Nils Ackermann School of Mathematics and Statistics, University of Sydney Equilibria and Connecting Orbits in Non-Dissipative Parabolic Equations Wednesday 12th April 14:05-14:55pm, Carslaw Building Room 373. For many years the existence of solutions of semilinear elliptic equations on bounded domains has been investigated using variational methods. If 0 is a solution and a superlinear term is present then the celebrated "Mountain Pass Theorem" by Ambrosetti and Rabinowitz and its variants usually yield at least one solution, in many cases several or even infinitely many. Here I present an alternative approach to the existence question for the elliptic equation, replacing the gradient flow of the energy functional by the flow of the associated parabolic problem. It turns out that one can nicely combine the topological ideas of the variational method with information on the geometric structure of stable and unstable manifolds in the parabolic problem. Due to the existence of blow-up in finite time the main difficulty here is the compactness issue, which has been resolved recently by Pavol Quittner. We obtain existence results for equilibria and connecting orbits, and information on their nodal properties. There is no known variational setting that yields the latter in the case of an indefinite superlinear term.