On a Fourth-Order Equation of Moore–Gibson–Thompson Type

• Published in: Milan journal of mathematics Vol. 85; no. 2; pp. 215 - 234
• Main Authors: Dell’Oro, Filippo, Pata, Vittorino
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2017
• Subjects: Analysis; Mathematics; Mathematics and Statistics; 47D06; 74D05; growth bound; Secondary 35G05; Moore–Gibson–Thompson equation with memory; solution semigroup; Primary 35B35; fourth-order equation; exponential stability
• ISSN: 1424-9286; 1424-9294; 1424-9294
• DOI: 10.1007/s00032-017-0270-0

An abstract version of the fourth-order equation ∂ t t t t u + α ∂ t t t u + β ∂ t t u - γ Δ ∂ t t u - δ Δ ∂ t u - ϱ Δ u = 0 subject to the homogeneous Dirichlet boundary condition is analyzed. Such a model encompasses the Moore–Gibson–Thompson equation with memory in presence of an exponential kernel. The stability properties of the related solution semigroup are investigated. In particular, a necessary and sufficient condition for exponential stability is established, in terms of the values of certain stability numbers depending on the strictly positive parameters α , β , γ , δ , ϱ .