2023/08/08 by Sangmin Park, Park, Sangmin, Robert L. Pego +1
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2308.04281
We study the asymptotic convergence of solutions as t→∞ of ∂t u=-f(u)+∫ f(u), a nonlocal differential equation that is formally a gradient flow in a constant-mass subspace of L2 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.