Some results on doubly nonlinear parabolic equations as dynamical systems

AbstractOne of the long term objectives of the dynamical systems approach to PDE's is to reduce them to finite dimensional ODE's. The first step in this direction is to show the existence of a finite dimensional compact set to which all solutions converge eventually, i.e. the existence of the global attractor. In this work, we deal with a class of equations that includes, on one hand, reaction diffusion equations, and on the other hand, porous media type equations. We show that, even under the effect of both non-linearities, these equations can be considered as dynamical system. With appropriate smoothness assumptions on the non-linearities, existence and finite dimensionality of the compact attractor is obtained