Local Repulsion from Unstable States in Reduced D.o.A

In this Subsection we prove Claim II.B.1 in the proof of Theorem B.2.

Proposition B.4. Forπ²< λ6= (πn)², n∈Ngiven andv∈ E^λ\ {φ⁺, φ⁻} there exists δu=δu(v)>0andκu=κu(v)>0 such that for0< σ6δu there isεu =εu(v, σ)>0 ensuring for0< ε6εu thatD^±(ε^γ)∩Bσ(v)6=∅, and we have forx∈ D^±(ε^γ)∩Bσ(v) andt>κ_uγ|lnε|that

u(t;x)∈B_σ^c(v)∩ D^±(ε^γ).

Letγ >0 be fixed throughout this Subsection. We study the local behaviour of the solution u(·;x) of the Chafee-Infante equation starting in x ∈ Bσ(v) for small σ >0 and an unstable statev∈ E^λ\ {φ⁺, φ⁻}. More precisely we are interested to determine an upper bound in terms of ε >0 of the time u(·;x) starting in x∈B_σ(v)∩ D^±(ε^γ) needs to leave Bσ(v)∩ D^±(ε^γ) for σ >0, ε >0 sufficiently small. For convenience we recall that the formal Chafee-Infante equation (2.6) for fixed Chafee-Infante parameter

π²< λ6= (πn)², n∈N, is given as

∂

∂tu(t, ζ) = ∂²

∂ζ²u(t, ζ) +f(u(t, ζ)) ζ∈[0,1], t >0, u(t,0) = u(t,1) = 0, t >0,

u(0, ζ) = x(ζ), ζ∈[0,1],

wheref(y) =−λ(y³−y).

Without loss of generality letv= 0∈ E^λ\ {φ⁺, φ⁻}. Hencef(v) = 0. We set B:D(B)⊂H →H, Bw=∂²w

∂ζ² +f⁰(v)w, w∈D(B) G:H →H, G(w) =f(w)−f⁰(v)w, w∈H.

ThusG(v) =G⁰(v) = 0. It is well-known that B has a discrete spectrum and a finite number of positive eigenvalues. Denote byω₀the smallest positive eigenvalue ofB. We denote byP_u:H →H⁺the orthogonal projection ontoH⁺, the span of the eigenvectors of the positive eigenvalues, and respectively Ps : H → H⁻, where H⁻ is the span of the eigenvectors of the negative eigenvalues. Since for the Chafee-Infante parameter π² < λ 6= (πn)², n ∈N, all steady states in E^λ are hyperbolic, 0 is not an eigenvalue and H =H⁺⊕H⁻. We denote byws =Psw and wu =Puw for w∈ H. Fort >0 andx∈ D^±(ε^γ) and the solutionu(t;x) of equation (2.6) we use in this subsection the notation

X(t;x) :=Psu(t;x), Y(t;x) :=Puu(t;x).

We write

g:H⁺⊕H⁻ →H⁻, g(w_u, w_s) :=P_sG(w), h:H⁺⊕H⁻→H⁺, h(w_u, w_s) :=P_uG(w)

and by (T(t))t>0 theC0-semigroup by the linearized operator B = ∆ +f⁰(v). Solving equation (2.6) for u(t;x) is then equivalent to solving the coupled system of projected equations

X(t) =T(t)x_s+

t

Z

0

T(t−ϑ)g(X(ϑ), Y(ϑ)) dϑ (B.20)

Y(t) =T(t)xu+

t

Z

0

T(t−ϑ)h(X(ϑ), Y(ϑ)) dϑ (B.21)

for (X(t), Y(t)) = (X(t;xs), Y(t;xu)).

For initial valuesxs∈H⁻ we have for allt>0

|T(t)xs|6e^−ω0t|xs|.

MoreoverT(t)P_u can be extended forx∈H⁺ to t60 with

|T(t)x_u|6e^ω0t|xu|.

Before giving the proof we sketch the so-calledLyapunov-Perron construction of the unstable manifold. More details can be found in Temam [1992], Chapter IX.

Definition B.5. Let (Ψ(t))_t>0be a dynamical system onH, i.e. a family of continuous operators Ψ(t) :H →H satisfying

Ψ(t+s) = Ψ(t)◦Ψ(s), t, s>0.

Letv∈H a fixed point, i.e. Ψ(t)v=v. Theunstable manifoldW^u(v)ofvis defined as W^u(v) :={w∈H | lim

t→−∞Ψ(t;w) =v}.

thestable manifoldW^s(v)of v by

W^s(v) :={w∈H | lim

t→∞Ψ(t;w) =v}.

