Optimal estimates of the dimension of attractors of a nonlinear wave equation

Explicit estimates of order $\gamma-d$ are obtained for the fractal dimension of an attractor of a nonlinear wave equation (or system) in a bounded domain $\Omega \in Rd$, $d \ge 1$, with a linear dissipative term with coefficient $\gamma > 0$. In the case $d \ge 3$, the key role is played by Lieb estimates for $Lp$-norms of systems with orthonormal gradients, based on the use of the Zwickel–Lieb–Rosenblum (CLR) inequality for negative eigenvalues of the Schrödinder operator. The cases $d = 1,2$ are surprisingly much more complicated. Lower bounds of the same order for the attractor dimension are also obtained for a nonlinear hyperbolic system with a nonlinearity containing a small non-gradient perturbation term, which means that in this case our bounds are optimal (for $d \ge 3$). The purely gradient case is fundamentally different. In particular, it turns out that the Lyapunov dimension of a non-trivial attractor is of order $\gamma - 1$ in all spatial dimensions $d \ge 1$.