Logarithmic Relaxation Times (Proof of Proposition 2.15)

-ϕ ϕ

ϕ ϕ

0

+

-+

-1 1

ϕ

ϕ

ϕ ϕ

0

-+

+

-1 1

ϕ

ϕ

+

-2

2

Figure B.1.: Sketch ofA^λ forλ∈(π²,(2π)²),λ∈((2π)²,(3π)²),λ∈((3π)²,(4π)²)

of E^λ for λ∈((π(n−1))²,(πn)²) is retained in A^λ for λ > (nπ)² as a substructure, but two new unstable fixed pointsφ^±_n appear in E^λ. In addition, exactly 2(n−1) new connecting orbits emerge in the attractor: 2(2n−3) ones linking the 2n−3 previously unstable fixed points{0, φ^±_j, j= 1, . . . , n−2}with each of the new ones{φ⁺_n, φ⁻_n}, and 4 trajectories directed from each the latter ones to each of the stable points{φ⁺, φ⁻}.

There is an extensive literature on further properties of attractors for reaction diffusion equations, see instance the survey article by Fiedler and Scheel [1982]. It turns out to be important in the proof of Proposition 2.15 that the longest cascade visitsn−1 fixed points and any cascade ends in one of the stable pointsφ^±. In particular the number of connecting orbits forλ∈((π(n−1))²,(πn)²)) is exactly

n

X

k=1

2(2k−1) = 2n².

B.2.2. Logarithmic Relaxation Times (Proof of Proposition 2.15)

We prove the statement of Proposition 2.15 for the finer topology on H related to the normk · kand then infer the result for the topology related to| · |∞. Letγ >0 be fixed.

Denote byD^± =D^± ⊂H₀¹(0,1) the domain of attraction of φ^± (i.e. φ⁺ or φ⁻). We defineD^±(ε^γ), ε >0 as subsets ofD^±in the same fashion asD^±(ε^γ), with the normk · k replacing| · |∞. By definition of D^±(ε^γ) and by| · |∞6k · k, we have forx∈D^±(ε^γ),

s∈ S andt>0

ku(t;x)−sk>|u(t;x)−s|∞> ε^γ,

hencex∈ D^±(ε^γ), which impliesD^±(ε^γ)⊂ D^±(ε^γ). In addition, by the same argument forv∈ {φ⁺, φ⁻}

|u(t;x)−v|∞6ku(t;x)−vk6 1 2ε^2γ.

Therefore it is sufficient to prove the statement of Proposition 2.15 in the topology of k · k, i.e. for initial valuesx∈ D^±(ε^γ) and the distancek · kinstead of| · |_∞.

For any setA⊂H andσ >0, we define theσ-neighborhood ofA by Uσ(A) := [

x∈A

Bσ(x).

For two fixed points v, w∈ E^λ of the Chafee-Infante equation that are connected in A^λ (C(v, w)6=∅) in the previous Subsection B.2.1 we recall the notation (v→w) :⇔

C(v, w)6=∅and denote forη, σ >0 by

Uσ(v, w) :=Uη(C(v, w)) theσ-tube around the heteroclinic orbitC(v, w)⊂ A^λ and by

U_η,σ⁻ (v, w) :=Uη(C(v, w))\(Bσ(v)∪Bσ(w)).

theη-tube around the heteroclinic orbitC(v, w)⊂ A^λdeprived of theσ-balls around the end pointsv andw.

Forη6^||v−w||₃ andσ >0 it follows for allv, w∈ E^λ that

U_η,σ⁻ (v, w)6=∅ ⇐⇒ v→w. (B.4)

For convenience we writev←wequivalently tow→v (⇔(C(v, w)6=∅). Define for σ >0 the maximal radius d(σ) so that the d(σ)-tubes around the heteroclinic orbits deprived of theσ-balls around the fixed points are all disjoint. More precisely forσ >0 we define

d(σ) := sup{h >0| for allv, w1, w2∈ E^λ,

ifw₁←v→w₂, thenU_h(v, w₁)∩U_h(v, w₂)∩B_σ^c(v) =∅, ifw₁→v←w₂, thenU_h(w₁, v)∩U_h(w₂, v)∩B_σ^c(v) =∅, ifw1→v→w2, thenUh(w1, v)∩Uh(v, w2)∩B_σ^c(v) =∅,

ifw1←v←w2, thenUh(v, w1)∩Uh(w2, v)∩B^c_σ(v) =∅}. (B.5) Forw1→v→w2, we sketchd(σ) in Figure B.2.

