Variational analysis of semilinear plate equation with free boundary conditions

Andrzej Nowakowski

doi:10.3934/dcds.2015.35.3133

Article Contents

2015, Volume 35, Issue 7: 3133-3154. Doi: 10.3934/dcds.2015.35.3133

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Variational analysis of semilinear plate equation with free boundary conditions

Andrzej Nowakowski^1,

1.
University of Lodz, Faculty of Math & Computer Sciences, Banacha 22, 90-238 Lodz

Received: August 31, 2013

Revised: October 31, 2014

Published: May 2017

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Abstract

We present a variational analysis for the semilinear equation of the vibrating plate $x_{t t}(t,y)+\Delta ^{2}x(t,y)+l(t,y,x(t,y))=0$ in a bounded domain and a free nonlinear boundary condition $\Delta x(t,y)=H_{x}(t,y,x(t,y))-Q_{x}(t,y,x(t,y))$. In this context new dual variational methods are developed. Applying a variational approach we discuss a stability of solutions with respect to initial conditions.

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Mathematics Subject Classification: Primary: 35L05; Secondary: 35L20.

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