Sharp anisotropic Gevrey regularization and directional microlocal smoothing for dissipative–dispersive Fourier evolutions

We establish a sharp Fourier–Gevrey regularization theory for a broad class of nonautonomous dissipative–dispersive evolution equations on Rⁿ R n . The dissipative symbol is allowed to have different fractional orders in different coordinate directions, while the dispersive symbol is only required to be real. We prove an exact moving-radius estimate in exponentially weighted Fourier spaces: each anisotropic regularity radius increases by precisely the time integral of the corresponding dissipative coefficient. For the model symbol ∑ ⱼ cⱼ(t)|ξ ⱼ|κ ⱼ ∑ j c j ( t ) | ξ j | κ j , this threshold is optimal in every coordinate, in the strong sense that the evolution operator is bounded between two Fourier–Gevrey spaces if and only if the target radius does not exceed the initial radius plus the accumulated dissipation. Explicit factorial derivative estimates identify the sharp anisotropic Gevrey indices 1/κ ⱼ 1 / κ j , including analytic and ultra-analytic regimes. We also treat inhomogeneous problems and complex lower-order Fourier perturbations. In the microlocal part we develop a nonquasianalytic anisotropic formulation for mixed exponents 0<κ ⱼ<1 0 < κ j < 1 . Writing κ _*=max ⱼκ ⱼ κ ∗ = max j κ j , qⱼ=κ _*/κ ⱼ q j = κ ∗ / κ j , and σ =1/κ _* σ = 1 / κ ∗ , we use q q -conic neighborhoods and the anisotropic Gevrey scale G^σ q G σ q , in the spirit of Zanghirati’s anisotropic microlocal theory. If the integrated dissipation is elliptic only in a