It is easy to see that due to the transversality of the fixed points there isδ_d>0 such that for all 0< σ6δd it follows

d(σ)>0. (B.6)

v

w

w

1

2 σ

d(σ)

d(σ)

d(σ)

Figure B.2.: Disjoint tubes around an unstable fixed pointw1→v→w2

Note that necessarilyd(σ)< σ.

Let us rewrite Proposition 2.15 for the finer topology generated byk · k.

Proposition B.2. Let the Chafee-Infante parameter π² < λ 6= (nπ)², forn ∈ N, be given. Then there exists Trec>0 and a constant κ >0 such that for eachγ >0 there exist constantsε0=ε0(γ)>0, such that for all 0< ε6ε0 andt>Trec+κγ|lnε|and x∈ D^±(ε^γ)

ku(t;x)−φ^±k6(1/2)ε^2γ.

Proof. The proof is structured into three parts. In Part I we discuss the absorbtion of the trajectories of the Chafee-Infante equation for any initial valuex∈Hby a neighborhood of the attractor in finite time. This is followed, in Part II, by a detailed discussion of the local behavior of the system when entering different parts of this neighborhood.

In other words, using the flow properities we analyze the behaviour of the solution for initial values taking values in the mentioned neighborhood of the attractor. Here we exploit the well-known shape of the attractor and the hyperbolicity of the fixed points.

In Part III we can finally use the gradient structure of the system in order to determine the global behavior by the local information gained in Part II.

We fixγ >0.

I. The global dynamics absorbed by a neighborhood of the attractor

Claim I.1: There is a universal relaxation time to enterUη(A^λ). For anyη >0there is a timeτ1=τ1(η)>0 such that for allt>τ1 andx∈H

u(t;x)∈Uη(A^λ).

Proof. By Temam [1992], Remark 1.4, p. 88 there existρ₁ =ρ₁(λ)>0 and a uniform upper boundt0=t0(λ)>0 such that for allt>t0 andx∈H

u(t;x)∈Bρ₁(0).

By the definition of a global attractor for eachη >0 and each bounded setA⊂H there is a timet1=t1(A, η, λ)>0 such that for all t>t1 andx∈A

u(t;x)∈Uη(A^λ)

holds true. The claim follows forA=B_ρ1(0) andτ₁:=t₁+t₂.

Claim I.2: There is a unique last entrance time for open neighborhoods of the attractor. For any open setO ⊃ A^λ and x /∈ O there is a unique θ1 =θ1(x,O)>0 such that

u(θ₁;x)∈∂O andu(t;x)∈ O for all t>θ₁. For allx∈H,η >0, open setsO ⊇Uη(A^λ) andx∈H\ Owe have

θ₁(x,O)6τ₁(η).

Proof. For the first part of the statement it suffices to write θ₁(x,O) := sup{t >0|u(t;x)∈ O}/ <∞,

sinceA^λis an attracting set. For the second part we use Claim I.1 and obtain θ1(x,O)6θ1(x, Uη(A))6τ1(η).

II. The local behavior in a neighborhood of the attractor inD^±(ε^γ) There exists a universal constantδb>0 such that for 0< σ6δb the ballsBσ(v), v∈ E^λ, are pairwise disjoint, (B.4) and (B.6) are satisfied, and B_σ(φ^±) ⊂ D^±. Then there is exists ε_b = εb(σ)>0 such that for 0< ε6εb,Bσ(φ^±)⊂ D^±(ε^γ). We shall exploit the segmented structure (B.3) of attractorA^λwhich is reflected in the structure of the surface∂Uσ(A^λ).

Due to (B.2) and the definition of U_σ,σ⁻ (v, w) we have the following decomposition in three disjoint sets ifσ >0 is small enough

Uσ(A^λ) =Bσ(φ^±)∪

[

v∈E^λ\{φ⁺,φ⁻}

Bσ(v)

∪

[

v,w∈E^λ v→w

U_σ,σ⁻ (C(v, w))

. (B.7)

By the choice of σ the balls Bσ(v), v ∈ E^λ appearing in (B.7) are pairwise disjoint.

However, in general there is noσ >0 such that the set [

v,w∈E^λ v→w

U_σ,σ⁻ (C(v, w))

becomes a disjoint union. Since we shall use this property we argue for 0< σ6δb and 0< η6d(σ) for the following modified neighborhoodU_η,σ(A^λ) ofA^λ we have

Uη,σ(A^λ) =Bσ(φ^±)∪

[

v∈E^λ\{φ⁺,φ⁻}

Bσ(v)

∪

[

v,w∈E^λ v→w

U_η,σ⁻ (C(v, w))

. (B.8)

Hence for 0< ε6εb, 0< σ6δb and 0< η6d(σ) U_η,σ(A^λ)∩ D^±(ε^γ)

=Bσ(φ^±)∪

[

v∈E^λ\{φ⁺,φ⁻}

Bσ(v)∩ D^±(ε^γ)

∪ [

v,w∈E^λ v→w

U_η,σ⁻ (C(v, w))∩ D^±(ε^γ)

,

(B.9) which by definition of d(σ) is a union of pairwise disjoint sets. In the sequel we shall further reduce the upper bounds forσ,η andεappropriately.

