The paper examines the parabolic equation \[ \frac{1}{\tau} v_t = v_{xx} - 2v + H(x-\Phi(t)) \] on \((0,1)\times \mathbb{R}_+\) with zero Neumann boundary conditions, \(H\) the Heaviside function and \(\Phi\) satisfying \(0< \Phi (0)<1\) and \[ \frac{d\Phi}{d t} = \frac{2 v(\Phi(t),t)-\frac{1}{2}}{\sqrt{( \frac{3}{4}-v(\Phi(t),t) )( v(\Phi(t),t)+\frac{1}{4})}}\;. \] Building on previous results by \textit{D. Hilhorst} {Y. Nishiura} and \textit{M. Mimur} [Proc. R. Soc. Edinb., Sect. A 118, No. 3/4, 355--378 (1991; Zbl 0752.35093)] and \textit{Y.-M. Lee} \textit{R. Schaaf} and \textit{R. C. Thomspon} [J. Comput. Appl. Math. 52, No. 1--3, 305--324 (1994; Zbl 0814.35151)], the authors prove that the solution exists for all \(0