<p style='text-indent:20px;'>This paper establishes the emergence of slowly moving transition layer solutions for the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian (nonlinear) evolution equation, <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula> are constants, driven by the action of a family of double-well potentials of the form <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ F(u) = \frac{1}{2\theta}|1-u^2|^\theta, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>indexed by <inline-formula><tex-math id="M5">\begin{document}$ \theta>1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \theta\in \mathbb{R} $\end{document}</tex-math></inline-formula> with minima at two pure phases <inline-formula><tex-math id="M7">\begin{document}$ u = \pm1 $\end{document}</tex-math></inline-formula>. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to <inline-formula><tex-math id="M8">\begin{document}$ \pm 1 $\end{document}</tex-math></inline-formula> except at a finite number of thin transitions of width <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>, persist for an exponentially long time in the critical case with <inline-formula><tex-math id="M10">\begin{document}$ \theta = p $\end{document}</tex-math></inline-formula>, and for an algebraically long time in the supercritical (or degenerate) case with <inline-formula><tex-math id="M11">\begin{document}$ \theta>p $\end{document}</tex-math></inline-formula>. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg–Landau type are established. In contrast, in the subcritical case with <inline-formula><tex-math id="M12">\begin{document}$ \theta<p $\end{document}</tex-math></inline-formula>, the transition layer solutions are stationary.