On the regularity and stability of the dual-phase-lag equation. (February 2020)

• Record Type: Journal Article
• Authors: Liu, Zhuangyi; Quintanilla, Ramon; Wang, Yang
• Is Part Of: Applied mathematics letters. Volume 100(2020)
• Publication Date: 2020-02
• Subjects: Dual-phase-lag heat equation -- Analyticity -- Exponential stability; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2019.106038
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: In this paper we consider the following linear partial differential equation which is usually seen as an approximation to the dual-phase-lag heat equation proposed by Tzou. T ̇ + τ q T ̈ + τ q 2 2 T ⃛ = κ △ T + κ τ T △ T ̇ + κ τ T 2 2 △ T ̈ on a bounded domain Ω in R n with smooth boundary. We obtain analyticity for the associated C 0 − semigroup. Moreover, we also obtain exponential stability of the solutions by spectrum analysis and Hurwitz criterion under one of the following conditions: (i). τ T τ q > 2 − 3 ; (ii). 2 − 3 ≥ τ T τ q > ( 1 + κ τ T λ 1 ) 2 + ( κ τ T λ 1 ) 2 + ( κ τ T λ 1 ) 3 − ( 1 + κ τ T λ 1 ) κ τ T λ 1 ( 1 + κ τ T λ 1 ), where λ 1 is the smallest eigenvalue of the negative Laplacian on Ω with Dirichlet boundary condition.