On global attractor for 2d Kirchhoff-Boussinesq model with supercritical nonlinearity
Dynamics for a class of nonlinear 2D Kirchhoff-Boussinesq models is studied. These nonlinear plate models are characterized by the presence of a nonlinear source that alone leads to finite-time blow up of solutions. In order to counteract, restorative forces are introduced, which however are of a supercritical nature. This raises natural questions such as: (i) wellposedness of finite energy (weak) solutions, (ii) their regularity, and (iii) long time behavior of both weak and strong solutions. It is shown that finite energy solutions do exist globally, are unique and satisfy Hadamard wellposedness criterium. In addition, weak solutions corresponding to "strong" initial data (i.e., strong solutions) enjoy, likewise, the full Hadamard wellposedness. The proof is based on logarithmic control of the lack of Sobolev's embedding. In addition to wellposedness, long time behavior is analyzed. Viscous damping added to the model controls long time behaviour of solutions. It is shown that both weak and (resp. strong) solutions admit compact global attractors in the finite energy norm, (resp. strong topology of strong solutions). The proof of long time behavior is based on Ball's method  and on recent asymptotic quasi-stability inequalities established in . These inequalities enable to prove that strong attractors are finite-dimensional and the corresponding trajectories can exhibit C∞ smoothness. © Taylor & Francis Group, LLC.
Communications in Partial Differential Equations
Chueshov, I., & Irena, L. (2011). On global attractor for 2d Kirchhoff-Boussinesq model with supercritical nonlinearity. Communications in Partial Differential Equations, 36 (1), 67-99. https://doi.org/10.1080/03605302.2010.484472