The strategy of the proof is the following. For σ, η, εto be determined in the sequel and x∈ D^±(ε^γ), we shall use the flow property of the solutionu(·;x) of the Chafee-Infante equation and treat the local behaviour ofu(t;y) fort>0 after having entered Uη/2(A^λ)⊂Uη,σ(A^λ), i.e. for initial conditions

y=u(θ1;x) for θ1=θ1(x, U_η/2(A^λ)∩ D^±(ε^γ)).

In Part II.A we treaty∈Bσ(φ^±), followed by the casey∈Bσ(v)∩ D^±(ε^γ) forv ∈ E^λ in Part II.B, and finally the situationy ∈U_η,σ⁻ (v, w) forv, w∈ E^λ withv →win Part II.C.

II.A: Local behavior in a ball around a stable state

Claim II.A.1: Close to a stable state there is exponential convergence. For any v∈ {φ⁺, φ⁻}there areδs=δs(v)>0,κs=κs(v)>0 andεs=εs(v)>0 such that for all 0< σ6δs, 0< ε6εs,y∈Bσ(v) and

θ2(y, v, σ, ε) :={t >0 |u(t;y)∈Bε^γ(v)}

the inequality

θ26κsγ|lnε| forθ2=θ2(y, v, σ, ε)

holds.

This will follow from Lemma B.3 below.

II.B: Local behavior inD^±(ε^γ)in a ball around an unstable state.

Claim II.B.1: Trajectories in D^±(ε^γ) leave balls around unstable states in at most logarithmic time inε. For eachv∈ E^λ\ {φ⁺, φ⁻}there exist constantsδu=δu(v)>0 and κ_u =κ_u(v)>0 such that for all 0 < σ6δ_u there isε_u =ε_u(v, σ)>0 such that for all 0< ε6εu we haveD^±(ε^γ)∩Bσ(v)6=∅, and for ally∈ D^±(ε^γ)∩Bσ(v) and

θ₃(y, v, σ, ε) := inf{t >0|u(t;y)∈/B_σ(v)∩ D^±(ε^γ)}

the inequality

θ36kuγ|lnε| for θ3=θ3(y, v, σ, ε) holds.

This results from Proposition B.4 below.

II.C: Local behavior in a tube around a connecting orbit. For arbitrary 0 < η 6 σ and v, w ∈ E^λ with v → w and y ∈ U_η,σ⁻ (v, w) we define the first exit time from U_η,σ⁻ (v, w)

τ3(y, η, σ, v, w) := inf{t >0 |u(t;x)∈/ U_η,σ⁻ (v, w)}.

Claim II.C.1: Tubes around connecting orbits are left at the edges. Forv, w∈ E^λ with v → w there is δ1 = δ1(v, w) > 0 such that for any 0 < σ 6 δ1 there is η₁=η₁(σ, v, w)>0 such that for all 0< η6η₁ andy∈U_η,σ⁻ (v, w) we have

u(τ₃;y)∈∂U_η,σ⁻ (v, w)∩(∂B_σ(v)∪∂B_σ(w)) forτ₃=τ₃(y, v, w, η, σ).

Proof. Since all trajectories with initial valuesx∈H\Sfinally enterB_σ(φ⁺) orB_σ(φ⁻) in finite time they have to leave the connecting tube in finite time. The latter is also true forx∈ Sdue to the convergence towards a fixed point onS. Henceτ3(y, v, w, η, σ)<∞ for allv, w∈ E^λ, 0< η6σ, andy∈U_η,σ⁻ (v, w).

Sinceτ3(y, v, w, η, σ)>0 and by definition ofy=u(θ1(x, Uη(A^λ));x), the trajectory cannot leaveU_η,σ⁻ (v, w) via its outer hull, because this is part of∂U_η(A^λ). ButU_η(A^λ) is positive invariant fort>θ₁(x, Uη(A^λ)). Hence the only possible exit locus is

u(τ₃;y)∈∂U_η⁻(v, w)∩(∂B_σ(v)∪∂B_σ(w)) for τ₃=τ₃(y, v, w, η, σ).

