Electron. J. Diff. Equ., Vol. 2010(2010), No. 137, pp. 1-14.

Heteroclinic solutions to an asymptotically autonomous second-order equation

Gregory S. Spradlin

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.

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Gregory S. Spradlin
Department of Mathematics, Embry-Riddle University
Daytona Beach, Florida 32114-3900, USA
email: spradlig@erau.edu

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