A singular energy line of potential well on evolutionary p-Laplacian with logarithmic source

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(·, t₀) is on the Nehari manifold \(\mathscr{N}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)=0, \Vert\nabla v\Vert_p^p\neq0 \big\}\) $$\mathcal{N}:=\{v\in W_0^{1,p}(\mathrm{\Omega }):I(v,t_0)=0,\parallel \mathrm{\nabla }v{\parallel }_p^p\ne 0\}$$ . This is quite different from the known results that the weak solutions may blow up as \(u(\cdot, t_0)\in \mathscr{N}^{-}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)<0\big\}\) $$u(\cdot ,t_0)\in {\mathcal{N}}^{-}:=\{v\in W_0^{1,p}(\mathrm{\Omega }):I(v,t_0)<0\}$$ and weak solutions may decay as \(u(\cdot, t_0)\in\mathscr{N}^{+}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)>0\big\}\) $$u(\cdot ,t_0)\in {\mathcal{N}}^{+}:=\{v\in W_0^{1,p}(\mathrm{\Omega }):I(v,t_0)>0\}$$ .