions to this
problem when $F(s)=|s|^p$ is a power non-linearity, looking for a
\textit{critical exponent} $p_c>1$ such that global small data solutions exist
in the supercritical case $p>p_c$, whereas no global weak solutions exist,
under suitable sign assumptions on the data, in the subcritical case $1<p<p_c$.
In the present paper we consider a more general non-linear term in the form
$F(s)=|s|^p\mu(|s|)$; for a large class of models, we provide an integral
condition on $\mu$ which allows to distinguish more precisely the region of
existence of a global (in time) small data solution from that in which the
problem admits no global (in time) weak solutions, refining the existing
results about the critical exponents for power type non-linearities.
