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Full-Text Articles in Ordinary Differential Equations and Applied Dynamics
A Non-Autonomous Second Order Boundary Value Problem On The Half-Line, Gregory S. Spradlin
A Non-Autonomous Second Order Boundary Value Problem On The Half-Line, Gregory S. Spradlin
Greg S. Spradlin Ph.D.
By variational arguments, the existence of a solution to a nonautonomous second-order boundary problem on the half-line is proven. The corresponding autonomous problem has no solution, revealing significant differences between the autonomous and the non-autonomous case.
Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin
Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin
Gregory S. Spradlin
We study the differential equation ¨x(t) = a(t)V' (x(t)), where V is a double-well potential with minima at x = ±1 and a(t) → l > 0 as |t| → ∞. It is proven that under certain additional assumptions on a, there exists a heteroclinic solution x to the differential equation with x(t) → −1 as t → −∞ and x(t) → 1 as t → ∞. 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| → ∞.