  In this paper we find the critical exponent for the global existence (in
time) of small data solutions to the Cauchy problem for the semilinear
dissipative evolution equations % \[ u_{tt}+(-\Delta)^\delta
u_{tt}+(-\Delta)^\alpha u+(-\Delta)^\theta u_t=|u_t|^p, \quad t\geq 0,\,\,
x\in\R^n,\] % with $p>1$, $2\theta \in [0, \alpha]$ and $\delta \in
(\theta,\alpha]$. We show that, under additional regularity
$\left(H^{\alpha+\delta}(\R^n)\cap L^{m}(\R^n) \right)\times
\left(H^{2\delta}(\R^n)\cap L^{m}(\R^n)\right) $ for initial data, with $m\in
(1,2]$, the critical exponent is given by $p_c=1+\frac{2m\theta}{n}$. The
nonexistence of global solutions in the subcritical cases is proved, in the
case of integers parameters $\alpha, \delta, \theta$, by using the test
function method (under suitable sign assumptions on the initial data).
