erezat kurimaiová presentation of master's thesis written

Spectral properties of the damped wave equation

Tereza KurimaiováPresentation of Master's thesis written under supervision of David Krejèiøík

Czech Technical University in Prague

Aspect'19

Tereza Kurimaiová (CTU in Prague) Spectral properties of damped wave equation Aspect'19 1 / 23

Contents

1 Motivation - Damped vibration of string

2 Damped wave operator on Ω ⊂ Rd with bounded damping and

Schrödinger operator

3 Results for the damped wave operator obtained using:

Lieb-Thirring inequalities

Buslaev-Faddeev-Zakharov trace formulae

Birman-Schwinger principle

4 Finite potential well

Motivation - Damped vibration of string

Let Ω = (0, L), a > 0 then we have the damped wave equation with the

damping a in the form

utt + aut − uxx = 0, x ∈ (0, L), t > 0

u = u1, x ∈ (0, L), t = 0

ut = u2, x ∈ (0, L), t = 0

u = 0, x = 0, L, t > 0

Denote

)a U(t) :=

then formally

dtU(t) =

(ututt

−aut + uxx

(0 I∂2

∂x2−a

(0 I∂2

∂x2−a

)U(t).

Choosing the Hilbert space

H :=(H10 (0, L)× L2(0, L), (·, ·)H

)with the inner product

(Ψ,Φ)H :=

(φ1φ2

dφ1dx

+ ψ2φ2

Denote

)a U(t) :=

then formally

dtU(t) =

(ututt

−aut + uxx

(0 I∂2

∂x2−a

(0 I∂2

∂x2−a

)U(t).

Choosing the Hilbert space

H :=(H10 (0, L)× L2(0, L), (·, ·)H

)with the inner product

(Ψ,Φ)H :=

(φ1φ2

dφ1dx

+ ψ2φ2

(0 Id2

dx2−a

), Dom(A) :=

(H2(0, L) ∩ H1

0 (0, L))× H1

0 (0, L).

We thus obtain an evolution problem

dtU(t) = AU(t), U(0) = U0

for A being densely dened, closed, unbounded, non-self-adjoint and

generating a C0-semigroup eAt .

There exists ω ∈ R, ‖eAt‖ ≤ eωt .

We dene ω0 as the smallest such ω. For the string it holds

ω0 = ωσ(A) := supRλ : λ ∈ σ(A),

Cox, S., and E. Zuazua. \The Rate at Which Energy Decays in a

Damped String." Communications in Partial Dierential Equations,

vol. 19, no. 1-2, 1994, pp. 213243.

(0 Id2

dx2−a

), Dom(A) :=

(H2(0, L) ∩ H1

0 (0, L))× H1

0 (0, L).

dtU(t) = AU(t), U(0) = U0

ω0 = ωσ(A) := supRλ : λ ∈ σ(A),

vol. 19, no. 1-2, 1994, pp. 213243.

(0 Id2

dx2−a

), Dom(A) :=

(H2(0, L) ∩ H1

0 (0, L))× H1

0 (0, L).

dtU(t) = AU(t), U(0) = U0

ω0 = ωσ(A) := supRλ : λ ∈ σ(A),

vol. 19, no. 1-2, 1994, pp. 213243.

(0 Id2

dx2−a

), Dom(A) :=

(H2(0, L) ∩ H1

0 (0, L))× H1

0 (0, L).

dtU(t) = AU(t), U(0) = U0

ω0 = ωσ(A) := supRλ : λ ∈ σ(A),

vol. 19, no. 1-2, 1994, pp. 213243.

Finding optimal damping

Let 0 6= Ψ =

)∈ Dom(A), AΨ = λΨ:

ψ2 = λψ1,d2

dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.

− d2

dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.

We obtain

λa + λ2 = −(nπ

and nally

σ(A) =

(−a±

√a2 − 4

)2)+∞

Let 0 6= Ψ =

ψ2 = λψ1,d2

dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.

− d2

dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.

We obtain

λa + λ2 = −(nπ

and nally

σ(A) =

(−a±

√a2 − 4

)2)+∞

Let 0 6= Ψ =

ψ2 = λψ1,d2

dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.

− d2

dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.

We obtain

λa + λ2 = −(nπ

and nally

σ(A) =

(−a±

√a2 − 4

)2)+∞

Let 0 6= Ψ =

ψ2 = λψ1,d2

dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.

− d2

dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.

We obtain

λa + λ2 = −(nπ

and nally

σ(A) =

(−a±

√a2 − 4

)2)+∞

Let 0 6= Ψ =

ψ2 = λψ1,d2

dx2ψ1 − aψ2 = λψ2, ψ1(0) = ψ1(L) = ψ2(0) = ψ2(L) = 0.

