Existence and uniqueness of weak solutions for a generalized thin film equation (Q1763868)

The authors consider the following fourth-order parabolic initial-boundary value problem \[ \begin{cases} u_t+ \text{div}(|\nabla\Delta u|^{p-2}\nabla\Delta u)= f-\text{div\,}g &\text{in }Q= \Omega\times [0,T],\\ u= 0,\;\Delta u= 0&\text{on }\Gamma=\partial\Omega\times (0,T),\\ u(x,0)= u_0(x) &\text{on }\Omega,\end{cases}\tag{1} \] where \(p> 1\) and \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary. The authors study the existence and uniqueness of weak solutions of (1) by employing the difference and variation methods. Moreover, they deduce some regularity results of weak solutions for (1). In (1) the authors assume that \(u_0\in H^1_0(\Omega)\), \(f\in L^{p'}(0,T,L^{(p_*)'}(\Omega))\), \(g\in (L^{p'}(Q))^N\), where \(p'\) denotes conjugate to \(p\) and \(p^*\) is defined by \[ p^*= \begin{cases} {np\over n-p}&\text{if }p< n,\\ q &\text{if }p= n,\;q\in (0,+\infty),\\ +\infty &\text{if }p> 0.\end{cases} \]