Heteroclinic Solutions to an Asymptotically Autonomous Second-Order Equation

We study the differential equation ẍ(t) = a(t)V ′(x(t)), where V is a double-well potential with minima at x = ±1 and a(t) → l > 0 as |t| → ∞. 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→ −∞ 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| → ∞.