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__Patterns, Nonlinear Dynamics and Applications - PANDA__

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__First meeting: Pattern formation in applications I: Monday 17th December
2001, DAMTP, Cambridge__

John Norbury |
*Patterns for inhomogeneous real Ginzburg-Landau equations* |

Abstract:

We extend the work of Rubinstein, Sternberg and Keller (1989 SIAM J.
Appl. Math. 49 116-33) and consider chemical reactions, phase transitions
or other processes governed by a semilinear reaction-diffusion equation
(with Neumann boundary conditions) for *u(x, t,*

*epsilon)* defined for *t > 0* and *x* being an
element of <(<Omega>)over bar> , a subset of *R*^{n}
by

*u(t) = epsilon del . (k(x)delu) + epsilon V*_{1}(u)

where *x* is an element of Omega, *epsilon* is a small parameter
and *V* is a bistable potential for *u*; here *V* and *k
*depend on *x* and *V* is even in *u*. Here one of the
stable minimizers is pointwise positive, and the fact that *V* is
even in *u* then gives that the other stable minimizer is negative.

The reaction rate *epsilon V*_{1}(u) is large, while the
diffusion coefficient is small. If the initial condition *u(x, 0) = phi
(x)* is positive in the open domain Omega (1), negative in the open
domain Omega (2) (with Omega (1) intersect Omega (2) = empty set), and
zero on a

surface Gamma (epsilon), a subset of Omega, with Omega (1) union Omega
(2) union Gamma (epsilon) = Omega, then *u* rapidly tends to
the positive stable state on Omega (1), and to the negative stable state
on Omega (2); an interface of width *O(epsilon)* develops at Gamma
(epsilon). Then each interface moves on a longer *O(1/epsilon)* timescale,
either towards a stable equilibrium position Gamma (epsilon) for epsilon
small, or away from unstable equilibrium positions. Here Gamma (0) is the
limit curve that arises from {Gamma (epsilon)} as *epsilon --> 0*.
The equilibrium locations for Gamma (0) are calculated from a geometric
geodesic condition, together with their local stability. Simple formulae
for these are derived, which depend only on the *x* variation in *V*
and *k* for epsilon small.

J. Norbury, *Mathematical Institute, 24-29 St Giles, Oxford, England.*

L.C. Yeh, *Department of Applied Mathematics, National Dong Hwa University,
Hualien, Taiwan*