General decay of solutions for a von Karman plate system with general type of relaxation functions on the boundary

Jum-Ran Kang; Jum-Ran Kang

doi:10.3934/math.2024114

AIMS Mathematics

2024, Volume 9, Issue 1: 2308-2325. doi: 10.3934/math.2024114

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General decay of solutions for a von Karman plate system with general type of relaxation functions on the boundary

Jum-Ran Kang ^,

Department of Applied Mathematics, Pukyong National University, Busan 48513, Republic of Korea

Received: 04 October 2023 Revised: 30 November 2023 Accepted: 05 December 2023 Published: 22 December 2023
MSC : 35B40, 35L05, 37L45, 74D99

In this paper, we investigate a von Karman plate system with general type of relaxation functions on the boundary. We derive the general decay rate result without requiring the assumption that the initial value $ w_0 \equiv 0 $ on the boundary, using the multiplier method and some properties of the convex functions. Here we consider the resolvent kernels $ k_i(i = 1, 2) $, namely $ k_i''(t) \geq - \xi_i(t) G_i(-k_i'(t)) $, where $ G_i $ are convex and increasing functions near the origin and $ \xi_i $ are positive nonincreasing functions. Moreover, the energy decay rates depend on the functions $ \xi_i $ and $ G_i. $ These general decay estimates allow for certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore improve earlier results in the literature.
- von Karman plate,
- general decay,
- memory term,
- relaxation function,
- convexity,
- boundary condition
Citation: Jum-Ran Kang. General decay of solutions for a von Karman plate system with general type of relaxation functions on the boundary[J]. AIMS Mathematics, 2024, 9(1): 2308-2325. doi: 10.3934/math.2024114

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Abstract

In this paper, we investigate a von Karman plate system with general type of relaxation functions on the boundary. We derive the general decay rate result without requiring the assumption that the initial value $ w_0 \equiv 0 $ on the boundary, using the multiplier method and some properties of the convex functions. Here we consider the resolvent kernels $ k_i(i = 1, 2) $, namely $ k_i''(t) \geq - \xi_i(t) G_i(-k_i'(t)) $, where $ G_i $ are convex and increasing functions near the origin and $ \xi_i $ are positive nonincreasing functions. Moreover, the energy decay rates depend on the functions $ \xi_i $ and $ G_i. $ These general decay estimates allow for certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore improve earlier results in the literature.

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