Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$

Abstract

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on $\mathbb{R}^N$: $u_{tt}-φ(x)Δ u_{t}-φ(x)M(\|\nabla u\|^{2})Δ u+f(u) = h(x)$. It proves that when the growth exponent $p$ of the nonlinearity $f(u) $ is up to the supercritical range: $ 1≤ p < p^{**}(\equiv \frac{N+4}{(N-4)^+})$, the related solution semigroup has in weighted energy space a (strong) global attractor and a partially strong exponential attractor, respectively. In particular, the partially strong exponential attractor becomes the strong one in non-supercritical case (i.e., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).