# Spectral Analysis, An Integral Mild Solution Formula and Asymptotic Dynamics of the Derivative Klein-Gordon Type Wave Equation

Volume 2, Number 1 (2011), 54 - 83

## Spectral Analysis, An Integral Mild Solution Formula and Asymptotic Dynamics of the Derivative Klein-Gordon Type Wave Equation

Communicated By:
Alain Haraux
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### Abstract

In this paper, we study a nonlinear wave problem (P,) of the form utt + ut − du = −g(ru)+ f (u) 2 X0 ,0 2 R, where ,d, 2 R+ \ {0} are constants, D(−d) := H2( )\H1 0( ), its linear problem well-posedness, behaviour of the spectrum of the wave differential operator in varied damping and diffusion constants, as well as the asymptotic dynamics defined by the derivative Klein- Gordon type wave problem. A particular integral mild solution formula is used to prove the local dynamics of the problem. Given a dissipative condition on f (u), and small damping constants for the problem (P,0) we prove an upper semicontinuity proposition on attractors {A,0 [A0,0 : > 0} Z1/2 as & 0, and via a monotone iteration technique we prove for the problem (P,) the existence of a global compact attractor A, Z1/2 H1 0( )\ L2( ) of finite fractal and Hausdorff dimensions.