Sharp bounds on the attractor dimensions for damped wave equations

We give explicit estimates of order $\gamma^{-d}$ (with logarithmic correction in the 1D case) for the fractal dimension of the attractor of the damped hyperbolic equation (or system) in a bounded domain $\Omega\subset \mathbb R^d$, $d\ge 1$, with linear damping coefficient $\gamma>0$. The key ingredient in the proof for $d\ge3$ is Lieb's bound for the $L^p$-norms of systems with orthonormal gradients based on the Cwikel-Lieb-Rozenblum (CLR) inequality for negative eigenvalues of the Schrödinger operator. The case $d=1$ is simpler, but contains a logarithmic correction term that seems to be inevitable. The 2D case is more difficult and is strongly based on the Strichartz-type estimates for the linear equation. Lower bounds of the same order for the dimension of the attractor are also obtained for a damped hyperbolic system with non-linearity containing a small non-gradient perturbation term, meaning that in this case our estimates are optimal for $d\ge2$ and contain a logarithmic discrepancy for $d=1$. Estimates for the various dimensions (Hausdorff, fractal, Lyapunov) of the attractor in purely gradient case are also given. We show, in particular, that the Lyapunov dimension of a non-trivial attractor is of the order $\gamma^{-1}$ in all spatial dimensions $d\ge 1$.