Existence and uniqueness of weak solutions for a generalized thin film equation
type: Item
instance of: scholarly article
title: Existence and uniqueness of weak solutions for a generalized thin film equation
zbMATH Open document ID: 1073.35102
author: Meng Xu
author: Shulin Zhou
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-22T00:00:00Z
review text: 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 \[ p^*= \begin{cases} {np\over n-p}&\text{if }p< n,\\ q &\text{if }p= n,\;q\in (0,+\infty),\\ +\infty &\text{if }p> 0.\end{cases} \]
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35K35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 2136707
zbMATH Keywords: Existence
zbMATH Keywords: Uniqueness
zbMATH Keywords: Regularity
zbMATH Keywords: Fourth-order parabolic equations
MaRDI profile type: Publication
full work available at URL: j.na.2004.01.013
OpenAlex ID: W2089804375
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DOI: 10.1016/J.NA.2004.01.013
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description: scientific article; zbMATH DE number 2136707