Sketch of the Lyapunov-Perron Construction of the Unstable Manifold: For a radius σ >0 let Bσ(v) a ball centered in the unstable solution v = 0. First of all we truncategandhby a functionψ^σ:H→H,ψ^σ∈ C^∞(H;H) such that

ψ^σ(x) =

(1 ifx∈Bσ(v) 0 ifx∈B^c_2σ(v).

We denote byL_σ>0 the common Lipschitz constant ofgand honB_2σ(v). Clearly Lσ→0, ifσ→0.

We want to construct the unstable manifoldW^u(v) as the graph of a bounded function Φu:H⁺→H⁻, which is Lipschitz continuous with Lipschitz constant L_Φu > 0 such that there isδ_Φu =δ_Φu(v)>0 such that for 0< σ6δ_Φu

W^u(v) ={h+ Φ_u(h)|h∈B_σ(v)∩H⁺}. (B.22) We assume the existence of Φu and that Φu ∈ C¹ in order to be allowed to apply the calculus. This is done by fixed point arguments, which can be found in Temam [1992], Chapter IX. However, we shall prove that if it exists it provides an exponentially attracting, invariant unstable manifold for the truncated equation onBσ(v).

x

ε^γε^γ D( )ε^γ D( )⁺ε^γ

v

-W (v)u

W (v)^u

σ

Figure B.3.: Sketch of the exit from a neighborhood an unstable point

Claim 1: Assume thatΦ_uexists, such that (B.22) is fulfilled, then there isδ₁=δ₁(v)>

0 such that for0< σ6δ1 the projection

R(t;y) :=Y(t;y) + Φ_u(Y(t;y))

satisfies (2.6) for ally∈B_σ(v)∩H⁺ andt∈Randlim_t→−∞R(t;y) = 0.

Proof. Letδl be the maximal positive number such that for 0< σ6δl

Lσ(1 +LΦ_u)6ω0

2 . (B.23)

Setδ₁=δ_Φu∧δ_l and fix 0< σ6δ₁. Under the assumption that Φ_u exists and (B.22) holds true the function Φu satisfies fory∈H⁺∩Bσ(0)

Φ_u(y) =

0

Z

−∞

T(−s)g(Φ_u(Y(s;y)), Y(s;y)) ds, (B.24)

Y(s;y) =T(s)y+

s

Z

0

T(s−r)h(Φu(Y(r;y)), Y(r;y)) dr, s60.

For details consult Temam [1992], Chapter IX. By the flow property we obtain fort∈R

Φu(Y(t;y)) =

t

Z

0

T(t−s)g(Φu(Y(s;y)), Y(s;y)) ds

Y(t;y) =T(t)y+

t

Z

0

T(t−s)h(Φu(Y(s;y)), Y(s;y)) ds.

Hence fort∈Randy∈Bσ(v)∩H⁺the functionR(t;y) = Φu(Y(t;y)) +Y(t;y) satisfies

R(t;y) =T(t)y+

t

Z

0

T(t−s)G(R(s;y)) ds, t60.

In addition, by the local Lipschitz continuity of ΦuandGinBσ(0) we obtain fort60

|Y(t;y)|6 e^ω0t|y|+

0

Z

t

e^ω0(t−ϑ)Lσ(1 +LΦ_u)|Y(ϑ;y)| dϑ.

By Gronwall’s Lemma

|Y(t;y)|6 e^{(ω0−Lσ(1+LΦu))t}|y|

and due to the choice ofσin (B.23) we obtain

|Y(t;y)|6e^ω20t|y| →0, t→ −∞.

Hence Φ_u(Y(t;y))→0, t→ −∞.

Claim 2: Assume thatΦu exists and (B.22) is true. Then there isδ2=δ2(v)>0such that for all0< σ6δ₂ such that for ally∈B_σ(v)∩H⁺ undt>0

|u(t;y)−P_u(u(t;y))−Φu(P_u(u(t;y)))|6e^−ω20t|Ps(y)−Φu(P_u(y))|.

In addition, for all y ∈ W^u(v) the global trajectories (u(t;y))t∈R are of the form u(t;y) =Y(t;y) + Φ_u(Y(t;y)).

