  We consider a quantized massless and minimally coupled scalar field on a
circular closed string with a time-dependent radius $R(t)$, whose undisturbed
dynamics is governed by the Nambu-Goto action. Within the semi-classical
treatment, the back-reaction of the quantum field onto the string dynamics is
taken into account in terms of the renormalized expectation value of the
energy-momentum tensor including the trace anomaly. The results indicate that
the back-reaction could prevent the collapse of the circle $R\downarrow0$ --
however, the semi-classical picture fails to describe the string dynamics at
the turning point (i.e., possible bounce) at finite values of $R$ and $\dot R$.
The fate of the closed string after that point (e.g., oscillation or eternal
acceleration) cannot be determined within the semi-classical picture and thus
probably requires the full quantum treatment. PACS: 04.62.+v, 03.70.+k,
11.15.Kc, 04.60.-m.
