Generation, propagation, and annihilation of metastable patterns

Abstract

We study the full-time dynamics of the initial value problem, for u ε = u ε ( x , t ) , u t ε = ε 2 u xx ε - F ′ ( u ε ) , x ∈ R , t > 0 , u ε ( x , 0 ) = u 0 ( x ) , x ∈ R , where 0 < ε ⪡ 1 and F ( u ) has global minimum 0 at u = ± 1 . We assume that u 0 ( · ) is bounded, continuous and independent of ε , changes sign finitely many times, and lim inf | x | → ∞ | u 0 ( x ) | > 0 . There are four stages in the dynamics. The first, phase separation of O ( | ln ε | ) long, develops phase regions where u ε ≈ ± 1 . The second, generation of metastable patterns of O ( ε - 1 ) long, allows u ε to enter an O ( e - ℓ / ε ) neighborhood of a standing wave profile near each interface x = z i where ℓ = min i { F ″ ( ± 1 ) ( z i + 1 - z i ) } is the shortest weighted length of phase regions. The third, super-slow motion of interfaces of O ( e ℓ / ε ) long, displays the O ( e - ℓ / ε ) speed interface motion governed by an approximate ode system. The fourth stage, annihilation of interfaces of O ( 1 ) long, interlaces with the third stage. Interfaces annihilate when they are close enough, and after every annihilation, new metastable patterns are developed to restore the super-slow motions of remaining interfaces. Eventually u ε approaches a stable equilibrium which has only one or none interface. The first and third stages have been studied. Here, we provide rigorous analysis for the second and fourth stages. To be self-contained, we also provide simplified analysis for the first and third stages, thereby completing the analytic treatment for the full-time behavior of the dynamics.