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 θ t ( t , x ) = ∆ θ ( t − τ ) , (1) or of hyperbolic type, u tt ( 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 d n dt n u ( t ) = Au ( t − τ ) , (3) n ∈ N fixed, ( − A ): linear operator in a Banach space having a sequence of real eigenvalues ( λ k ) k such that 0 < λ k → ∞ 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, u tt ( t , x ) − au xx ( t − τ, x ) + b θ x ( t , x ) = 0 , (4) θ t ( t , x ) − d θ xx ( t , x ) + bu tx ( t , x ) = 0 . (5) Schr¨ odinger type coupled to parabolic equation, u tt ( t , x ) + a ∆ 2 u ( t − τ, x ) + b ∆ θ ( t , x ) = 0 , (6) θ t ( t , x ) − d ∆ θ ( t , x ) − b ∆ u t ( t , x ) = 0 . (7)

Introduction Results Sketch of the proof Further results References Coupled systems α - β -system with delay: u tt ( t ) + aAu ( t − τ ) − bA β θ ( t ) = 0 , (8) θ t ( t ) + dA α θ ( t ) + bA β u t ( t ) = 0 , (9) u , θ : [0 , ∞ ) → H , A : self-adjoint having a countable complete orthonormal system of eigenfunctions ( φ j ) j with corresponding eigenvalues 0 < λ j → ∞ as j → ∞ .

Introduction Results Sketch of the proof Further results References Coupled systems τ = 0: Area of smoothing: α ✻ 1 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 3 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 4 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 1 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 2 ✁ ✁ ✁ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 1 ✁ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ✁ 4 ✁ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ✁ ✁ ✁ ✁ ❆ ✁ ✲ 1 1 3 1 β 4 2 4

Introduction Results Sketch of the proof Further results References Coupled systems τ = 0: Area of analyticity: α ✻ 1 ✁ � ✁ � α = 2 β − 1 ✁ � A an 2 ■ ❘ ✁ � α = β ✁ � 1 ✁ � 2 ✲ 1 3 1 β 2 4

Introduction Results Sketch of the proof Further results References Coupled systems Today τ > 0: Area of ill-posedness for (8), (9): A ill := { ( β, α ) | 0 < β ≤ α ≤ 1 , ( β, α ) � = (1 , 1) } . (10) α ✻ 1 � � 3 � 4 � � 1 � 2 � � 1 � 4 � � � ✲ 1 1 3 1 β 4 2 4

Introduction Results Sketch of the proof Further results References Theorem on ill-posedness 2. Results Initial conditions: u ( s ) = u 0 ( s ) , ( − τ ≤ s ≤ 0) , u t (0) = u 1 , θ (0) = 0 0 . (11) Theorem Let ( β, α ) ∈ A ill . Then the delay problem (8), (9), (11) is ill-posed. There exists a sequence ( u j , θ j ) j of solutions with norm � u j ( t ) � H tending to infinity (as j → ∞ ) for any fixed t, while for the initial data the norms sup − τ ≤ s ≤ 0 � ( u 0 j ( s ) , u 1 j , θ 0 ) � H 3 remain bounded.

Introduction Results Sketch of the proof Further results References Sketch of the proof 3. Sketch of the proof Ansatz: u = u j ( t ) = h j ( t ) ϕ j , (12) θ = θ j ( t ) = g j ( t ) ϕ j . (13) Plug in: j ( t )+ b 2 λ 2 β h ′′′ j ( t )+ d λ α j h ′′ j h ′ j ( t )+ a λ j h ′ j ( t − τ )+ ad λ 1+ α h j ( t − τ ) = 0 , j (14) j g j ( t ) = − b λ β g ′ j ( t ) + d λ α j h ′ j ( t ) , (15) with 1 ( h ′′ g j (0) := j (0) + a λ j h j ( − τ )) . (16) b λ β j

Introduction Results Sketch of the proof Further results References Sketch of the proof Ansatz: h j ( t ) = 1 e ω j t , (17) ω 2 j For a solution sufficient and necessary: ω 3 j ω 2 j + ( b 2 λ 2 β + a λ j e − τω j ) ω j = − ad λ 1+ α e − τω j . j + d λ α (18) j j Find subsequence ( ω j k ) k with Re ω j k → ∞ as k → ∞ , (19) λ 2 − β e − τω jk j k sup | | < ∞ , (20) ω 2 k j k for t > 0 : | e ω jk t | → ∞ as k → ∞ . (21) ω 2 j k

Introduction Results Sketch of the proof Further results References Sketch of the proof Case 1: α < 2 β . (18) is equivalent to ω 2 d ω j + a = − ad � e − τω j � j b 2 λ 1 − 2 β b 2 λ 1+ α − 2 β e − τω j . ω j 1+ + b 2 λ 2 β b 2 λ 2 β − α j j j j (22) Ansatz: ω j = µ j (1 + ζ j ) , (23) where | ζ j | < 1 2 and µ j = − ad b 2 λ 1+2 α − β e − τµ j . (24) j (24) has a subsequence ( µ j k ) k of solutions with Re µ j k → ∞ (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 j k ) (1 − e − τµζ ) + ( q ( ζ ) + ζ + ζ q ( ζ )) = 0 , (25) � �� � � �� � =: f ( ζ ) =: g ( ζ ) where q ( ζ ) := µ 2 (1 + ζ ) 2 + d µ (1 + ζ ) b 2 λ 2 β − α + a b 2 λ 1 − 2 β e − τµ (1+ ζ ) . (26) b 2 λ 2 β 1 f , g are holomorphic in B (0 , 10 τ | µ | ) with a single zero of f there. | g ( ζ ) | ≤ C (27) | µ | , 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 ≥ 2 β . (18) is now equivalent to λ α − 2 β ω + a λ 1 − α b 2 ω ω 2 � e − τω � = − a λ e − τω . 1 + d λ α + (28) d ω Ansatz (23) again, now µ solving µ 2 = − a λ e − 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: u tt ( t ) + aAu ( t ) − bA β θ ( t ) = 0 , (30) θ t ( t ) + dA α θ ( t − τ ) + bA β u t ( t ) = 0 , (31) u (0) = u 0 , u t (0) = u 1 , θ ( s ) = θ 0 ( s ) , ( − τ ≤ s ≤ 0) . (32) Theorem Let ( β, α ) ∈ A ill . Then the delay problem (30), (31), (32) is ill-posed. There exists a sequence ( u j , θ j ) j of solutions with norm � u j ( t ) � H tending to infinity (as j → ∞ ) for any fixed t, while for the initial data the norms sup − τ ≤ s ≤ 0 � ( u 0 j , u 1 j , θ 0 ( s )) � H 3 remain bounded.

Introduction Results Sketch of the proof Further results References Further results 2. [Fischer]: – Additional damping u t in (8) does not lead to well-posedness. – Additional damping A γ u t 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 L 2 -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].