It is well-known that the mean curvature flow applied to a time-dependent graph $\gamma := (x,\rho(x,t))$ leads to the PDE $$\partial_t \rho = \frac{\partial_{xx} \rho}{1+|\partial_x \rho|^2} = \partial_x \left(\arctan(\partial_x \rho)\right).$$ I know that we can apply the classical diffusion equation even if the initial datum is a Dirac mass, i.e., the Cauchy problem $$\partial_t \rho = \partial_{xx} \rho, \quad \rho(x,0) = \delta_0(x) $$ is well-posed and has a analytical solution $\rho(x,t) = \frac{1}{\sqrt{4\pi t}}\,\mathrm{e}^{-\frac{x^2}{4t}}$. But I am not sure whether it makes sense to associate the graphical MCF with the Dirac mass initial condition. In a nutshell, I am wondering whether the Cauchy problem $$\partial_t \rho = \frac{\partial_{xx} \rho}{1+|\partial_x \rho|^2} = \partial_x \left(\arctan(\partial_x \rho)\right), \quad \rho(x,0) = \delta_0(x) $$ is well-defined, and I appreciate any help!
Edit: it is suggested in one of comments below that to get a MCF with Dirac mass as initial condition, you need a sequence $\rho_n$ of pdf as initial conditions which converges to $\delta_0$ in distribution, and somehow show that the sequence of corresponding MCF has a limit. Sure, the Dirac mass located at $0$ has many "smooth approximations", for instance $$\delta_0(x) = \lim_{\epsilon \to 0^+} \frac{\mathrm{e}^{-\frac{x^2}{2\epsilon}}}{\sqrt{2\pi\epsilon}} \quad \text{or} \quad \delta_0(x) = \lim_{\epsilon \to 0^+} \frac{1}{\pi}\frac{\epsilon}{\epsilon^2 + x^2}.$$ So the question is, can one predict the long time behavior of the solution of the Cauchy problem defined through $$\partial_t \rho^{\epsilon} = \frac{\partial_{xx} \rho^{\epsilon}}{1+|\partial_x \rho^{\epsilon}|^2} = \partial_x \left(\arctan(\partial_x \rho^{\epsilon})\right), \quad \rho^{\epsilon}(x,0) = \frac{\mathrm{e}^{-\frac{x^2}{2\epsilon}}}{\sqrt{2\pi\epsilon}} $$ for each pre-fixed $\epsilon > 0 $?