Variational approach to stability of semilinear wave equation with nonlinear boundary conditions

Andrzej Nowakowski

doi:10.3934/dcdsb.2014.19.2603

Article Contents

2014, Volume 19, Issue 8: 2603-2616. Doi: 10.3934/dcdsb.2014.19.2603

This issue Previous Article Periodic solutions to differential equations with a generalized p-Laplacian Next Article Generalized fractional isoperimetric problem of several variables

Variational approach to stability of semilinear wave equation with nonlinear boundary conditions

Andrzej Nowakowski^1,

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

Received: September 30, 2013

Revised: March 31, 2014

Published: September 2014

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Abstract

We discuss solvability for the semilinear equation of the vibrating string $ x_{tt}(t,y)-\Delta x(t,y)=F_{x}(t,y,x(t,y))-G_{x}(t,y,x(t,y))$ in bounded domain and same type of nonlinearity on the boundary. To this effect we derive new variational methods one for the boundary equation the second for interior equation. Next we discuss stability of solutions with respect to initial conditions basing on variational approach.

Keywords:

Dual variational method,
semi-linear wave equation,
mass and sources terms in the domaln and on the boundary.

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

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