On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source
Abstract
:1. Introduction
2. Preliminaries
- (H1)
- The relaxation function is a twice-differentiable and bounded function satisfying andThere exist constants depending on such that
- (H2)
- The exponent p satisfies for and for .
- (H3)
- Assume that l satisfies whenever and whenever .
3. Global Existence and Energy Decay
- (i)
- , ;
- (ii)
- is increasing in the interval , decreasing in the interval and takes its maximum at where ;
- (iii)
- for , for and .
3.1. Global Existence for Low Initial Energy
3.2. General Decay of Global Solution
4. An Example and Numerical Study
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Belhadji, B.; Alzabut, J.; Samei, M.E.; Fatima, N. On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source. Mathematics 2022, 10, 4194. https://doi.org/10.3390/math10224194
Belhadji B, Alzabut J, Samei ME, Fatima N. On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source. Mathematics. 2022; 10(22):4194. https://doi.org/10.3390/math10224194
Chicago/Turabian StyleBelhadji, Bochra, Jehad Alzabut, Mohammad Esmael Samei, and Nahid Fatima. 2022. "On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source" Mathematics 10, no. 22: 4194. https://doi.org/10.3390/math10224194
APA StyleBelhadji, B., Alzabut, J., Samei, M. E., & Fatima, N. (2022). On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source. Mathematics, 10(22), 4194. https://doi.org/10.3390/math10224194