Two classes of coupled second order evolution equations with indirect damping: Lack of exponential decay

Kun-Peng Jin; Can Liu; Kun-Peng Jin; Can Liu

doi:10.3934/nhm.2025044

Networks and Heterogeneous Media

2025, Volume 20, Issue 3: 1010-1025. doi: 10.3934/nhm.2025044

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Two classes of coupled second order evolution equations with indirect damping: Lack of exponential decay

Kun-Peng Jin ^{1,2
,
,},
Can Liu ¹

1.
Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, China
2.
Guangxi Key Lab of Human-machine Interaction and Intelligent Decision, Nanning Normal University, Nanning 530100, China

Received: 21 April 2025 Revised: 02 September 2025 Accepted: 15 September 2025 Published: 19 September 2025

In this work, we consider two classes of coupled second-order abstract evolution equations with past history only acting in one of the equations of the systems, where the coupling terms are composed of fractional powers of operators. With the aid of the semigroup method and operator theory, we successfully establish that the coupled systems have no exponential decay rate for some fractional powers of coupled operators. Simultaneously, some applications of our abstract results in concrete models of the real world are given. It can be seen that our theorems cover and generalize the previous related results.
- coupled systems,
- indirect damping,
- exponential decay,
- past history
Citation: Kun-Peng Jin, Can Liu. Two classes of coupled second order evolution equations with indirect damping: Lack of exponential decay[J]. Networks and Heterogeneous Media, 2025, 20(3): 1010-1025. doi: 10.3934/nhm.2025044

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Abstract

In this work, we consider two classes of coupled second-order abstract evolution equations with past history only acting in one of the equations of the systems, where the coupling terms are composed of fractional powers of operators. With the aid of the semigroup method and operator theory, we successfully establish that the coupled systems have no exponential decay rate for some fractional powers of coupled operators. Simultaneously, some applications of our abstract results in concrete models of the real world are given. It can be seen that our theorems cover and generalize the previous related results.

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