We construct some particular kind of solution to the
two - dimensional equivariant wave map problem with
inhomogeneous source term in space-time domain of type
$\Omega_\alpha(t) = {x \in \mathbb R^2 : |x|^\alpha < t}$,
where $\alpha\in (0, 1]$. More precisely, we take the initial data
$(u_0, u_1)$ at time T in the space $H^{1+\epsilon} \times H^\epsilon$
with some $\epsilon > 0$. The source
term is in $L^1((0, T); H^\epsilon(\Omega_\alpha(t)))$ and
we show that the $H^{1+\epsilon}$ -norm of the solution blows-up,
when $t \rightarrow 0_+$ and $\alpha\in (0, 1 − \epsilon)$.