Proof. Let 0< σ6δ2,δ2=δΦ_u∧δlsuch that (B.23) is true andy∈Bσ(v)∩H⁺. Since Y(s;Y(t;y)) =Y(t+s;y) and by (B.24)

Φu(Y(t;y)) =

0

Z

−∞

T(−s)g(Φu(Y(s;Y(t;y))), Y(s;Y(t;y))) ds

it follows

Φu(Y(t;y)) =

t

Z

−∞

T(t−s)g(Φu(Y(s;y));Y(s;y)) ds.

Thus Φu(Y(t;y)) is a mild solution of d

dtΦu(Y(t;y)) =BΦu(Y(t;y)) +g(Φu(Y(t;y)), Y(t;y)) (B.25) with the respective initial condition. Since by the chain rule

d

dtΦu(Y(t;y)) = (∇Φu)(Y(t;y))∂

∂tY(t;y)

= (∇Φ_u)(Y(t;y))(BY(t;y) +h(X(t;y), Y(t;y)) (B.26) we obtain att= 0 that

(∇Φu)(y)(By+h(Φu(y), y))−BΦu(y)−G(Φu(y), y) = 0 for ally∈H.

Let (X(t;y);Y(t;y))t>0be the associated solution ofu(t;y). In the next calculation we omit the arguments for convenience. By identification of the right-hand side of (B.25) and (B.26) we obtain

d

dt(X−Φu(Y)) =BX+g(X, Y)− ∇Φu(Y)(BY +h(X, Y))

=B(X−Φu(Y)) +g(X, Y)−g(Φu(Y), Y) +∇Φu(Y)h(Φu(Y), Y)− ∇Φu(Y)h(X, Y)

− ∇Φu(Y)(BY)− ∇Φu(Y)h(Φ_u(Y), Y) +B(Φ_u(Y)) +g(Φ_u(Y), Y)

=B(X−Φu(Y)) +g(X, Y)−g(Φu(Y), Y) +∇Φu(Y)(h(Φu(Y), Y)−h(X, Y)).

ThereforeZ(t;y) =X(t;y)−Φ_u(Y(t;y)) satisfies fort>0

Z(t;y) =T(t)Z(0;y) +

t

Z

0

T(t−s) (g(X(s;y), Y(s;y))−g(Φu(Y(s;y)), Y(s;y))) ds

+

t

Z

0

T(t−s)∇Φu(Y(s;y)) (h(Φu(Y(s;y)), Y(s;y))−h(X(s;y), Y(s;y))) ds.

Taking theL²-norm we use the Lipschitz continuity of Φ_u andGtruncated by ψ^σ we

arrive at

|Z(t;y)|

6e^−ω0t|Z(0;y)|+

t

Z

0

e^{−ω0(t−s)}Lσ|Z(s;y)|ds+

t

Z

0

e^{−ω0(t−s)}LΦ_uLσ|Z(s;y)| ds.

By Gronwall’s Lemma we obtain

|Z(t;y)| 6 e^{−(ω0+Lσ(1+LΦu))t}|Z(0;y)|.

Thus for 0< σ6δlsuch that (B.23) is true andy∈Bσ(v)∩H⁺, we have exponential estimate

|Z(t;y)| 6 |Z(0;y)|e^−ω20t fort>0.

In other words fory∈Bσ(v)∩H⁺

|X(t;y)−Φu(Y(t;y))| 6 e^−ω20t|X(0;y)−Φu(Y(0;y))|, fort>0. (B.27) Hence the graph of Φ_u is exponentially attracting for t → ∞and invariant. For y ∈ Bσ(v)∩H⁺ let (X(t;y), Y(t;y)) be a solution defined on R⁻ which converges to the unstable state 0 fort→ −∞, it is in particular bounded by a constant,M >0, say

|X(t;y)|+|Y(t;y)|6M, t60.

Fort060 and allt>t0we have

|X(t;y)−Φ_u(Y(t;y))| 6 e^{−ω20(t−t0)}|X(t₀;y)−Φ(Y(t₀;y))| 6 M e^{−ω20(t−t0)}. Since the left-hand side does not depend on t₀ we can pass to the limit t₀ → −∞

implying

X(t;y) = Φu(Y(t;y)) fort∈R, z∈Bσ(v)∩H⁺.

Hence a global solution ((X(t;y), Y(t;y))_t∈R fory ∈B_σ(0)∩H⁺ lives on the unstable manifold given as the local graph of ΦuinBσ(v). This finishes the proof of Claim 2.

Remark B.6. The whole construction so far is entirely carried out in the topology of L²(0,1).

We can now prove Proposition B.4.

Proof. Fixγ >0. We show that close to an unstable statev∈ E^λthere are timest>t0

for appropriatet₀ and initial valuesZ₀ for which we obtain an affine upper bound for the respective unstable projectionY(t;x) =Puu(t;x) of the solution of (2.6). This will be used for the argument in the second part.

