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Full-Text Articles in Physical Sciences and Mathematics
Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin
Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin
Mathematics - Daytona Beach
We study the differential equation ẍ(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| → 1. 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 → -1 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| → ∞ © 2010 Texas State University - …
An Elliptic Equation With No Monotonicity Condition On The Nonlinearity, Gregory S. Spradlin
An Elliptic Equation With No Monotonicity Condition On The Nonlinearity, Gregory S. Spradlin
Mathematics - Daytona Beach
An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used. © EDP Sciences, SMAI 2006.