On inverse problem for a class of fourth order strongly damped wave equations

Nam Danh Hua Quoc1,∗, Can Nguyen Huu2,∗∗, Au Vo Van3,∗∗∗, and Binh Tran Thanh4,∗∗∗∗ 1Department of Scientific Research Management, Thu Dau Mot University, Thu Dau Mot City, Binh Duong Province, Viet Nam 2Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam 3Faculty of General Sciences, Can Tho University of Technology, Can Tho City, Viet Nam 4Faculty of Applied Mathematics, Sai Gon University, Ho Chi Minh City, Viet Nam


Introduction
Throughout this paper, H will denote a Hilbert space endowed with the inner product •, • and the norm • .Let T be a positive number and Ω be an open, bounded and connected domain in Ω ∈ R d (d ≥ 1) with a smooth boundary ∂Ω.We consider the problem of finding a function u : [0, T ] → H satisfes the initial boundary value problem for a class of fourth order strongly damped linear wave equations subject to the conditions There exists a sequence of positive real eigenvalues of the operator −∆, which denoted by {λ k } k≥1 where And γ > 2 is a positive constant, satisfies (γ 2 − 4)λ 1 − 4 > 0.
In practice, the data (ϕ, χ) ∈ H ×H is obtained by measurement contaminated with noise.Hence, instead of (ϕ, χ), we have the observation data (ϕ α , χ α ) satisfying where the constant α > 0 represents a bound on the measurement error.
In the last years, the class of fourth order strongly damped wave equations has been investigated by many researchers, for example, existence, global classical solution, attractor, well-posedness, decay estimates, blowup, controllability, bootstrapping (e.g., [3]- [10]).Another important aspect of the qualitative study for the solutions of strongly damped wave equations is ill-posedness (in the sense of Hadamard).
In 2013, Xu Runzhang and Yanbing Yang [11] considered the global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations but they did not regularize it.Our work provide a regularization method for the solution of the ill-posed problem (1.1) -(1.2).
There are three sections in this paper.In Section 1, we have introduced our problem.The exact solution and ill-posedness of the inverse problem (1.1) -(1.2) are given in Section 2.Moreover, in Subsection 2.2, we give an example to show that the problem (1.1) -(1.2) is not stable.In Section 3, we develop a regularization method based on the truncated Fourier method and the stability estimate is established.

Exact solution and ill-posedness of the inverse problem 2.1 Exact solution
Assume that the problem (1.1) -(1.2) has a unique solution in the series form (2.6) Then u(x, t) satisfies the following system where For any γ > 2 satisfies (γ 2 − 4)λ 1 − 4 > 0, the solution of (2.7) is given by: Let us define the operators G(t) and K(t) as follow and for w ∈ H. From above observations, we can rewrite (2.8) as (2.12)

Ill-posedness of the inverse problem
In this subsection, we give an example which shows that the solution u * (x, t) of problem (1.1) − (1.2) (if it exists) is not stable.
• Let ϕ * , χ * , and R * be defined as follows Then u * satisfy the integral equation We have where This leads to

.15)
On the other hand, we get Using the inequality | exp(−y) − exp(−z)| ≤ |y − z|, for y, z > 0 and noting that µ As m → ∞, we get

Fourier's truncation method
Let N = k ∈ N such that k ≥ 1 and λ k ≤ Λ(α) .Without loss of generality, we assume that Λ(α) be a regularization parameter satisfying For w ∈ H, let us define the truncated version of (2.10) and (2.11) as and Note that we pick Λ(α) so large that G, K are retained.
Let us define the regularized solution by Fourier's truncation method as follows We have the following lemma which shall be useful in next results. Proof.
• From (2.9) and (3.19), let w ∈ H , we have where µ On the other hand, using the inequality (a • From (2.9) and (3.20), let w ∈ H , we have This completes the proof of Lemma 3.1.Before we formulate the main theorem, we also introduce the Gevrey class of functions of order p > 0 and index r > 0, see e.g.[1], defined by the spectrum of the Laplacian is denoted by and their norms given by where T max = max 2 + 4T γ 2 , T , N α is the integer part of Λ(α) and γ > 2 is a positive constant, satisfies (γ 2 − 4)λ 1 − 4 > 0.
Moreover, there exists an orthonormal basis {ξ k } k≥1 , where ξ k is an eigenfunction corresponding to λ k Conferences 20, 02006 (2018) https://doi.org/10.1051/itmconf/20182002006ICM 2018 It is easy to see that the function f Thus the problem (1.1)-(1.2) is ill-posed in the sense of Hadamard.Next section, we propose a regularization method for stabilizing the problem.