Search

The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:utt+Δ2u+(-Δ)αut-Δf(u)=g(x), with α∈(0,1). For general bounded domain Ω⊂RN(N≥1), we show that there exists a critical exponent pα≡N+2(2α-1)(N-2)+ depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: 1≤pα, (i) the weak solutions of the equations are of additionally global smoothness when t>0; (i) the related dynamical system possesses a global attractor Aα and an exponential attractor Aαexp in natural energy space for each α∈(0,1), respectively; (ⅲ) the family of global attractors {Aα} is upper semicontinuous at each point α0∈(0,1], i.e., for any neighborhood U of Aα0,Aα⊂U when |α-α0|≪1. These results extend those for structural damping case: α∈[1,2) in [31,32]. [ABSTRACT FROM AUTHOR]

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on RN: utt-φ(x)Δut-φ(x)M(∥∇u∥2)Δu+f(u)=h(x). It proves that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: 1≤p

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. [ABSTRACT FROM AUTHOR]

The paper investigates the well-posedness and longtime dynamics of Boussinesq type equations with fractional damping: u t t + Δ 2 u + ( − Δ ) α u t − Δ f ( u ) = g ( x ) , with α ∈ ( 1 , 2 ) . The main results focus on the relations among the dissipative exponent α , the growth exponent p of nonlinearity f ( u ) and the well-posedness and the longtime dynamics of the equations. We find a new critical exponent p α ≡ N + 2 ( 2 α − 1 ) ( N − 2 ( 2 α − 1 ) ) + rather than p ∗ ≡ N + 2 N − 2 ( N ≥ 3 ) as known before and show that when 1 ≤ p < p α : (i) The equations are like parabolic, that is, not only the IBVP of the equations are well-posedness, but also their weak solutions are of higher global regularity as t > 0 . (ii) The related solution semigroup has a global attractor A α in natural energy space, and also has an exponential attractor A e x p α in the sense of partially strong topology. In particular, when 1 ≤ p < p α ′ ≡ N + 2 α ( N − 2 α ) + ( < p α ) , the partially strong topology becomes the strong one. (iii) For any α 0 ∈ [ 1 , 2 ) , the family of global attractors A α is upper semicontinuous at the point α 0 . [ABSTRACT FROM AUTHOR]

The paper investigates the upper semicontinuity of global attractors A α of a family of semilinear wave equations with gentle dissipation: u t t − △ u + γ ( − △ ) α u t + f ( u ) = g ( x ) , with α ∈ ( 0 , 1 ∕ 2 ) . It is a continuation of researches in Yang et al. (2016), where the existence of global attractors of the equations has been established. We further show that for any neighborhood U of A α 0 , α 0 ∈ ( 0 , 1 ∕ 2 ) , A α ⊂ U when 0 < α − α 0 ≪ 1 . [ABSTRACT FROM AUTHOR]

The paper studies the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on R N : u t t − Δ u t − M ( ‖ ∇ u ‖ 2 ) Δ u + u t + g ( x , u ) = f ( x ) . We establish the well-posedness, the existence of the global and exponential attractors in natural energy space H = H 1 ( R N ) × L 2 ( R N ) in critical nonlinearity case. These results improve not only the recent ones achieved by Yang (2007) [32] , but also ones achieved by Conti, Pata and Squassina (2005) [9] to some extent. [ABSTRACT FROM AUTHOR]