We study the dynamics of pattern formation in the one-dimensional partial differential equation
uu - (W¢(ux ))x - uxxt + u = 0\text (u = u(x,t),\text x Î (0,1),\text t > 0)u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)
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proposed recently by
Ball, Holmes, James, Pego &
Swart [BHJPS] as a mathematical
E[u,ut ] = ò01 (\fracut2 2 + W(ux ) + \fracu2 2)dx.E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}
Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of
Swart &
Holmes [SH]:
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 While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (u
nvn) of E[u,v] in the Sobolev space W
1
p(0, 1)×L2(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W
1
p(0,1)×L2(0,1) of all solutions (u(t),ut(t)) with low initial energy as time t   ).
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