Long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation (Q1862919)

The authors consider the following generalization of the Kolmogorov-Spiegel-Sivashinsky equation: \[ u_t+\alpha\Delta^2 u+\beta\Delta u + \nabla\cdot \vec f(u)+\Delta\phi(u)+\nu u+h(x)=0 \tag{1} \] in the three-dimensional bounded domain \(\Omega:=(0,L_1)\times(0,L_2)\times(0,L_3)\) endowed by periodic boundary conditions. Here \(\Delta\) is the Laplacian, \(\vec f\)\ and \(\phi\) are given nonlinear interaction functions, \(h\) are the external forces and \(\alpha,\beta,\nu>0\) are physical parameters which are assumed to satisfy the following relation: \[ 4\alpha\nu>\beta^2. \] The authors show that, under the appropriate assumptions on the functions \(\vec f\) and \(\phi\), the longtime behavior of solutions of (1) can be described in terms of a global attractor \(\mathcal A\) with finite Hausdorff and fractal dimensions. Moreover (under the additional assumptions on \(\vec f\), \(\phi\) and \(h\)), they prove the Gervey regularity of solutions of equation (1) and (based on this fact) construct the approximate inertial manifold with the exponential order of approximation for that problem.