1. Claim: For v ∈ E^λ\ {φ⁺, φ⁻} there isδ3 =δ3(v) >0 such that for 0 < σ 6δ3, there areε1 =ε1(σ)>0 and C=C(σ)>0 which ensures that for all 0< ε6ε1 and x∈ D^±(ε^γ)∩B_σ(v)there is t₀ =t₀(ε, σ)>0 exists Z₀=Z₀(t₀, x)∈ D^±(ε^γ)∩B_σ(v) such that fort06t6sthe inequality

|Y(t;Z₀)| 6 e^{−ω20(s−t)}|Y(s;Z₀)|+Cε^γ. is satisfied.

Without loss of generality, we still consider v = 0. Fix δ₃ = δ₃(v) = δ₁∧δ₂ and 0< σ6δ3 such that Claim 1 and Claim 2 are true. Fix ε1 = ε1(σ) > 0 small enough such that for 0 < ε 6 ε1 we have Bσ(v)∩ D^±(ε^γ) 6= ∅ and pick an element x∈Bσ(v)∩ D^±(ε^γ). This is justified by Lemma (2.13). Then we can bound the first exit time fromB_σ(v)∩ D^±(ε^γ) in terms of 0< ε6ε₁. Let Φ_u be the generating graph of the unstable manifold as constructed above with Lipschitz constant LΦ_u. Due to estimate (B.27) we obtain

|X(t;x)−Φu(Y(t;x))| 6 e^−ω20t|Ps(x)−Φu(Pu(x))| 6 1

8ε^γ (B.28) for

t>t₀:=−2γ

ω₀ln(ε) + 2

ω₀ln(8σ) (B.29)

and setZ₀=Z₀(x, ε, σ) = (X(t₀;x), Y(t₀;x)). In other words

|Ps(Z₀)−Φ_u(P_u(Z₀))|6 1

8ε^γ. (B.30)

Note that on the finite dimensional subspaceH⁺ the linearised semigroup (T(t) _H+) is defined for allt∈R. For anyt, s∈R

Y(t;Z0) = T(t−s)Y(s;Z0) +

t

Z

s

T(t−r)h(X(r;Z0), Y(r;Z0)) dr

=T(t−s)Y(s;Z₀) +

t

Z

s

T(t−r)h(Φu(Y(r;Z₀)), Y(r;Z₀)) dr

+

t

Z

s

T(t−r) (h(X(r;Z0);Y(r;Z0))−h(Φu(Y(r;Z0)), Y(r;Z0)) dr.

Note that Φu(v) = 0 andT(t)

_H+ is a contraction fort60 with operator norm T(t)

_H+

_L(H+)=e^ω0t.

Hence we obtain fort06t6swith the help of (B.28)

|Y(t;Z₀)|6e^ω0(t−s)|Y(s;Z₀)|+

s

Z

t

e^ω0(t−r)L_σ(1 +L_Φu)|Y(r, Z₀)|dr

+

s

Z

t

e^ω0(t−r)Lσ|Φu(Y(r;Z0))−X(r;Z0)|dr

6e^ω0(t−s)|Y(s;Z0)|+Lσ(1 +LΦ_u)

s

Z

t

e^ω0(t−r)|Y(r, Z0)| dr

+ L_σε^γ 8

s

Z

t

e^3ω20(t−r)dr.

Hence Gronwall’s Lemma implies for Ψ(t) :=e^−ω0t|Y(t;Z₀)|the estimate

|Y(t;Z₀)|6e^{(ω0−Lσ(1+LΦu))(t−s)}|Y(s;Z₀)|+L_σ 8

2 3ω0

ε^γ fort₀6t6s (B.31) Since 0< σ6δl andLσ(1 +L_Φu)<^ω₂⁰ we obtain the desired result forC= ^L₈^σ_3ω²

0.

2. Claim: For v ∈ E^λ there is δ_u = δ_u(v) > 0 such that for 0 < σ 6 δ_u there is εu =εu(σ) >0 ensuring that for 0 < ε 6εu we have D^±(ε^γ)∩Bσ(v) 6= ∅, and for x∈ D^±(ε^γ)∩Bσ(v)ands>κuγ|lnε| that

u(s;x)∈ D/ ^±(ε^γ)∩Bσ(v).

