Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

Wassila Ghecham; Salah-Eddine Rebiai; Fatima Zohra Sidiali

doi:10.1515/jaa-2021-2051

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Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

Wassila Ghecham , Salah-Eddine Rebiai and Fatima Zohra Sidiali

Published/Copyright: April 2, 2021

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From the journal Journal of Applied Analysis Volume 28 Issue 1

Abstract

A wave equation in a bounded and smooth domain of ℝ n with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

Keywords: Wave equation; time delays; nonlinear boundary feedback; stabilization; decay rate

MSC 2010: 35L05; 93D15

A Appendix

Lemma A.1.

The operator T given by

T ⁢ η = λ ⁢ η + 1 λ ⁢ A ⁢ η + α 1 ⁢ AN ⁡ f ^ ⁢ ( η ) + α 2 ⁢ AN ⁡ g ^ ⁢ ( η )

is surjective from V = D ⁢ ( A 1 / 2 ) = H Γ 1 1 ⁢ ( Ω ) onto V ′ = ( D ⁢ ( A 1 / 2 ) ) ′ = ( H Γ 1 1 ⁢ ( Ω ) ) ′ .

Proof.

For η ∈ V , let

B η = α 1 AN f ^ ( η ) + α 2 AN g ^ ( ( η ) and C η = λ η + 1 λ A η .

Then

T ⁢ η = B ⁢ η + C ⁢ η .

According to Barbu [3, Corollary 3, p. 49], in order to establish surjectivity of the operator T, it is sufficient to prove that B is monotone and hemicontinuous, C is maximal monotone and B + C is coercive.

Monotonicity of B. Let η , v ∈ V and set ℬ = - B . Then

〈 ℬ η - ℬ v , η - v 〉 V ′ × V = - α 1 〈 A N ( f ^ ( η ) - f ^ ( v ) , η - v 〉 V ′ × V - α 2 〈 A N ( g ^ ( η ) - g ^ ( v ) ) , η - v 〉 V ′ × V

= - α 1 ⁢ 〈 f ^ ⁢ ( η ) - f ^ ⁢ ( v ) , N ∗ ⁢ A ∗ ⁢ ( η - v ) 〉 L 2 ⁢ ( Γ 2 ) - α 2 ⁢ 〈 g ^ ⁢ ( η ) - g ^ ⁢ ( v ) , N ∗ ⁢ A ∗ ⁢ ( η - v ) 〉 L 2 ⁢ ( Γ 2 )

≤ - ( α 1 ⁢ r - α 2 ⁢ L ⁢ e - λ ⁢ τ ) ⁢ ∥ ( η | Γ 2 ) - ( v | Γ 2 ) ∥ L 2 ⁢ ( Γ 2 ) 2 .

By (2.19), we conclude that

(A.1) 〈 ℬ ⁢ η - ℬ ⁢ v , η - v 〉 V ′ × V ≤ 0 .

Thus B is monotone.

Hemicontinuity of B. Let η , v , w ∈ V . We will prove that the function

t ↦ 〈 B ⁢ ( η + t ⁢ v ) , w 〉 V ′ × V

is continuous. Indeed, we have

| 〈 B ⁢ ( η + t ⁢ v ) - B ⁢ ( η + t 0 ⁢ v ) , w 〉 V ′ × V |

= | 〈 α 1 ⁢ AN ⁡ ( f ^ ⁢ ( η + t ⁢ v ) - f ^ ⁢ ( η + t 0 ⁢ v ) ) , w 〉 V ′ × V + 〈 α 2 ⁢ AN ⁡ ( g ^ ⁢ ( η + t ⁢ v ) - g ^ ⁢ ( η + t 0 ⁢ v ) ) , w 〉 V ′ × V |

≤ | 〈 α 1 ⁢ ( f ^ ⁢ ( η + t ⁢ v ) - f ^ ⁢ ( η + t 0 ⁢ v ) ) , N ∗ ⁢ A ∗ ⁢ w 〉 L 2 ⁢ ( Γ 2 ) | + | 〈 α 2 ⁢ ( g ^ ⁢ ( η + t ⁢ v ) - g ^ ⁢ ( η + t 0 ⁢ v ) ) , N ∗ ⁢ A ∗ ⁢ w 〉 L 2 ⁢ ( Γ 2 ) |

≤ c ⁢ ∥ w ∥ L 2 ⁢ ( Γ 2 ) ⁢ { ∥ f ^ ⁢ ( η + t ⁢ v ) - f ^ ⁢ ( η + t 0 ⁢ v ) ∥ L 2 ⁢ ( Γ 2 ) + ∥ g ^ ⁢ ( η + t ⁢ v ) - g ^ ⁢ ( η + t 0 ⁢ v ) ∥ L 2 ⁢ ( Γ 2 ) } .

From assumption (2.13), we have

| 〈 B ⁢ ( η + t ⁢ v ) - B ⁢ ( η + t 0 ⁢ v ) , w 〉 V ′ × V | ≤ c ⁢ ∥ w ∥ L 2 ⁢ ( Γ 2 ) ⁢ ( ∥ f ^ ⁢ ( η + t ⁢ v ) - f ^ ⁢ ( η + t 0 ⁢ v ) ∥ L 2 ⁢ ( Γ 2 ) + L ⁢ ∥ ( t - t 0 ) ⁢ ( v | Γ 2 ) ∥ L 2 ⁢ ( Γ 2 ) ) .

The continuity of f allows us to deduce that

| 〈 B ⁢ ( η + t ⁢ v ) - B ⁢ ( η + t 0 ⁢ v ) , w 〉 V ′ × V | < ϵ ~

for | t - t 0 | < δ ~ . This proves the continuity of the function t ↦ 〈 B ⁢ ( η + t ⁢ v ) , w 〉 .

Maximal monotonicity of C. For η , v ∈ V , we have

〈 C ⁢ η - C ⁢ v , η - v 〉 V ′ × V = 〈 λ ⁢ η + 1 λ ⁢ A ⁢ η - λ ⁢ v - 1 λ ⁢ A ⁢ v , η - v 〉 V ′ × V

= λ ⁢ ∥ η - v ∥ V 2 + 1 λ ⁢ 〈 A ⁢ ( η - v ) , η - v 〉 V ′ × V

(A.2) ≥ λ ⁢ ∥ η - v ∥ V 2 ,

so that C is monotone. Since the operator

A : V → V ′

is continuous, the operator C is also continuous. Then it is maximal monotone.

Coercivity of B + C . This follows from (A.1) and (A.2). ∎

Acknowledgements

The authors would like to thank the anonymous referee for his valuable comments.

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Received: 2019-12-16

Revised: 2020-11-04

Accepted: 2020-11-06

Published Online: 2021-04-02

Published in Print: 2022-06-01

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Articles in the same Issue

https://doi.org/10.1515/jaa-2021-2051

Keywords for this article

Wave equation; time delays; nonlinear boundary feedback; stabilization; decay rate