Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms

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