Existence of nontrivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms
type: Item
instance of: scholarly article
title: Existence of nontrivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms
zbMATH Open document ID: 1090.35021
author: Rui Huang
author: Yifu Wang
author: Yuanyuan Ke
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2006-01-26T00:00:00Z
review text: Using the Leray-Schauder degree theory, the authors show the existence of nontrivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms by considering the following periodic boundary value problem \[ \begin{alignedat}{2} \frac{\partial u}{\partial t}-\Delta u^m & =(a-\Phi[u])u,\quad & (x,t) & \in Q_{\omega},\\ u(x,t) & =0, & (x,t) & \in \partial\Omega\times [0,\omega],\tag{1}\\ u(x,0) & =u(x,\omega), & x & \in \Omega,\end{alignedat} \] where \(m>1,\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary, \(Q_{\omega}=\Omega\times(0,\omega),\;\Phi[u]:L^2(\Omega)^+ \to[0,\infty)\), \( L^2(\Omega)^+=\{u\in L^2(\Omega):u\geq 0,\;\text{a. e. in }\Omega\}\). Let \(0<\varepsilon\leq \frac{1}{2}\) be constant, they first showed the existence of nontrivial nonnegative solution \(u_{\varepsilon}\) to the corresponding regularized periodic boundary value problem \[ \begin{alignedat}{2} \frac{\partial u_{\varepsilon}}{\partial t}-\text{div}((mu_{\varepsilon}^{m-1}+\varepsilon)\nabla u_{\varepsilon}) & =(a-\Phi[u_{\varepsilon}])u_{\varepsilon}, \quad &(x,t) & \in Q_{\omega},\\ u_{\varepsilon}(x,t) & =0, &(x,t) & \in \partial\Omega\times [0,\omega],\\ u_{\varepsilon}(x,0) & =u_{\epsilon}(x,\omega), &x & \in \Omega,\end{alignedat} \] and then passing to a limit by letting \(\varepsilon\to 0\) to get the solution \(u\) of (1).
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 5001891
zbMATH Keywords: degenerate equation
zbMATH Keywords: nonlocal term
zbMATH Keywords: existence of periodic solution
zbMATH Keywords: Leray-Schauder's degree
zbMATH Keywords: regularized periodic boundary value problem
reviewed by: Peixuan Weng
MaRDI profile type: Publication
full work available at URL: dcdsb.2005.5.1005
OpenAlex ID: W2032164288
DOI: 10.3934/DCDSB.2005.5.1005
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description: scientific article; zbMATH DE number 5001891