Heteroclinic Solutions to an Asymptotically Autonomous Second-Order Equation

We study the differential equation ẍ(t) = a(t)V \u27(x(t)), where V is a double-well potential with minima at x = ±1 and a(t) →l \u3e 0 as |t| → 1. It is proven that under certain additional assumptions on a, there exists a heteroclinic solution x to the differential equation with x(t) → -1 as t → -1 and x(t) → 1 as t → ∞. The assumptions allow l - a(t) to change sign for arbitrarily large values of |t|, and do not restrict the decay rate of |l -a(t)| as |t| → ∞ © 2010 Texas State University - San Marcos