Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 137, pp. 1-14.

Title: Heteroclinic solutions to an asymptotically autonomous
       second-order equation

Author: Gregory S. Spradlin (Embry-Riddle Univ., Daytona Beach, FL, USA)

Abstract:  
 We study the differential equation $\ddot{x}(t) = a(t)V'(x(t))$,
 where $V$ is a double-well potential with minima at
 $x = \pm 1$ and $a(t) \to l > 0$ as $|t| \to \infty$.
 It is proven that under certain additional assumptions on $a$,
 there exists a heteroclinic solution $x$ to the differential
 equation with $x(t) \to -1$ as $t \to -\infty$ and
 $x(t) \to 1$ as $t \to \infty$.  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 $|t| \to \infty$.
 
Submitted May 21, 2009. Published September 23, 2010.
Math Subject Classifications: 34B40, 34C37.
Key Words: Heteroclinic; non-autonomous equation;
           bounded solution; variational methods.