When starting a bounty, one is invited to provide a short description of what is asked for. The usefulness of this notice is evident; the task of writing or improving a purposeful answer is significantly alleviated by the elucidation of the expectations. In several cases, this notice is used to motivate why the present question deserves more attention, ask for certain extensions or generalisations, or present new interesting ideas.
This is a really neat question. Solution appears to exist up to time t=5.547 or so. It exhibits oscillations of rapidly increasing in frequency and amplitude up to the point of singularity. If it helps, the quantity w^4/4+(w'')^2/2 is a convex function, which with these initial condition is increasing. But I could not prove that it becomes infinite in finite time.
What is more, I see little disadvantage to keeping these notices; they constitute part of the natural development of a topic. At last, I wonder whether it is possible to view descriptions of old bounties; if not, then I ask for having this as an option.