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Very slow phase separation in one dimension. (English) Zbl 0991.35515
Rascle, Michel (ed.) et al., PDEs and continuum models of phase transitions. Proceedings of an NSF-CNRS joint seminar held in Nice, France, January 18-22, 1988. Berlin etc.: Springer-Verlag. Lect. Notes Phys. 344, 216-226 (1989).
Summary: We consider a scalar Landau-Ginzburg equation with nonconserved order parameter: \(u_t=\epsilon^2u_{xx}-F'(u)\) for \(0\leq x\leq 1\) with Neumann boundary conditions, where \(F\) is a double-well free-energy density with equal minima at \(u=ffi1\). Given typical mixed-phase initial data, then over a short time there form domains near \(+1\) and \(-1\) in which \(u\) alternately lies. Collapse of these domains typically occurs on a time scale which is exponentially large, proportional to \(\exp(Al/\epsilon)\) where \(A^2=F"(ffi1)\) and \(l\) is the domain length. We infer a (nonrigorous) scaling law for the coarsening process in one dimension. Physically, the motion of an interface between domains is driven by the tiny potential jump across it. For our rigorous results, we take a dynamical systems approach. In geometric terms, we find thin channels which attract the flow and along which certain slow variables dominate. Proofs will appear elsewhere.
For the entire collection see [Zbl 0717.76011].

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)