Lower bound of blow-up time to a fourth order parabolic equation modeling epitaxial thin film growth

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

“…Recently, there have been several studies on the fourth‐order parabolic equation [4, 12, 13, 17]. We find that the boundary conditions of the fourth‐order parabolic equations basically have the following three cases.…”

p-Biharmonic Pseudo-Parabolic Equation with Logarithmic Non linearity

“…Lower bound of Blow-up time to a fourth order parabolic equation modelling epitaxial thin film growth Recently, higher-order equations have gained much importance in studies. Lower bound of Blow-up time to a fourth order parabolic equation modelling epitaxial thin film growth studied by Liu et.al [3]. The p-biharmonic equation…”

Well‐posedness and asymptotic behavior for a p‐biharmonic pseudo‐parabolic equation with logarithmic nonlinearity of the gradient type

“…In 2021, Liu et al. [13] investigated the following equation modeling epitaxial thin film growth: $$$$\begin{eqnarray} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll}u_{t}+\Delta ^2 u=-\textrm {div}{\left(|\nabla u|^{q-2}\nabla u\log |\nabla u|\right)}, & x\in \Omega ,t>0, \\[3pt] u(x,t)=\Delta u(x,t)=0, & x\in \partial \Omega ,t>0, \\[3pt] u(x,0)=u_{0}(x), & x\in \Omega , \end{array} \right.} \end{eqnarray}$$$$where $$\Delta ^2 u$$ denotes the capillarity‐driven surface diffusion, $$\textrm {div}\left(|\nabla u|^{q-2}\nabla u\log |\nabla u|\right)$$ represents the upward hopping of atoms affected by molecular, ion, etc.…”