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Periodic solution for a plate operator. (English) Zbl 1173.35087
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 $$w=u+u_0.$$
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)
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References:
 [1] Agmon, S., The $$L^p$$ approach to the Dirichlet problem I, Ann. sc. norm. sup. Pisa, 405-448, (1959) [2] Browder, F.E., () [3] Lions, J.L., () [4] Lions, J.L., (), (Nouvelle Présentation Dunod 2002) [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 London
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