Convergence and nonconvergence in a nonlocal gradient flow

Abstract We study the asymptotic convergence as of solutions of , a nonlocal differential equation that is formally a gradient flow in a constant‐mass subspace of 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 . Curiously, the exponential rate of convergence in the finite‐value case can jump from order to arbitrarily small values upon perturbation of parameters.