Stability for a class of nonlinear pseudo-differential equations

We study a class of nonlinear evolutionary equations generated by a pseudo-differential operator with the elliptic principal symbol and with nonlinearities of the form G(ux)G(ux) where cη2≤G(η)≤Cη2cη2≤G(η)≤Cη2 for large |η||η|. We demonstrate existence of a universal absorbing set, and a compact attractor, and show that the attractor is of a finite Hausdorff dimension. The stabilization mechanism is similar to the nonlinear saturation well known for the Kuramoto–Sivashinsky equation.