− d2

dx2ψ1 = (−λa− λ2)ψ1, ψ1(0) = ψ1(L) = 0.

We obtain

λa + λ2 = −(nπ

and nally

σ(A) =

(−a±

√a2 − 4

)2)+∞

ω0(A) = supRλ : λ ∈ σ(A) =

2, a ≤ 2π

√a2 − 4

)2, a >

mina≥0

ω0(A) = −πL

ω0(A) = supRλ : λ ∈ σ(A) =

2, a ≤ 2π

√a2 − 4

)2, a >

mina≥0

ω0(A) = −πL

ω0(A) = supRλ : λ ∈ σ(A) =

2, a ≤ 2π

√a2 − 4

)2, a >

mina≥0

ω0(A) = −πL

Damped wave operator on Ω ⊂ Rd with bounded damping

Let Ω ⊂ Rd and a ∈ L∞(Ω). We choose

H :=(H10 (Ω)× L2(Ω), (·, ·)H

)where

(Ψ,Φ)H :=

(φ1φ2

∫Ω∇ψ1∇φ1 + ψ1φ1 + ψ2φ2

and dene the damped wave operator

(0 I∆ −a

), Dom(A) := Dom(−∆)× H1

0 (Ω)

Damped wave operator on Ω ⊂ Rd with bounded damping

Let Ω ⊂ Rd and a ∈ L∞(Ω). We choose

H :=(H10 (Ω)× L2(Ω), (·, ·)H

)where

(Ψ,Φ)H :=

(φ1φ2

∫Ω∇ψ1∇φ1 + ψ1φ1 + ψ2φ2

and dene the damped wave operator

(0 I∆ −a

), Dom(A) := Dom(−∆)× H1

0 (Ω)

A is again densely dened, closed, unbounded, non-self-adjoint and

generates a C0-semigroup eAt

If the damping a changes sign, then there lies a positive point in the

spectrum and thus there exists an unstable solution:

Freitas, P., and Krejèiøík D. \Instability Results for the Damped Wave

Equation in Unbounded Domains." Journal of Dierential Equations,

vol. 211, no. 1, 2005, pp. 168186

Our goal: better localization of the spectrum

vol. 211, no. 1, 2005, pp. 168186

On L2(Ω) we dene

Sµψ := −∆ψ + µVψ, Dom(Sµ) := ψ ∈ H10 (Ω) : ∆ψ ∈ L2(Ω)

where V ∈ L∞(Ω), V −−−−−→|x |→+∞

0 and µ ∈ R

For α > 0 and a ≡ αV we have

−(µα

)2∈ σp(Sµ)⇐⇒ −∆ψ + µVψ = −

)2ψ ⇐⇒

⇐⇒ Aα(ψµαψ

(ψµαψ

)⇐⇒ µ

α∈ σp(Aα)

On L2(Ω) we dene

Sµψ := −∆ψ + µVψ, Dom(Sµ) := ψ ∈ H10 (Ω) : ∆ψ ∈ L2(Ω)

where V ∈ L∞(Ω), V −−−−−→|x |→+∞

0 and µ ∈ R

For α > 0 and a ≡ αV we have

−(µα

)2∈ σp(Sµ)⇐⇒ −∆ψ + µVψ = −

)2ψ ⇐⇒

⇐⇒ Aα(ψµαψ

(ψµαψ

)⇐⇒ µ

α∈ σp(Aα)

Let Ω = Rd . For the negative point spectrum of the Schrödinger operator

there exists the Lieb-Thirring inequalities

Nµ∑n=1

|λn(µ)|γ ≤ Lγ,d

(µV )γ+ d

Theorem (1)

Let A be the damped wave operator with damping V . If V∓ ∈ Ld(Rd) and∫Rd

V d∓ <

L d2,d

then A has no positive, respectively negative eigenvalues.

Theorem (2)

Let A be the damped wave operator with damping V . Let µ be its

positive, respectively negative eigenvalue and V∓ ∈ Lγ+ d2 (Rd). Then

|µ|γ−d2 ≤ Lγ,d

Vγ+ d

2∓ .

Nµ∑n=1

(µV )γ+ d

Theorem (1)

V d∓ <

L d2,d

Theorem (2)

Vγ+ d

2∓ .

Nµ∑n=1

(µV )γ+ d

Theorem (1)

V d∓ <

L d2,d

Theorem (2)

Vγ+ d

2∓ .

