  We study the problem of existence of orbits connecting stationary points for
the nonlinear heat and strongly damped wave equations being at resonance at
infinity. The main difficulty lies in the fact that the problems may have no
solutions for general nonlinearity. To address this question we introduce
geometrical assumptions for the nonlinear term and use them to prove index
formulas expressing the Conley index of associated semiflows. We also prove
that the geometrical assumptions are generalizations of the well known
Landesman- Lazer and strong resonance conditions. Obtained index formulas are
used to derive criteria determining the existence of orbits connecting
stationary points.
