Study of the well-posedness and decay rates for Rao–Nakra sandwich beam models subject to a single internal infinite memory and Dirichlet–Neumann boundary conditions

• Published in: Computational & applied mathematics Vol. 44; no. 2
• Main Author: Guesmia, Aissa
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.02.2025
• Subjects: Asymptotic properties; Beams (structural); Boundary conditions; Decay rate; Euler-Bernoulli equation; Mathematics; Parameters; Polynomials; Sandwich structures; Systems stability; Wave equations; Wave propagation; Well posed problems
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-024-03033-6

The Rao–Nakra sandwich beam models under study in this paper consist of coupled two wave equations and one Euler–Bernoulli equation on an open bounded interval of R subject to homogeneous Dirichlet–Neumann boundary conditions. The objective of this paper is to investigate the well-posedness and asymptotic behavior of solutions when a single internal infinite memory term is present either (i) on one wave equation or (ii) on the Euler–Bernoulli equation. The involved operator in the memory term is -∂xxm with m∈{0,1} in case (i), and m∈{0,1,2} in case (ii). It is proved in this paper that the system is well posed and can be indirectly stabilized polynomially (for strong solutions) independently from the parameter m, and the coefficients of the system in case (i), and if and only if the speeds of wave propagation of the two wave equations are different in case (ii), where the considered memory kernel is exponentially decreasing. We prove also that the same conditions guarantee the strong stability of the system (for weak solutions). However, in both cases (i) and (ii) and independently from the parameter m and the coefficients of the system, we prove that this single internal infinite memory term does not stabilize exponentially the whole considered Rao–Nakra sandwich beam models. On the other hand, the polynomial decay rates are given explicitly in terms of the parameter m. Our results are new because they are the first to demonstrate exponential, polynomial, and strong stability for Rao–Nakra sandwich beams with a single internal infinite memory.