Convergence and nonconvergence in a nonlocal gradient flow

Abstract

We study the asymptotic convergence as $t\to \infty$ of solutions of ${\partial }_{t}u=-f\left(u\right)+\int f\left(u\right)$, a nonlocal differential equation that is formally a gradient flow in a constant-mass subspace of $L^2$ arising from simplified models of phase transitions. In case the solution takes finitely many values, we provide a new proof of stabilization that uses a Łojasiewicz-type gradient inequality near a degenerate curve of equilibria. Solutions with infinitely many values in general need not converge to equilibrium, however, which we demonstrate by providing counterexamples for piecewise linear and cubic functions $f$. Curiously, the exponential rate of convergence in the finite-value case can jump from order $O\left(1\right)$ to arbitrarily small values upon perturbation of parameters.