  In this paper, we discuss a test function method to obtain nonexistence of
global-in-time solutions for higher order evolution equations with fractional
derivatives and a power nonlinearity, under a sign condition on the initial
data. In order to deal with fractional powers of the Laplace operator, we
introduce a suitable test function and a suitable class of weak solutions. The
optimality of the nonexistence result provided is guaranteed by both scaling
arguments and counterexamples. In particular, our manuscript provides the
counterpart of nonexistence for several recent results of global existence of
small data solutions to the following problem: \[ \begin{cases} u_{tt} +
(-\Delta)^{\theta}u_t + (-\Delta)^{\sigma} u = f(u,u_t),& t>0, \ x\in\mathbb
R^n,\\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x) \end{cases} \] with $f=|u|^p$ or
$f=|u_t|^p$, where $\theta\geq0$ and $\sigma>0$ are fractional powers.
