Heteroclinic solutions to an asymptotically autonomous second-order equation

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 $a(t) o l > 0$ as $|t| o infty$. It is proven that under certain additional assumptions on $a$, there exists a heteroclinic solution $x$ to the differential equation with $x(t) o -1$ as $t o -infty$ and $x(t) o 1$ as $t o infty$. 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 $|t| o infty$.

Keywords

• Heteroclinic
• non-autonomous equation
• bounded solution
• variational methods