Geodesic
Well-Posedness and General Decay of Solutions for the Heat Equation with a Time Varying Delay Term
Kragujevac Journal of Mathematics, Tome 46 (2022) no. 2, p. 267

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We consider the nonlinear heat equation in a bounded domain with a time varying delay term $u_{t}+\Delta ^{2}u - J(t)ıt^{t}_{0} g(t-s)\Delta ^{2}u (s)ds+ lpha K(t)u+ \beta K(t)ueft( t-au(t) \right) = 0, $ with initial conditions. By introducing suitable energy and Lyapunov functionals, under some assumptions, we then prove a general decay result of the energy associated of this system under some conditions.
Classification : 35L05, 35L15 35L70, 35B05
Keywords: Heat equation, time varying delay, energy decay, Lyapunov functional, global existence, viscoelastic term 
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     author = {Abdelkader Braik and Abderrahmane Beniani and Yamina Miloudi},
     title = {Well-Posedness and {General} {Decay} of {Solutions} for the {Heat} {Equation} with a {Time} {Varying} {Delay} {Term}},
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Abdelkader Braik; Abderrahmane Beniani; Yamina Miloudi. Well-Posedness and General Decay of Solutions for the Heat Equation with a Time Varying Delay Term. Kragujevac Journal of Mathematics, Tome 46 (2022) no. 2, p. 267 . http://geodesic.mathdoc.fr/item/KJM_2022_46_2_a6/