Publication 23-CNA-011
Convergence and non-convergence in a nonlocal gradient flow
Sangmin Park
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
sangminp@andrew.cmu.edu
Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@cmu.edu
Abstract: We study the asymptotic convergence of solutions as $t\rightarrow\infty$ of $\partial_t u=-f(u)+\int f(u)$, 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 $O(1)$ to arbitrarily small values upon perturbation of parameters.
Dedicated to Sir John Ball
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