  Let $\Omega$ be a $\mathcal C^2$-bounded domain of $\mathbb R^d$, $d=2,3$,
and fix $Q=(0,T)\times\Omega$ with $T\in(0,+\infty]$. In the present paper we
consider a Dirichlet initial-boundary value problem associated to the
semilinear fractional wave equation $\partial_t^\alpha u+\mathcal A u=f_b(u)$
in $Q$ where $1<\alpha<2$, $\partial_t^\alpha$ corresponds to the Caputo
fractional derivative of order $\alpha$, $\mathcal A$ is an elliptic operator
and the nonlinearity $f_b\in \mathcal C^1( \mathbb R)$ satisfies $f_b(0)=0$ and
$|f_b'(u)|\leq C|u|^{b-1}$ for some $b>1$. We first provide a definition of
local weak solutions of this problem by applying some properties of the
associated linear equation $\partial_t^\alpha u+\mathcal A u=f(t,x)$ in $Q$.
Then, we prove existence of local solutions of the semilinear fractional wave
equation for some suitable values of $b>1$. Moreover, we obtain an explicit
dependence of the time of existence of solutions with respect to the initial
data that allows longer time of existence for small initial data.
