Ill-posedness of Coupled Systems with Delay Reinhard Racke - - PowerPoint PPT Presentation

Ill-posedness of Coupled Systems with Delay

Reinhard Racke

University of Konstanz

Mathematics in Savoie 2015, Chamb´ ery, June 15–18, 2015

Introduction Results Sketch of the proof Further results References

Simplest delay equations are ill-posed

• 1. Introduction

Simplest delay equations of parabolic type θt(t, x) = ∆θ(t − τ), (1)

• r of hyperbolic type,

utt(t, x) = ∆u(t − τ), (2) (τ > 0: delay parameter) are ill-posed: There is a sequence of initial data remaining bounded, while the corresponding solutions, at a fixed time, go to infinity in an exponential manner.

Introduction Results Sketch of the proof Further results References

Simplest delay equations are ill-posed

Same for dn dtn u(t) = Au(t − τ), (3) n ∈ N fixed, (−A): linear operator in a Banach space having a sequence of real eigenvalues (λk)k such that 0 < λk → ∞ as k → ∞. Adding certain non-delay terms, e.g. ∆θ(t, x) on the right-hand side of (1), is – for example – sufficient to obtain a well-posed problem.

Introduction Results Sketch of the proof Further results References

Some related work

Nicaise, Ammari, Fridman, Gerbi, Pignotti, Valein; Said-Houari, Laskri, Kirane, Anwar; recently Pokojovy, Khusainov, R., Fischer

Introduction Results Sketch of the proof Further results References

Coupled systems

Today: Coupled systems of different types Hyperbolic-parabolic system, utt(t, x) − auxx(t − τ, x) + bθx(t, x) = 0, (4) θt(t, x) − dθxx(t, x) + butx(t, x) = 0. (5) Schr¨

• dinger type coupled to parabolic equation,

utt(t, x) + a∆2u(t − τ, x) + b∆θ(t, x) = 0, (6) θt(t, x) − d∆θ(t, x) − b∆ut(t, x) = 0. (7)

Introduction Results Sketch of the proof Further results References

Coupled systems

α-β-system with delay: utt(t) + aAu(t − τ) − bAβθ(t) = 0, (8) θt(t) + dAαθ(t) + bAβut(t) = 0, (9) u, θ : [0, ∞) → H, A: self-adjoint having a countable complete

• rthonormal system of eigenfunctions (φj)j with corresponding

eigenvalues 0 < λj → ∞ as j → ∞.

Introduction Results Sketch of the proof Further results References

Coupled systems

τ = 0: Area of smoothing:

✻ ✲ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

1 4 1 2 3 4

1 β

1 4 1 2 3 4

1 α

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

Introduction Results Sketch of the proof Further results References

Coupled systems

τ = 0: Area of analyticity:

✻ ✲

1 2 3 4

1 β

1 2

1 α

• ✁

✁ ✁ ✁ ✁ ✁

Aan

❘ α = 2β − 1

2

■ α = β

Introduction Results Sketch of the proof Further results References

Coupled systems

Today τ > 0: Area of ill-posedness for (8), (9): Aill := {(β, α) | 0 < β ≤ α ≤ 1, (β, α) = (1, 1)}. (10)

✻ ✲

1 4 1 2 3 4

1 β

1 4 1 2 3 4

1 α

Introduction Results Sketch of the proof Further results References

Theorem on ill-posedness

• 2. Results

Initial conditions: u(s) = u0(s), (−τ ≤ s ≤ 0), ut(0) = u1, θ(0) = 00. (11) Theorem Let (β, α) ∈ Aill. Then the delay problem (8), (9), (11) is ill-posed. There exists a sequence (uj, θj)j of solutions with norm uj(t)H tending to infinity (as j → ∞) for any fixed t, while for the initial data the norms sup−τ≤s≤0 (u0

j (s), u1 j , θ0)H3 remain bounded.

Introduction Results Sketch of the proof Further results References

Sketch of the proof

• 3. Sketch of the proof

Ansatz: u = uj(t) = hj(t)ϕj, (12) θ = θj(t) = gj(t)ϕj. (13) Plug in: h′′′

j (t)+dλα j h′′ j (t)+b2λ2β j h′ j(t)+aλjh′ j(t−τ)+adλ1+α j

hj(t−τ) = 0, (14) g′

j (t) + dλα j gj(t) = −bλβ j h′ j(t),

(15) with gj(0) := 1 bλβ

j

(h′′

j (0) + aλjhj(−τ)).

