Anisotropic absolute-instability threshold for positive diffusion (${\mu }_a=\frac{1}{4}{\mathbf{c}}^T{\Gamma }^{-1}\mathbf{c}$)

For ${\eta }_t+\mathbf{c}\cdot \nabla \eta =\mu \eta +\nabla \cdot (\Gamma \nabla \eta )$ with symmetric positive definite $\Gamma$, the dispersion relation is $\omega =\mathbf{c}\cdot \mathbf{k}+i(\mu -{\mathbf{k}}^T\Gamma \mathbf{k})$. Its zero-group velocity saddle is ${\mathbf{k}}_0=-i{\Gamma }^{-1}\mathbf{c}/2$. The anisotropic heat kernel has fixed-position exponential rate $\mu -{\mathbf{c}}^T{\Gamma }^{-1}\mathbf{c}/4$, so this accessible saddle gives the actual threshold. At fixed speed $q$, the threshold ranges from $q^2/(4d_{\max })$ to $q^2/(4d_{\min })$, where $d_{\min },d_{\max }$ are the eigenvalues of $\Gamma$. Absence of absolute hydrodynamic instability in every direction requires the upper bound $\mu \le q^2/(4d_{\max })$.