Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems
type: Item
instance of: scholarly article
title: Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems
zbMATH Open document ID: 0938.35083
DOI: 10.1016/S0362-546X(98)00172-2
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2000-06-12T00:00:00Z
review text: In the article under review, the author studies uniform boundedness of solutions and gradients to the initial boundary value problem for the degenerate parabolic system: \[ \frac{\partial u_i}{\partial t} = \text{div}(|Du_i|^{p_i - 1}Du_i) + f_i(x,\vec u),\quad t>0,\;x\in\Omega, \] \[ u_i\frac{\partial u_i}{\partial n}\leq 0 \quad\text{on } \partial\Omega, \qquad u_i(0,x) = u_i^0\quad\text{in } \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(p_i\geq 1\), \(u_i^0\) are bounded functions given on \(\Omega\). It is assumed that there exists positive constants \(\alpha\), \(k_1\), \(k_2\) such that \[ \sum_{i=1}^r f_i(x,\vec u)|u_i|^{\lambda - 1}u_i\leq k_1\sum_{i=1}^r|u_i|^{\alpha + \lambda} + k_2 \] for all \(\lambda\geq 1\) and \((x,\vec u)\in\Omega\times \mathbb{R}^r\). The case when the diffusion terms are given by the \(p\)-Laplacians are also considered. By using the theory of monotone operators, the local existence and uniqueness results can be easily obtained for the problem. The author introduces the notions following: (\(A_q\)) Uniformly \(L^q\)- dissipative, (\(B_q\)) Pointwise \(L^q\)-dissipative and (\(C_q\)) Globally \(L^q\)- bounded. The studied question is whether one of these conditions holds for \(q=\infty\) if it is verified for some finite \(q\). The first results regards the boundedness of solutions and reads as: If there exists \(q=p_*>\max\{1, (n/(p+1))(\alpha - p)\}\) where \(p=\min_i p_i \geq 1\) and \(n\equiv N\) if \(N>p+1\), otherwise any fixed \(n>p+1\), such that: (i) the system satisfies (\(A_q\)), (resp. (\(B_q\))) then its also satisfies (\(A_q\)), (resp. (\(B_q\))) for \(q = \infty\); (ii) the solution of the problem satisfies (\(C_q\)) then it also satisfies (\(C_q\)) for \(q=\infty\). If the solution of the system satisfies Neumann boundary conditions, then the estimates of the gradients in the \(L^{\infty}\)-norm are uniformly with respect to the initial data. This type …
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 37L30
zbMATH DE Number: 1383337
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: global attractor
zbMATH Keywords: dissipativity
zbMATH Keywords: uniform estimates
author: Dung Le
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00172-2
OpenAlex ID: W2085865016
Wikidata QID: Q127957085
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description: scientific article; zbMATH DE number 1383337