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.