Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities
type: Item
instance of: scholarly article
title: Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities
zbMATH Open document ID: 1031.35013
DOI: 10.1016/S0362-546X(03)00101-9
author: Norimichi Hirano
author: Wan Se Kim
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-07-01T00:00:00Z
review text: Let \(N\geq 2\) and \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with a smooth boundary \(\partial\Omega\) and \(h: \mathbb{R}\times\overline\Omega\to \mathbb{R}\) be a continuous function such that for each \(x\in\Omega\), \(h(\cdot, x)\) is \(T\)-periodic and continuous, and \(h> 0\) on \(\mathbb{R}\times \Omega\). In this paper, the authors are interested in multiple existence of \(T\)-periodic solutions of the following problem: \[ \begin{alignedat}{2} & u_t-\Delta_x u= g(u)+ h(t,x)\quad &&\text{in }(0,T)\times\Omega,\\ & u=0\quad &&\text{on }(0,T)\times\partial\Omega,\tag{1}\\ & u(0)= u(T)\quad &&\text{in }\overline\Omega,\end{alignedat} \] where \(\Delta_x\) denotes the Laplacian on \(\mathbb{R}^N\). For a proof of solvability of (1) the authors use topological arguments for variational formulation of (1).
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1945034
zbMATH Keywords: sublinear grow
zbMATH Keywords: stability
zbMATH Keywords: unstability
zbMATH Keywords: topological arguments
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: Q3915354
cites work: A Perturbation Method in Critical Point Theory and Applications
cites work: Sublinear elliptic equations in \(\mathbb{R}{}^ n\)
cites work: Remarks on sublinear elliptic equations
cites work: Q3966483
cites work: Existence of stable and unstable periodic solutions for semilinear parabolic problems
cites work: Q3998030
cites work: Multiplicity and stability result for semilinear parabolic equations
cites work: Non-radial solutions with group invariance for the sublinear Emden-Fowler equation.
cites work: Q4955480
cites work: Semigroups of linear operators and applications to partial differential equations
full work available at URL: s0362-546x(03)00101-9
OpenAlex ID: W2071266519
description: scientific article; zbMATH DE number 1945034