Marinho, Alexandro O. Periodic solution for a plate operator. (English) Zbl 1173.35087 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 3, 1349-1364 (2009). The author investigates the existence and uniqueness of periodic solutions problem for the nonlinear plate problem: \[ w''+\Delta^2w+|w'|^{p-2}w'=f\quad \text{in }Q, \]\[ w=\frac{\partial w}{\partial \nu}=0\quad \text{on }\Sigma,\quad w(0)=w(T),\quad w'(0)=w'(T), \]where \(Q=\Omega\times (0,T)\), \(\Omega\subset \mathbb{R}^2\), \(\Sigma=\Gamma\times (0,T)\), \(\Gamma=\partial\Omega\). A solution \(w\) has the form \(w=u+u_0\), where \(u\) is a weak solution of the problem \[ \frac d{dt}\left(u''+\Delta^2u+|u'|^{p-2}u'\right)=\frac {df}{dt}\quad \text{in }Q, \]\[ u=\frac{\partial u}{\partial \nu}=\quad \text{on }\Sigma,\quad u(0)=u(T),\quad u'(0)=u'(T),\quad \int_0^Tu(s)\,ds=0\;\text{ in }\Omega \]and \(u_0\) solves the Dirichlet-Neumann problem \(\Delta^2u_0=g_0\) in \(\Omega\), \(u_0=\frac{\partial u_0}{\partial \nu}=\;\text{on }\Gamma.\)The main result is the existence and the uniqueness of a periodic solution \(w=u+u_0.\) Reviewer: Igor Bock (Bratislava) MSC: 35L75 Higher-order nonlinear hyperbolic equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 35B10 Periodic solutions to PDEs 35L35 Initial-boundary value problems for higher-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:nonlinear plate equation; elliptic regularization PDFBibTeX XMLCite \textit{A. O. Marinho}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 3, 1349--1364 (2009; Zbl 1173.35087) Full Text: DOI References: [1] Agmon, S., The \(L^p\) approach to the Dirichlet problem I, Ann. Sc. Norm. Sup. Pisa, 405-448 (1959) · Zbl 0093.10601 [2] Browder, F. E., (Problems Non Linéaires, Press de L’Universite de Montreal (1966)) [3] Lions, J. L., (Equations Differenhelles Operationelles dans des Espaces de Hilbert (1963), CIME: CIME Verena) [4] Lions, J. L., (Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod: Dunod Paris), (Nouvelle Présentation Dunod 2002) · Zbl 0189.40603 [5] Prodi, G., Soluzioni Periodiche de l’equazioni della onde con terme dissipative non linéaire, Rend. Sem. Mat. Padova, 35, 38-49 (1968) [6] Tanabe, H., Evolution Equation (1979), Pitman: Pitman London This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.