Claim II.C.2: The exit time from connecting tubes is uniformly bounded. For each pair of connected orbitsv, w∈ E^λ withv →w there isδh =δh(v, w) such that for all 0 < σ 6 δh there exists η2 = η2(σ, v, w) > 0 such that for 0 < η 6 η2 there exists

τ4=τ4(v, w, σ, η)>0 such that for ally∈U_η,σ⁻ (v, w) we have u(τ4;y)∈B_σ/2(w).

In other words for

θ₄(y, v, w, η, σ) := inf{t >0| u(t;y)∈B_σ/2(w)}

we have the obtain the estimate

θ4(y, v, w, η, σ)6τ4(v, w, η, σ) for ally∈U_η,σ⁻ (v, w).

This results from Lemma B.7.

III: Global dynamics by the local behavior and the gradient structure

Claim III.1: Along a cascade of steady states trajectories visit each unstable state at most once.

1. For any ”cascade” of length k>2 inE^λ, v₁ → . . .→v_k, v_i ∈ E^λ, we have that vi6=vj fori6=j. Hence it has no loops.

2. There is δg >0 such that for 0 < σ 6δg and any cascadev1→. . .→vk in E^λ the trajectory t 7→ u(t;x) for x ∈ D^±(ε^γ) can only move forward along σ-balls centered at fixed points in the cascade.

3. For all v₁, v₂ ∈ E^λ with v₁ → v₂ there is δ₂ = δ₂(v₁, v₂) > 0, such that for 0 < σ 6δ2 there exists η3=η3(v1, v2, σ)>0 and ε0=ε0(v, w, σ)>0 such that for all 0< η6η3, 0< ε6ε0 andy∈U_η,σ⁻ (v1, v2)∩ D^±(ε^γ) there isv3∈ E^λ with v₂→v₃such that for

θ5(y, v1, v2, v3, η, σ) := inf{t >0 |u(t;y)∈U_η,σ⁻ (v2, v3)}

the inequality

θ5(y, v1, v2, v3, η, σ)6τ4(v1, v2, η, σ) +κu(v2)γ|lnε|

holds.

Proof. 1. Consider the energy functional

E(z) =

1

Z

0

1 2

ζ

Z

0

∂z

∂ξ(ξ)

2

dξ+λ z⁴(ζ)−z²(ζ)

dζ, z∈H.

First note that for all steady states v, w∈ E^λ, v→w we haveE(v)>E(w). As t 7→E(u(t;x)) is non-increasing, E(v)>E(w). The equality E(v) =E(w) would

imply that 0 = d

dtE(u(t;x)) =∇E(u(t;x))∂u

∂t(t) =

h∆u(t;x) +f(u(t;x)),∂u

∂t(t;x)i=−

∂u

∂t(t;x)

2

holds for anyx∈C(v, w) andt∈Rimplying thatC(v, w)⊂ E^λ, which is absurd.

Hence forv →w we haveE(v)>E(w). Thus for any cascadev₁ →. . .→v_k we obtain

E(v1)>· · ·>E(vk).

2. Due to the continuity of H =H₀¹(0,1)3z 7→E(z)∈Rthere isδ_g>0 such that for 0< σ6δg and each cascadev1→. . .→vk in E^λ we have

sup

w∈Bσ(v1)

E(w)> inf

w∈Bσ(v1)

E(w)>· · ·> sup

w∈Bσ(vk)

E(w)> inf

w∈Bσ(vk)

E(w).

Since t → E(u(t;x)) is non-increasing, each trajectory (u(t;x))t>0 that visits B_σ(v_i) in a cascade cannot come back to the previous oneB_σ(v_i−1).

3. By Claim II.C.2 for all v₁, v₂∈ E^λ withv₁→v₂ there existsδ_h =δ_h(v₁, v₁) such that for 0 < σ 6δh there is η2=η2(v1, v2, σ)>0 which ensures for 0< η 6η2

the existence of τ4=τ4(v1, w2, η, σ)>0 such that fory∈U_η,σ⁻ (v1, v2) we have u(τ4;y)∈B_σ/2(v2).

Due to the positive invariance of Uη(A^λ) andD^±(ε^γ) for anyε >0 we obtain u(τ₄;y)∈B_σ/2(w)∩U_η2(A^λ)∩ D^±(ε^γ).

By Claim II.B.1 and the positive invariance of U_σ(A^λ), for v₂∈ E^λ\ {φ⁺, φ⁻} there are constantsκ_u=κ_u(v₂)>0 andδ_u=δ_u(v₂)>0 such that for 0< σ6 δu

2 there is ε0 >0 such that for 0 < ε 6ε0 and z ∈ Bσ(v2)∩ D^±(ε^γ) we have for t>κuγ|lnε|

u(t;z)∈B^c_2σ(v₂)∩ D^±(ε^γ).

