| Contents |
| We are concerned with the longtime dynamics of the Kirchhoff
equation with strong dissipation $ u_{tt}-M(\|\nabla u\|^2)\Delta u- \Delta
u_t+h(u_t)+g(u)=f(x)$. We prove that
the IBVP of the equation
admits a unique global weak solution, the related dynamical
system has a finite dimensional
global attractor and an exponential attractor provided that $h(s)$ is
quasi-monotone and both growth exponents $p$ and $q$ of the
nonlinearities $h(s)$ and $g(s)$ are up to critical range, that is,
$1\leq p, q\leq p^*=\frac{N+2}{(N-2)^+}$. Moreover, the optimal regularity of the global attractor is established within further restrictions that the above-mentioned growth exponents are equal and up to subcritical range.
In particular, when $M(s)=1$, we further show that
the IBVP of the equation
possesses a global weak solution provided that the growth exponents $p$ and $ q$ are fully
supercritical, that is, $p=q>p^*$, and the related generalized semiflow has in natural energy space endowed with strong topology a
global attractor. |
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