Global attractors and their deconstructions provide us with a versatile tool for studying the long-time behavior of solutions to reaction-diffusion equations. The majority of global attractor theory assumes that the equation in question is dissipative, but recent work has shown that similar results can be obtained for non-dissipative reaction-diffusion equations which exhibit infinite-time blow-up of solutions without any finite-time blow-up. Such phenomena appear in the analysis of various physical models ranging from suspension bridges to Rayleigh-Benard convection to tumor growth. In this talk I will discuss my recent results in this area and some of the geometric methods which yield them, including bifurcation diagrams, Fucik spectra, and nodal properties, as well as the relation of this work to the analysis of non-smooth dynamical systems. This is joint work with Kristen Moore and Juliette Hell.