Blow-up of solutions to a nonlinear wave equation

We study the solutions to the the radial 2-dimensional wave equation $$displaylines{ chi_{tt}-{1over r}chi_r-chi_{rr}+{{sinh2chi}over {2r^2}}=g, cr chi(1, r)=chi_{circ}in {dot H}^{gamma}_{ m rad},quad chi_t(1, r)=chi_1 in {dot H}^{gamma-1}_{ m rad}, }$$ where $r=|x|$ and $x$ in $mathbb{R}^2$. We show that this Cauchy problem, with values into a hyperbolic space, is ill posed in subcritical Sobolev spaces. In particular, we construct a function $g(t, r)$ in the space $L^p([0,1]L_{ m rad}^q)$, with ${1over p}+{2over q}=3-gamma$, $0 Keywords Wave equation --- blow-up --- hyperbolic space.