Houda Fadel, Djamila Benterki, Salim A. Messaoudi
Abstract:
In this work, we consider the coupled viscoelastic wave system
$$\displaylines{
u_{tt} - \Delta u + \int_0^t g(t - s) \Delta u \, ds + \alpha v = 0, \quad \text{in } \Omega \times (0, T), \cr
v_{tt} - \Delta v + \alpha u = 0, \quad \text{in } \Omega \times (0, T), \cr
u = v = 0, \quad \text{on } \partial \Omega \times (0, T), \\
u(0) = u_0, \quad u_t(0) = u_1, \quad v(0) = v_0, \quad v_t(0) = v_1, \quad \text{in } \Omega,
}$$
where \( \Omega \) is a bounded domain in \( \mathbb{R}^N \), \( \alpha > 0 \), and
the initial data belong to suitable spaces.
The relaxation function \(g \) satisfies \(g'(t) \leq -\xi(t) G(g(t))\), where
\(G\) is an increasing and convex function near the origin and
\(\xi\) is a non-increasing function.
We first prove the well-posedness of the problem, and then we establish a general
decay rate of the system energy, highlighting the influence of the
relaxation function on the stability of the solutions.
Numerical tests were also conducted to validate our theoretical findings.
Submitted September 24, 2025. Published February 19, 2026.
Math Subject Classifications: 35L51, 35B35, 35B40, 49K40, 93D15.
Key Words: Viscoelasticity; energy method; wave equation; general stability.
DOI: 10.58997/ejde.2026.16
Show me the PDF file (537 KB), TEX file for this article.
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Houda Fadel Laboratory of Mathematical Analysis and Applications Department of Mathematics University Mohamed El Bachir El Ibrahimi Bordj Bou Arreridj, Algeria email: houda.fadel@univ-bba.dz |
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Djamila Benterki Laboratory of Mathematical Analysis and Applications Department of Mathematics University Mohamed El Bachir El Ibrahimi Bordj Bou Arreridj, Algeria email: djamila.benterki@univ-bba.dz |
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Salim A. Messaoudi Department of Mathematics College of Sciences University of Sharjah, Sharjah, UAE email: smessaoudi@sharjah.ac.ae |
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