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Example text
4 be true. 53) and E sup t∈[0,T0 ε−2 ] Ps Res(εw)(t) p ≤ Cε3p−2 . Furthermore Pc Res(εw) is differentiable with ∂t Pc Res(εw)(t) = ε4 [Ac ψ + Bc (ψ)](ε2 t) . 2. 4. 5in 46 ws-book975x65 Amplitude Equations for SPDEs Proof. 54) t [Ac ψ + Bc (ψ)](ε2 τ )dτ . 55) cancel. Thus, t Pc Res(εw)(t) = ε4 [Ac ψ + Bc (ψ)](ε2 τ )dτ . 46) of ψ together with the −α , X). 9 to bound the 4p-th moment of a for p ≥ 1. 53) follows easily. Furthermore, the derivative of Pc Res is obvious. 57) 0 t +ε3 e(t−τ )L [As (a + εψ) + 2Bs (a, ψ) + εBs (ψ)](ε2 τ )dτ .
However it exhibits a complete non-orthogonal basis of eigenfunctions. For simplicity of presentation, we do not focus on that technical point (cf. 1). For a detailed discussion see [Bl¨o05a]. In this example L and A do not commute, and hence Pc A = APc , which is in contrast to most of the other examples stated above, where we have equality. This does not cause major technical difficulties, and we allow for quite general A in the abstract setting (cf. 4). 5in ws-book975x65 Chapter 2 Amplitude Equations on Bounded Domains On bounded domains, we can approximate on long time-scales the essential dynamics of an SPDE near a change of stability by the amplitude equation.
For the attractivity our main goal is to verify that there is a time tε > 0 such that u(tε ) = εaε + ε3 ψε , where aε ∈ N and ψε ∈ Ps X are both of order O(1). 3) in X. 3) in X independent of the initial condition. Especially, τe = ∞ almost surely for the maximal time of existence of u. 5in ws-book975x65 Bounded Domains that E u(0) q 33 ≤ δεq for all ε ∈ (0, 1) implies sup E u(t) p ≤ Cεp for all sufficiently small ε > 0. 13) t≥0 Additionally, for tε = that 2 ω ln(ε−1 ) and all p ∈ [4, q/3] there is a constant C > 0 such sup E Ps u(t) t≥tε p ≤ Cε3p for all sufficiently small ε > 0.