Homotopy invariants methods in the global dynamics of strongly damped wave equation

Piotr  Kokocki

doi:10.3934/dcds.2016.36.3227

Article Contents

2016, Volume 36, Issue 6: 3227-3250. Doi: 10.3934/dcds.2016.36.3227

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Homotopy invariants methods in the global dynamics of strongly damped wave equation

Piotr Kokocki^1,

1.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń

Received: April 30, 2015

Revised: September 30, 2015

Published: May 2017

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Abstract

We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and determine the Conley index of the set $K_\infty$, that is the union of the bounded orbits of this equation.

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Mathematics Subject Classification: Primary: 37B30, 47J35; Secondary: 35B34.

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