(16)

Introduction Results Sketch of the proof Further results References

Sketch of the proof

Ansatz: hj(t) = 1 ω2

j

eωjt, (17) For a solution sufficient and necessary: ω3

j + dλα j ω2 j + (b2λ2β j

+ aλj e−τωj)ωj = −adλ1+α

j

e−τωj. (18) Find subsequence (ωjk)k with Re ωjk → ∞ as k → ∞, (19) sup

k

| λ2−β

jk

e−τωjk ω2

jk

| < ∞, (20) for t > 0 : |eωjk t ω2

jk

| → ∞ as k → ∞. (21)

Introduction Results Sketch of the proof Further results References

Sketch of the proof

Case 1: α < 2β. (18) is equivalent to ωj

• 1+

ω2

j

b2λ2β

j

+ dωj b2λ2β−α

j

+ a b2 λ1−2β

j

e−τωj = −ad b2 λ1+α−2β

j

e−τωj. (22) Ansatz: ωj = µj(1 + ζj), (23) where |ζj| < 1

2 and

µj = −ad b2 λ1+2α−β

j

e−τµj. (24) (24) has a subsequence (µjk)k of solutions with Re µjk → ∞ (see Dreher, Quintanilla, R.).

Introduction Results Sketch of the proof Further results References

Sketch of the proof

(22) is equivalent to (omitting indices j or jk) (1 − e−τµζ)

• =:f (ζ)

+ (q(ζ) + ζ + ζq(ζ))

• =:g(ζ)

= 0, (25) where q(ζ) := µ2(1 + ζ)2 b2λ2β + dµ(1 + ζ) b2λ2β−α + a b2 λ1−2β e−τµ(1+ζ). (26) f , g are holomorphic in B(0,

1 10τ|µ|) with a single zero of f there.

|g(ζ)| ≤ C |µ|, (27) using the information on α, β. Rouch´ e’s theorem gives the desired zero ζ.

Introduction Results Sketch of the proof Further results References

Sketch of the proof

Case 2: α ≥ 2β. (18) is now equivalent to ω2 1 + ω dλα + b2 λα−2βω + aλ1−α dω e−τω = −aλe−τω. (28) Ansatz (23) again, now µ solving µ2 = −aλe−τµ. (29) Proceed as in Case 1.

Introduction Results Sketch of the proof Further results References

Further results

• 4. Further results
• 1. Delay in the second equation:

utt(t) + aAu(t) − bAβθ(t) = 0, (30) θt(t) + dAαθ(t − τ) + bAβut(t) = 0, (31) u(0) = u0, ut(0) = u1, θ(s) = θ0(s), (−τ ≤ s ≤ 0). (32) Theorem Let (β, α) ∈ Aill. Then the delay problem (30), (31), (32) is ill-posed. There exists a sequence (uj, θj)j of solutions with norm uj(t)H tending to infinity (as j → ∞) for any fixed t, while for the initial data the norms sup−τ≤s≤0 (u0

j , u1 j , θ0(s))H3 remain

bounded.

Introduction Results Sketch of the proof Further results References

Further results

• 2. [Fischer]:

– Additional damping ut in (8) does not lead to well-posedness. – Additional damping Aγut in (8) leads to ill-posedness in regions depending on α, β, γ. – Delay (only) in coupling terms might lead to well-posedness.

Introduction Results Sketch of the proof Further results References

References

• 5. References
• 1. Fischer, L.: Instabilit¨

at gekoppelter Systeme mit Delay. Bachelor

• thesis. University of Konstanz (2014).
• 2. Khusainov, D., Pokojovy, M., Racke, R.: Strong and mild extra-

polated L2-solutions to the heat equation with constant delay. SIAM J. Math. Anal. (to appear).

• 3. Racke, R.: Instability in coupled systems with delay.
• Comm. Pure Appl. Anal. 11 (2012), 1753–1773.
• 4. References in [3].