Abstract: We consider large-time behaviors of weak solutions to the evolutionary $p$-Laplacian with logarithmic source of time-dependent coefficient. We find that the weak solutions may neither decay nor blow up, provided that the initial data $u(\cdot,t_0)$ is on the Nehari manifold $\mathscr{N}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)=0, \|\nabla v\|_p^p\neq0 \big\}$. This is quite different from the known results that the weak solutions may blow up as $u(\cdot, t_0)\in \mathscr{N}^{-}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)<0\big\}$ and weak solutions may decay as $u(\cdot, t_0)\in\mathscr{N}^{+}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)>0\big\}$.

Key words: logarithmic source, singular energy line, large-time behavior