On a Hartree-type nonlinearity wave equation with distributed delay combined with a fractional condition

Salah Boulaaras; Abdelbaki Choucha; Salah Boulaaras; Abdelbaki Choucha

doi:10.3934/nhm.2026012

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 243-260. doi: 10.3934/nhm.2026012

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On a Hartree-type nonlinearity wave equation with distributed delay combined with a fractional condition

Salah Boulaaras ^{1
,
,},
Abdelbaki Choucha ^2,3

1.
Department of Mathematics, College of Sciences, Qassim University, Buraydah 51452, Saudi Arabia
2.
Department of Material Sciences, Faculty of Sciences, Amar Teleji Laghouat University, Laghouat 03000, Algeria
3.
Laboratory of Mathematics and Applied Sciences, Ghardaia University, Ghardaia 47000, Algeria

Received: 18 January 2026 Revised: 06 February 2026 Accepted: 25 February 2026 Published: 03 March 2026
35B40, 35L70, 76Exx, 93D20

This work focused on the analysis of a nonlinear wave equation of Hartree-type that includes a distributed delay term, where the delay effects are governed with fractional conditions. Such a formulation allows the model to incorporate long-range memory effects and anomalous dissipation phenomena, which are characteristic of complex media. The model captures complex memory and nonlocal interaction effects that arise in various physical systems, such as quantum mechanics and nonlinear optics. In particular, the fractional delay mechanism provides a more accurate description of hereditary effects than classical integer-order delay models. We worked under a framework that allows for initial data with negative energy and imposed suitable assumptions on the kernel functions and nonlinear terms. Using energy methods and a concavity argument, we rigorously proved that the solution to the system cannot exist globally in time and must blow up in finite time. Compared with the classical Hartree wave equation without delay or fractional effects, our results show that the combined presence of distributed delay and fractional damping significantly enhances the instability mechanism.
- nonlinear equations,
- blow up,
- fractional damping,
- distributed delay,
- Hartree-type nonlinearity
Citation: Salah Boulaaras, Abdelbaki Choucha. On a Hartree-type nonlinearity wave equation with distributed delay combined with a fractional condition[J]. Networks and Heterogeneous Media, 2026, 21(1): 243-260. doi: 10.3934/nhm.2026012

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Abstract

This work focused on the analysis of a nonlinear wave equation of Hartree-type that includes a distributed delay term, where the delay effects are governed with fractional conditions. Such a formulation allows the model to incorporate long-range memory effects and anomalous dissipation phenomena, which are characteristic of complex media. The model captures complex memory and nonlocal interaction effects that arise in various physical systems, such as quantum mechanics and nonlinear optics. In particular, the fractional delay mechanism provides a more accurate description of hereditary effects than classical integer-order delay models. We worked under a framework that allows for initial data with negative energy and imposed suitable assumptions on the kernel functions and nonlinear terms. Using energy methods and a concavity argument, we rigorously proved that the solution to the system cannot exist globally in time and must blow up in finite time. Compared with the classical Hartree wave equation without delay or fractional effects, our results show that the combined presence of distributed delay and fractional damping significantly enhances the instability mechanism.

References

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