Heteroclinic solutions to asymptotically autonomous equations via continuation methods

By means of a continuation argument, we prove 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, which is definitively monotone with respect to |t|, converges to a positive limit l as |t| diverges and fulfils one of the two following assumptions: a(t) never goes below l, or a(t)-l converges to 0, as |t| diverges, more slowly than a suitable exponential term.