Solutions, Stability, and Reduction Methods for Slowly Non-Dissipative Evolutionary Equations

In the much-studied realm of dissipative evolutionary equations, mathematicians classically turn to the tools of bifurcation and stability properties, inertial manifolds, and global attractors in order to reduce an infinite-dimensional dynamical system to its finite-dimensional ruling structures. But these techniques have traditionally either become problematic or simply ceased to be applicable when addressing non-dissipative evolutionary equations, those equations for which solutions may grow to infinite norm.

In this talk I present recent results wherein non-compact analogues to these and other tools were constructed in order to study the realm of slowly non-dissipative PDEs, wherein solutions grow to infinite norm only in infinite time. I will show that, despite their more complicated natures, the non-dissipative forms of these tools allow us to prove extensive results on the asymptotic and geometric properties of solutions to such equations. I will discuss the construction of the ‘completed inertial manifold’ and ‘non-compact global attractor’, and show how these in particular allow us to produce equivalent results for a class of slowly non-dissipative equations as have been achieved for dissipative equations.