TY - JOUR
T1 - Coarsening in an integro-differential model of phase transitions
AU - Duncan, Dugald B.
AU - Grinfeld, Michael
AU - Stoleriu, Iulian
PY - 2000
Y1 - 2000
N2 - Coarsening of solutions of the integro-differential equation ut = e ?O J(|x - y|)(u(y) - u(x)) dy - f(u), x e O, where O ? Rn, J(·) = 0, e > 0 and f(u) = u3 - u (or similar bistable nonlinear term), is examined, and compared with results for the Allen-Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when e is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen-Cahn equation. The special case J(·) = 1 is explored in detail, giving insight into the behaviour in the more general case J(·) = 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.
AB - Coarsening of solutions of the integro-differential equation ut = e ?O J(|x - y|)(u(y) - u(x)) dy - f(u), x e O, where O ? Rn, J(·) = 0, e > 0 and f(u) = u3 - u (or similar bistable nonlinear term), is examined, and compared with results for the Allen-Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when e is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen-Cahn equation. The special case J(·) = 1 is explored in detail, giving insight into the behaviour in the more general case J(·) = 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.
UR - http://www.scopus.com/inward/record.url?scp=0034556058&partnerID=8YFLogxK
M3 - Article
SN - 1469-4425
VL - 11
SP - 561
EP - 572
JO - European Journal of Applied Mathematics
JF - European Journal of Applied Mathematics
IS - 6
ER -