Global attractor of the Kirchhoff wave models with strong nonlinear damping

Abstract

The paper investigates the existence of global attractor of Kirchhoff type equations with strong nonlinear damping: u t t − σ ( ‖ ∇ u ‖ 2 ) Δ u t − ϕ ( ‖ ∇ u ‖ 2 ) Δ u + f ( u ) = h ( x ) . It proves that when the growth exponent p of the nonlinearity f ( u ) is up to the supercritical range: N + 2 N − 2 ≡ p ∗ < p < p ∗ ∗ ≡ N + 4 ( N − 4 ) + ( N ≥ 3 ) , the related solution semigroup still has a global attractor (rather than a partially strong one as known before) in natural energy space.