Finite dimensional dynamics for Kolmogorov-Petrovsky-Piskunov equation.
Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.
We study the existence and long-time behavior of weak solutions to Newton-Boussinesq equations in two-dimensional domains satisfying the Poincaré inequality. We prove the existence of a unique minimal finite-dimensional pullback -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms.
Using the asymptotic a priori estimate method, we prove the existence of a pullback -attractor for a reaction-diffusion equation with an inverse-square potential in a bounded domain of (N ≥ 3), with the nonlinearity of polynomial type and a suitable exponential growth of the external force. Then under some additional conditions, we show that the pullback -attractor has a finite fractal dimension and is upper semicontinuous with respect to the parameter in the potential.