  We consider the fourth-order Schr\"odinger equation $$ i\partial_tu+\Delta^2
u+\mu\Delta u+\lambda|u|^\alpha u=0, $$ where $\alpha>0,\mu=\pm1$ or $0$ and
$\lambda\in\mathbb{C}$. Firstly, we prove local well-posedness in
$H^4\left(\R^N\right)$ in both $H^4$ subcritical and critical case: $\alpha>0$,
$(N-8)\alpha\leq8$. Then, for any given compact set $K\subset\mathbb{R}^N$, we
construct $H^4(\R^N)$ solutions that are defined on $(-T, 0)$ for some $T>0$,
and blow up exactly on $K$ at $t=0$.