Let Ω = R. For the negative point spectrum of the Schrödinger operator

we have an upper bound using the Buslaev-Faddeev-Zakharov trace

formulaeNµ∑n=1

|λn(µ)|12 ≥ −µ

Zakharov, V. E., and L. D. Faddeev. \KortewegDe Vries Equation: A

Completely Integrable Hamiltonian System." Fifty Years of Mathematical

Physics, 2016, pp. 277284.

Theorem (3)

Let A be the damped wave operator with damping V ∈ L1(R, |x |dx). Letµ be its real eigenvalue. If µ > 0 and

∫R V < −4 or µ < 0 and

∫R V > 4

|µ| ≥(∫

R|V (x)||x | dx

)−1.

Let Ω = R. For the negative point spectrum of the Schrödinger operator

we have an upper bound using the Buslaev-Faddeev-Zakharov trace

formulaeNµ∑n=1

|λn(µ)|12 ≥ −µ

Zakharov, V. E., and L. D. Faddeev. \KortewegDe Vries Equation: A

Completely Integrable Hamiltonian System." Fifty Years of Mathematical

Physics, 2016, pp. 277284.

Theorem (3)

Let A be the damped wave operator with damping V ∈ L1(R, |x |dx). Letµ be its real eigenvalue. If µ > 0 and

∫R V < −4 or µ < 0 and

∫R V > 4

|µ| ≥(∫

R|V (x)||x | dx

)−1.

Theorem (4)

Let Aα be the damped wave operator with damping αV , V ∈ L1(R, |x |dx)and it holds

∫R V ≶ 0. Then for µ ≷ 0 such that

|µ| <(∫

R |V (x)||x | dx)−1

there exists exactly one α satisfying

(∫RV∓

)−1≤ α ≤ ∓4

(∫RV

)−1such that µ

α is an eigenvalue Aα.

Now assume Ω = Rd and V ∈ L∞(Rd) is complex.

µ ∈ σp(A) =⇒ −∆ψ + µVψ = −µ2ψ =⇒ (−∆ + µ2I )ψ = −µV 12|V |

=⇒ V−112

(−∆ + µ2I )|V |−12 |V |

12ψ = −µ|V |

=⇒ µ|V |12 (−∆ + µ2I )−1V 1

12ψ = −|V |

=⇒ Kµ|V |12ψ = −|V |

Theorem (5, BS principle for the damped wave operator)

Let A be the damped wave operator with damping V . For µ ∈ C, Rµ 6= 0

it holds

µ ∈ σp(A)⇒ −1 ∈ σp(Kµ).

=⇒ V−112

(−∆ + µ2I )|V |−12 |V |

12ψ = −µ|V |

=⇒ µ|V |12 (−∆ + µ2I )−1V 1

12ψ = −|V |

=⇒ Kµ|V |12ψ = −|V |

it holds

µ ∈ σp(A)⇒ −1 ∈ σp(Kµ).

=⇒ V−112

(−∆ + µ2I )|V |−12 |V |

12ψ = −µ|V |

=⇒ µ|V |12 (−∆ + µ2I )−1V 1

12ψ = −|V |

=⇒ Kµ|V |12ψ = −|V |

it holds

µ ∈ σp(A)⇒ −1 ∈ σp(Kµ).

Theorem (6)

Let d = 1 and A be the damped wave operator with damping V ∈ L1(R).If ‖V ‖L1 < 2 then σp(A) ⊂ µ ∈ C : Rµ ≤ 0.

Theorem (7)

Let d = 3 and A be the damped wave operator with damping

V ∈ L32 (R3). Then

σp(A) ⊂

µ ∈ C : Rµ ≤ 0 ∨ |µ| ≥ 4 3

3√2 ‖V ‖

Theorem (6)

Let d = 1 and A be the damped wave operator with damping V ∈ L1(R).If ‖V ‖L1 < 2 then σp(A) ⊂ µ ∈ C : Rµ ≤ 0.

Theorem (7)

Let d = 3 and A be the damped wave operator with damping

V ∈ L32 (R3). Then

σp(A) ⊂

µ ∈ C : Rµ ≤ 0 ∨ |µ| ≥ 4 3

3√2 ‖V ‖

Finite potential well

V (x) =

0, x < −ba, −b < x < b

0, x > b.

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5a

LT(3/2)

LT(5/2)

Figure: The dependence of the bounds for the eigenvalues of A with b = 1

and a ∈ (−4,−1.1)

Conclusions

We analyzed the damped wave equation with both real and complex

damping

Using the correspondence between the family of Schrödinger

operators and the damped wave operator we found various spectral

bounds for the damped wave operator which provided us with

information about the behavior of the system

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