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PreprintCC BY 4.0arXiv

Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases

  • Houria Chellaoua
  • Yamna Boukhatem
  • Cristina Pignotti
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Abstract

In this paper, we establish the well-posedness and exponential stability of a class of semilinear neutral abstract evolution equations with constant time delay. Two damping mechanisms are considered: frictional damping of the form CC*ut and viscoelastic damping with memory. Under suitable assumptions, we prove global well-posedness and exponential decay of solutions for sufficiently small initial data. This result is of considerable importance, as it provides a unified framework for the analysis of a wide class of neutral systems arising in structural mechanics, seismic isolation, and control theory. The abstract results are illustrated by some concrete examples. The analysis relies on a combination of semigroup theory, energy estimates, Duhamel's formula, and Gronwall-type arguments.

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Cite

BibTeXRIS
BibTeX
@misc{chellaoua2026wellposedness,
  title = {{Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases}},
  author = {Houria Chellaoua and Yamna Boukhatem and Cristina Pignotti},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08986},
  eprint = {2609.08986},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08986},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases
AU  - Houria Chellaoua
AU  - Yamna Boukhatem
AU  - Cristina Pignotti
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08986
UR  - https://arxiv.org/abs/2609.08986
AB  - In this paper, we establish the well-posedness and exponential stability of a class of semilinear neutral abstract evolution equations with constant time delay. Two damping mechanisms are considered: frictional damping of the form CC^*u_t and viscoelastic damping with memory. Under suitable assumptions, we prove global well-posedness and exponential decay of solutions for sufficiently small initial data. This result is of considerable importance, as it provides a unified framework for the analysis of a wide class of neutral systems arising in structural mechanics, seismic isolation, and control theory. The abstract results are illustrated by some concrete examples. The analysis relies on a combination of semigroup theory, energy estimates, Duhamel's formula, and Gronwall-type arguments.
ER  -