Denote by Φs:H⁺→H⁻the function that generates the local stable manifold ofvas a graph. More precisely we assume the existence ofδΦ_s >0 such that for all 0< σ6δΦ_s

W^s(v) ={h+ Φs(h)| h∈Bσ(v)∩H⁻}

according to an analogue construction as for Φu. In the same way as for Φu we denote byLΦ_s the Lipschitz constant of Φs onBσ(v). SinceH⁺ is finite dimensional there is M >0 such thatkyk6M|y|for ally∈H⁺. Since the Lipschitz constantL_σofGon a ballB_σ(v)∩H⁺ becomes arbitrarily small for σ→0 we may choose δ₄>0 such that for 0< σ6δ4

L_σ< 4 M ∧3ω0

M ∧1.

Fixδ_u=δ_Φs∧δ₃∧δ₄and 0< σ6δ_u. We setε_u:=ε₁∧exp(−^{ln(256M σ}_3γ ²⁾) and 0< ε6ε_u. By constructionD^±(ε^γ) is invariant for allε >0 under the solution flow and such that (X(t;z), Y(t;z)) ∈ D^±(ε^γ) for all t > 0 and z ∈ D^±(ε^γ). Hence for all t > 0 and

z∈ D^±(ε^γ) by construction (X(t;z),Φs(X(t;z)))∈ S. Thus forz∈ D^±(ε^γ)∩Bσ(v) kY(t;z)−Φs(X(t;z))k > dist ((X(t;z), Y(t;z);S)>ε^γ. (B.32) By definition the functionsY(·;x) and Φ_s(X(·;x)),x∈H, take values inH⁺, which is finite dimensional, such that all norms are equivalent there. Hence there is a constant M >0 such thatkyk6M|y| for ally∈H⁺ and

kY(t;z)−Φs(X(t;z))k 6 M|Y(t;z)−Φs(X(t;z))|. (B.33) UsingLσ61 and estimate (B.30) in the proof of Claim 1 we obtain for 0< ε6ε0and t>t0(v, σ, ε),x∈ D^±(ε^γ)∩Bσ(v) andz=Z0=Z0(t0, x)∈ D^±(ε^γ)∩Bσ(v)

|Y(t;Z₀)−Φ_s(X(t;Z₀))| 6 |Y(t;Z₀)|+|Φs(X(t;Z₀)|6|Y(t;Z₀)|+L_σ|X(t;Z₀)|

6|Y(t;Z0)|+Lσ|X(t;Z0)−Φu(Y(t;Z0))|+|Φu(Y(t;Z0))|

6 (1 +L_σ)|Y(t;Z₀)|+L_σ

8 ε^γ. (B.34) Since in additionLσ< _M⁴ by the choice ofσ, we can combine (B.34), (B.33) and (B.32) and obtain

ε^γ

2M 6 1

1 +L_σ 1

M −Lσ

8

ε^γ 6|Y(t;Z0)|. (B.35) Applying inequality (B.31) from Claim 1 to the right-hand side of (B.35) we obtain for s>t>t₀andZ₀that

ε^γ 1

2M − L_σ 12ω0

6 e^ω20(t−s)|Y(s;Z₀)|.

Since alsoL_σ< ^3ω_M⁰ we may estimate fors>t>t₀and Z₀ ε^γ

4M 6e^ω20(t−s)|Y(s;Z₀)|. (B.36)

Thus for 0< ε6εu,x∈ D^±(ε^γ)∩Bσ(v) ands>t>t0

|u(s+t₀;x)|=|u(s;Z₀)| > |Y(s;Z₀)| > e^ω20(s−t) ε^γ 4M Hence

s−t> 2 ω0

(ln(σ4M)−γln(ε)) (B.37)

implies

|u(s+t₀;x)|>σ. (B.38)

Addingt>t0 to (B.37) we may eliminatet and obtain by (B.29) that s> 2

ω0

(ln(σ4M)−γln(ε)) + 2t0

= 2 ω0

(ln(σ4M)−γln(ε)) + 4 ω0

(−γln(ε) + ln 8−lnσ)

= 6γ ω0

|ln(ε)|+ 2 ω0

ln(256M σ²)

(B.39) implies the desired result

ku(s;x)k > |u(s;x)|>σ.

Sinceε_u6exp(−^{ln(256M σ}_3γ ²⁾) inequality (B.39) simpifies for 0< ε6ε_uto s > 12γ

ω₀ γ|ln(ε)|.

Setκu(v) =^12γ_ω

0. This finishes the proof.

Im Dokument Metastability of the Chafee-Infante equation with small heavy-tailed Lévy Noise (Seite 166-176)