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Publication 23-CNA-011
Sangmin Park Robert L. Pego Abstract: We study the asymptotic convergence of solutions 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.
Get the paper in its entirety as 23-CNA-011.pdf |