Blow-up in the Parabolic Scalar Curvature Equation

The $\mathit{\text{parabolic}}\mathit{\text{scalar}}\mathit{\text{curvature}}\mathit{\text{equation}}$ is a reaction-diffusion type equation on an $(n-1)$-manifold $\mathrm{\Sigma }$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\times \mathrm{\Sigma }$ and a family of metrics $\gamma (r)$ on $\mathrm{\Sigma }$, when the coefficients of this equation are appropriately defined in terms of $\gamma$ and $R$, positive solutions give metrics of prescribed scalar curvature $R$ on $[r_0,r_1)\times \mathrm{\Sigma }$ in the form $$g=u^{2}d{r}^2+r^2\gamma \mathrm{.}$$ If the area element of $r^2\gamma$ is expanding for increasing $r$, then the equation is parabolic, and the basic existence problem is to take positive initial data at some $r=r_0$ and solve for $u$ on the maximal interval of existence, which above was implicitly assumed to be $I=[r_0,r_1)$; one often hopes that $r_1=\mathrm{\infty }$. However, the case of greatest physical interest, $R>0$, often leads to blow-up in finite time so that $r_1<\mathrm{\infty }$. It is the purpose of the present work to investigate the situation in which the blow-up nonetheless occurs in such a way that $g$ is continuously extendible to $\bar{M}=[r_0,r_1]\times \mathrm{\Sigma }$ as a manifold with totally geodesic outer boundary at $r=r_1$.