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 $u(\cdot, t_0)\in\mathscr{N}^{+}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)>0\big\}$.
Gege LIU
,
Jingxue YIN
,
Yong LUO
. A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 363
-384
.
DOI: 10.1007/s10473-025-0206-7
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