Spatially Extended Differential Equations in a Potential: On the Borders of the Basins of Attraction Related to Local Minima

We consider a class of non-linear parabolic or hyperbolic partial differential equations on unbounded domains. These equations can be viewed as spatial extensions of dissipative oscillators in a potential. We prove that the basins of attraction of the homogeneous stationary solutions corresponding to local minima of the potential are open, and we describe the asymptotic behavior of a class of solutions which belong to the borders of these basins. We also study the basin of attraction of a homogeneous stationary solution corresponding to a global minimum of the potential.