On global existence and blow up of weak solutions for a wave equation with mixed local and nonlocal propagation

Jixian Cui; Jixian Cui

doi:10.3934/math.20251121

AIMS Mathematics

2025, Volume 10, Issue 11: 25329-25345. doi: 10.3934/math.20251121

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On global existence and blow up of weak solutions for a wave equation with mixed local and nonlocal propagation

Jixian Cui ^,

School of Science, Qiqihar University, Qiqihar, China

Received: 11 July 2025 Revised: 16 October 2025 Accepted: 30 October 2025 Published: 04 November 2025
MSC : 35R11, 35A15, 45K05

In this paper, we investigate the initial boundary value problem of a wave equation with mixed local and nonlocal propagation. First, we introduce the spatial framework to study the wave equation, which is the intersection of a classical Sobolev space and a fractional Sobolev space. By the Mountain Pass Theorem, we obtain the attainability of the optimal embedding constant from the introduced space to the suitable Lebesgue space. Second, by introducing a family of potential wells in the introduced space, we obtain the existence of a global weak solution through the utilization of potential well theory. At last, by analyzing the properties of the energy functional, we show that any weak solution must blow up at the existence time.
- potential well,
- mixed local and nonlocal propagation,
- wave equation,
- global solution,
- blow up
Citation: Jixian Cui. On global existence and blow up of weak solutions for a wave equation with mixed local and nonlocal propagation[J]. AIMS Mathematics, 2025, 10(11): 25329-25345. doi: 10.3934/math.20251121

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Abstract

In this paper, we investigate the initial boundary value problem of a wave equation with mixed local and nonlocal propagation. First, we introduce the spatial framework to study the wave equation, which is the intersection of a classical Sobolev space and a fractional Sobolev space. By the Mountain Pass Theorem, we obtain the attainability of the optimal embedding constant from the introduced space to the suitable Lebesgue space. Second, by introducing a family of potential wells in the introduced space, we obtain the existence of a global weak solution through the utilization of potential well theory. At last, by analyzing the properties of the energy functional, we show that any weak solution must blow up at the existence time.

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