Finite-time blow-up of solutions for a Hartree type wave equation involving distributed delay, fractional conditions, and infinite memory effects
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Abstract
This paper studies a Hartree-type wave equation featuring a distributed delay term and memory effects governed by a past history. The problem is formulated with coupling through fractional boundary conditions. Under suitable assumptions and for negative initial energy, we prove that solutions blow up in finite time.
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References
[1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Academic Press, (2003), https://doi.org/10.1002/mma.4267.
[2] A.M. Al-Mehdi, M. M. Al-Gharabli, S. A. Messaoudi, New general decay result for a system of viscoelastic wave equation with past hystory, Communications On Pure and Applied Analysis, 20 (2021), Doi:10.3934/cpaa.2020273.
[3] R. Aounallah, A. Choucha, S. Boulaaras and A. Zarai, Asymptotic behavior of a viscoelastic wave equation with a delay in internal fractional feedback, Archives of Control Sciences, 34 2 (2024), 379-413. https://doi.10.24425/acs.2024.149665
[4] R. Aounallah, A. Benaissa, A. Zarai, Blow-up and asymptotic behavior for a wave equation with a time delay condition of fractional type, Rend. Circ. Mat. Palermo II., 70 (2021), 1061-1081.
[5] R. Aounallah, A stability result of a Timoshenko beam system with a delay term in the internal fractional feedback, J. Pseudo-Differ. Oper. Appl., (2024), 15-45. https://doi.org/10.1007/s11868-024-00615-0
[6] R. Aounallah, A. Choucha, S. Boulaaras and A. Zarai, Asymptotic behavior of a viscoelastic wave equation with a delay in internal fractional feedback, Archives of Control Sciences, 34 (2024), 379-413.
[7] J. Ball, Remarks on Blow-Up and Nonexistence Theorems for Nonlinear Evolutions Equation, Quarterly Journal of Mathematics, 28 (1977), 473-486.
[8] A. Benaissa, H. Benkhedda, Global existence and energy decay of solutions to a wave equation with a dynamic boundary dissipation of fractional derivative type, Z. Anal. Anwend., 37 (2018), 315-339.
[9] S. Boulaaras, A. Choucha, D. Ouchenane, B. Cherif, Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms, Adv. Differ. Equ., 2020 (2020), 310.
[10] A. Choucha, S. Boulaaras, On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions, Bull. Iran. Math. Soc., 50 55 (2024). https://doi.org/10.1007/s41980-024-00897-6
[11] A. Choucha, S. Boulaaras, B. Djafari-Rouhani, R. Guefaifia, A. Alharbi, Global existence and general decay for a nonlinear wave equation with acoustic and fractional boundary conditions coupling by source and delay terms, Results in Applied Mathematics, 23 (2024), 100476.
[12] A. Choucha, S. Boulaaras, R. Jan, R. Alharbi, Blow-up and decay of solutions for a viscoelastic Kirchhoff-type equation with distributed delay and variable exponents, Math Meth Appl Sci, 47 8 (2024). https://doi.org/10.1002/mma.9950
[13] B.D. Coleman and W. Noll, Foundations of linear viscoelasticity. Reviews of Modern Physics, Reviews of Modern Physics, 33 2 (1961), 239.
[14] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), Nos. 9-10, .297-308.
[15] H. Dai, H. Zhang, Exponential growth for wave equation with fractional boundary dissipation and boundary source term, Boundary Value Prob., 1 (2014), 1-8. https://doi.org/10.1186/s13661-014-0138-y
[16] A. Djebabla, A. Choucha, D. Ouchenane, K. Zennir, Explicit stability for a porous thermoelastic system with second sound and distributed delay term, International Journal of Applied and Computational Mathematics, 7 2, 50 (2021).
[17] A. Fidan, E. Pişkin, S.A. Adam Saad, L. Alkhalifa, & M. Abdalla, Existence and blow up of solution for a Petrovsky equations of Hartree type, Discrete and Continuous Dynamical Systems - Series S, (2025). https://doi.org/10.3934/dcdss.2025026
[18] A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite memory, Math. Model. Anal., 25 (2020), 351-373.
[19] A. Guesmia and N. Tatar, Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay, Hal-Inria, (2015).
[20] M. Kafini, and Salim A. Messaoudi, Local Existence and Blow Up of Solutions to a Logarithmic Nonlinear Wave Equation with Delay, Applicable Analysis, 99 (2020), 530-547. https://doi.org/10.1080/00036811.2018.1504029
[21] M. Kirane, N-E. Tatar, Exponential growth for fractionally damped wave equation, Z. Anal. Anwend, 22 (2003), 167-178. https://doi.org/10.1080/00036811.2018.1504029
[22] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Applied Mathematics, 57 (1977), 93-105.
[23] E. Lieb and M. Loss, Analysis, American Mathemetical Society Graduate Studies in Mathematics, (1977).
[24] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Redding, (2006).
[25] B. Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematical Control and Information, 23 (2006), 237-257. https://doi.org/10.1093/imamci/dni056
[26] G. P. Menzala and W. A. Strauss, On a wave equation with a cubic convolutions, Journal of Differential Equations, 43 (1982), 93-105.
[27] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Diff. Int. Equs., 21 (9-10) (2008), 935-958.
[28] S. I. Pekar, Untersuchungen Uuber die Elektronentheorie der Kristalle, De Gruyter, (1954).
[29] A. Pazy. Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag; 1983.
[30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, New York: Springer-Verlag, (1983).
[31] Z. Shen, F. Gao and M. Yang, On critical Choquard equation with potential well, Discrete and Continuous Dynamical Systems, 38 (2018), 3567-3593.
[32] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, (2011). https://doi.org/10.1007/978-3-642-14003-7
[33] H. Zhang and X. Su, Initial boundary value problem for a class of wave equations of Hartree type, Stud Appl Math, 149 (2022), 798-814. DOI:10.1111/sapm.12521
[34] K. Zennir, D. Ouchenane, A. Choucha, Stability for thermo-elastic Bresse system of second sound with past history and delay term, International Journal of Modelling, Identification and Control, 36 4 (2020), 315-328.