A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations

We consider the semilinear reaction diffusion equation $\partial_t\phi-\nu\Delta\phi-V(x)\phi+f(\phi)=0$, $\nu>0$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume the standard Allen-Cahn-type nonlinearity, while the potential $V$ is either the inverse square potential $V(x)=\delta |x|^{-2}$ or the borderline potential $V(x)=\delta \mathrm{dist}(x,\partial\Omega)^{-2}$, $\delta\geq 0$ (thus including the classical Allen-Cahn equation as a special case when $\delta=0$). In the subcritical cases $\delta=0$, $N\geq 1$ and $0<\mu:=\frac{\delta}{\nu}<\mu^*$, $N\geq 3$ (where $\mu^*$ is the optimal constant of Hardy and Hardy-type inequalities), we present a new estimate on the dimension of the global attractor. This estimate comes out by an improved lower bound for sums of eigenvalues of the Laplacian by A. D. Melas (Proc. Amer. Math. Soc. \textbf{131} (2003), 631-636). The estimate is sharp, revealing the existence of (an explicitly given) threshold value for the ratio of the volume to the moment of inertia of $\Omega$ on which the dimension of the attractor may considerably change. Consideration is also given on the finite dimensionality of the global attractor in the critical case $\mu=\mu^*$.