Periodic solutions of evolution \(p\)-Laplacian equations with nonlinear sources
type: Item
instance of: scholarly article
title: Periodic solutions of evolution \(p\)-Laplacian equations with nonlinear sources
zbMATH Open document ID: 0932.35127
author: Yifu Wang
author: Jingxue Yin
author: Zhoqun Wu
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-28T00:00:00Z
review text: Time-periodic solutions to the \(p\)-Laplacian evolution equations \[ \partial u/\partial t= \text{div} (|\nabla u |^{p-2}\nabla u)+ m( x, t) u^\alpha \quad \text{in } \Omega\times \mathbb{R}\tag{1} \] and \[ \partial u/\partial t= \sum_{i=1}^N \partial/\partial x_i (|\partial u/\partial x_i|^{p-2}\partial u/\partial x_i)+ m(x,t)u^\alpha \quad \text{in } \Omega \times \mathbb{R}\tag{2} \] are sought on a bounded convex domain \(\Omega \subset \mathbb{R}^N\), for \(N\), and \(p \geq 2\), zero Dirichlet conditions being required on the smooth boundary \( \partial \Omega \). The source term \(m(x,t)\) is assumed to be positive, continuous on \(\bar \Omega \times \mathbb{R} \) and to be \(t\)-periodic, with some period \(\omega > 0\). For the case of strongly nonlinear sources, namely \(\alpha > p-1\), a blow-up (scaling) argument is employed to deduce the existence of nonnegative continuous periodic solutions to (1), provided \(p,\alpha + 1 < p ( 1 + 1/N)\). The same argument applies also to (2), the solution in the former case satisfying \(u \in C(0, \omega; W_0^{1,p}(\Omega)) \cap C_\omega (\overline {Q})\), \(\partial u /\partial t \in L^2(Q)\) while in the latter it is \(u \in C(0, \omega; W_0^{1,0}(\Omega)) \cap C_\omega (\bar Q)\), \(\partial u /\partial t \in L^2(Q)\) which holds. Observe that due to the degeneracy of the equations being considered, the solutions exist in a weak sense with respect to the time variable.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1218657
zbMATH Keywords: nonnegative continuous periodic solutions
zbMATH Keywords: time-periodic solutions
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: strongly nonlinear sources
zbMATH Keywords: bounded convex domain
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5783
OpenAlex ID: W1987213353
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DOI: 10.1006/JMAA.1997.5783
description: scientific article; zbMATH DE number 1218657