We now fix

δ2(v1, v2) = 1

2(δb∧δh(v1, v2)∧δ1(v1, v2)∧δu(v2)∧δg(v1, v2)) and 0< σ6δ₂(v₁, v₂). Then there exists

η3=η3(v1, v2, σ) =η1(v1, v2, σ)∧η2(v1, v2, σ)∧d(σ)

with d(σ) defined in (B.5), for which we fix 0< η≤η3. We denote N(λ) =|E^λ|.

By Proposition B.1 are pairwise disjoint inU_η(A^λ), which is positive invariant. Hence, by continuity of t7→u(t;x), leavingUη(A^λ)∩B2σ(v₂) is equivalent to enter one of the connecting

We conclude the desired result. We fix δ0:=δu(0)∧

such that for all 0 < ε 6 ε0 the statements of the Claims I.1, II.A.1, II.B.1, II.C.1, III.C.2 are true simultaneously for allv∈ E^λ involved. Define

θ6(x) := inf{t >0| u(t;x)∈B_(1/2)ε2γ(φ^±)}.

Forx∈ D^±(ε^γ) by Claim I.2

u(t+θ₁(x, U_η/2(A^λ));x)∈U_η/2(A^λ)⊂Uη,σ(A^λ) (B.13) for allt>0. We have to distinguish three cases according to whereu(θ(x, U_η/2(A^λ);x) lies. Either

u(θ1(x, U_η/2(A^λ);x)∈Bσ(φ^±) (B.14) or there isv∈ E^λ\ {φ⁺, φ⁻}such that

u(θ₁(x, U_η/2(A^λ);x)∈B_σ(v)∩U_η,σ(A^λ) (B.15) or there arev, w∈ E^λsuch thatv→w and

u(θ1(x, U_η/2(A^λ);x)∈U_η,σ⁻ (v, w). (B.16) Case (B.14) is contained in the two preceding cases. For convenience we first treat case (B.16). By (B.13) and Claim III.1.2 and III.1.3 for x ∈ D^±(ε^γ) there is k = k(x) ∈ {1, . . . , N(λ)} and a cascade of steady states v₁ → . . . → v_k → v_k+1 = φ^± in E^λ of lengthk such that the trajectory u(·;x) visits all balls B_σ(v_i) or radius σonce along the cascade. The casek= 1 means that there is an unstable statev∈ E^λ withv→φ^± and

u(θ₁(x, U_η/2(A^λ));x)∈U_η,σ⁻ (v, φ^±).

Hence

θ6(x) =θ1(x, Uη/2(A^λ)) +θ4(u(θ1(x, Uη/2(A^λ);x), v, φ^±, η, σ) +θ2(u(θ4(u(θ1(x, U_η/2(A^λ);x), v, φ^±, η, σ);x), φ^±, σ)

6τ1(η/2) +τ4(v, φ^±, η, σ) +κs(φ^±)γ|lnε|. (B.17) This covers the case (B.14). For the cases k >2 we have to introduce the following global versions ofθ5form Claim III.1.3

σ0(x, η, σ) =θ1(x, U_η/2(A^λ))

σ_i(x, η, σ) =θ₅(u(σ_i−1(x, η, σ);x), v_i, v_i+1, v_i+2, η, σ), i∈ {1, . . . , k−1}.

In this case we may estimate

θ6(x) =θ1(x, Uη/2(A^λ)) +

k(x)

X

i=1

σi(x, η, σ) +θ4(u(σk(x)(x, η, σ);x), vk, φ^±, η, σ) +θ₂(u(θ₄(u(σ_k(x)(x, η, σ);x), v_k, φ^±, η, σ);x), φ^±, σ)

6τ1(η/2) +

k(x)

X

i=1

τ4(vi, vi+1, η, σ) +κu(vi+1)γ|lnε|

+τ4(vk(x), vk(x)+1, η, σ) +κs(φ^±)γ|lnε|

6τ1(η/2) +N(λ) max

v,w∈E^λ v→w

τ4(v, w, η, σ)

+N(λ)

max

v∈E^λ\{φ⁺,φ⁻|

κu(v) + max

{v∈{φ⁺,φ⁻}κs(v)

γ|lnε|

=:Trec+κγ|lnε|.

The expression in the preceding line is also an upper bound for (B.17) covering the case (B.14). Case (B.15) can be obtained analogously with a slightly shifted summation, for which the identical upper bound is attained. This finishes the proof.

Im Dokument Metastability of the Chafee-Infante equation with small heavy-tailed Lévy Noise (Seite 154-164)