Blow-up dynamics of hyperbolic equations with Hartree-type nonlinearities and internal fractional damping: Analytical results and numerical illustration

Muhammad Fahim Aslam; Jianghao Hao; Iqra Kanwal; Imran Shabir Chuhan; Mohamed Balegh; Zayd Hajjej; Muhammad Fahim Aslam; Jianghao Hao; Iqra Kanwal; Imran Shabir Chuhan; Mohamed Balegh; Zayd Hajjej

doi:10.3934/math.20261195

AIMS Mathematics

2026, Volume 11, Issue 9: 30162-30193. doi: 10.3934/math.20261195

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Research article

Blow-up dynamics of hyperbolic equations with Hartree-type nonlinearities and internal fractional damping: Analytical results and numerical illustration

1.
Department of Mathematics, University of Kotli Azad Jammu and Kashmir (UOKAJK), Kotli, Pakistan
2.
School of Mathematics and Statistics, Shanxi University, Taiyuan, China
3.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
4.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 11 July 2026 Revised: 24 August 2026 Accepted: 03 September 2026 Published: 16 September 2026
MSC : 26A33, 35L05, 35B44, 65M50

In this work, we studied a class of nonlinear wave equations involving a Hartree-type nonlocal source term together with an interior exponentially tempered fractional damping mechanism. The fractional operator was treated through a diffusive representation, which allows the problem to be reformulated as an augmented system suitable for semigroup analysis. Under appropriate assumptions on the exponent and the initial data, we first established local well-posedness by proving the local Lipschitz continuity of the nonlocal Hartree term in the energy space. We then proved the global existence of the solution under suitable assumptions. After that, we investigated the long-time behavior of solutions. In particular, by combining energy estimates with a negative initial energy argument, we derived a finite-time blow-up result. Finally, numerical simulations for a regularized one-dimensional analogue were presented to illustrate qualitatively the rapid-growth mechanism. These computations are not a direct numerical approximation of the multidimensional problem, but they retain its Hartree-type nonlocal interaction, fractional-memory damping, and negative-energy structure.
- fractional damping,
- wave equation,
- Hartree-type nonlinearity,
- finite-time blow-up,
- numerical simulations
Citation: Muhammad Fahim Aslam, Jianghao Hao, Iqra Kanwal, Imran Shabir Chuhan, Mohamed Balegh, Zayd Hajjej. Blow-up dynamics of hyperbolic equations with Hartree-type nonlinearities and internal fractional damping: Analytical results and numerical illustration[J]. AIMS Mathematics, 2026, 11(9): 30162-30193. doi: 10.3934/math.20261195

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Abstract

In this work, we studied a class of nonlinear wave equations involving a Hartree-type nonlocal source term together with an interior exponentially tempered fractional damping mechanism. The fractional operator was treated through a diffusive representation, which allows the problem to be reformulated as an augmented system suitable for semigroup analysis. Under appropriate assumptions on the exponent and the initial data, we first established local well-posedness by proving the local Lipschitz continuity of the nonlocal Hartree term in the energy space. We then proved the global existence of the solution under suitable assumptions. After that, we investigated the long-time behavior of solutions. In particular, by combining energy estimates with a negative initial energy argument, we derived a finite-time blow-up result. Finally, numerical simulations for a regularized one-dimensional analogue were presented to illustrate qualitatively the rapid-growth mechanism. These computations are not a direct numerical approximation of the multidimensional problem, but they retain its Hartree-type nonlocal interaction, fractional-memory damping, and negative-energy structure.

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