Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind x''=a(t)V'(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t diverges to the left and to the right, to two positive limits, and never goes below the minimum between them. Furthermore, the convergence to the maximum between them is supposed to be not too fast (slower than a suitable exponential term)