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

摘要

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = 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 → ±∞, to two positive limits l _* and l *, and never goes below min(l _*, l *). Furthermore, the convergence to max(l _*, l *) is supposed to be not too fast (slower than a suitable exponential term).