Abstract
In this current work, we are interested in a system of two singular one-dimensional nonlinear equations with a viscoelastic, general source and distributed delay terms. The existence of a global solution is established by the theory of potential well, and by using the energy method with the function of Lyapunov, we prove the general decay result of our system.
Highlights
We are interested in the following system: 8 >>>>>>>>>>>>>>
0, where Q = ð0, LÞ × ð0, TÞ, L < ∞, T < ∞, g1ð:Þ, g2ð:Þ: R+ ⟶ R+, μ1, μ3 > 0, the second integral represents the distributed delay and μ2, μ4 : 1⁄2τ1, τ2 ⟶ R are bounded functions, where τ1, τ2 are two real numbers satisfying 0 ≤ τ1 < τ2, and f1ð:, :Þ, f2ð:,:Þ: R2 ⟶ R are defined functions later. These problems that arise in oneð1Þ dimensional elasticity have been studied and developed with regard to viscosity with long-term memory
We present in the second section the definitions, basics, and theories of function spaces that are required throughout the rest of the paper
Summary
We are interested in the following system:. 0, where Q = ð0, LÞ × ð0, TÞ, L < ∞, T < ∞, g1ð:Þ, g2ð:Þ: R+ ⟶ R+, μ1, μ3 > 0, the second integral represents the distributed delay and μ2, μ4 : 1⁄2τ1, τ2 ⟶ R are bounded functions, where τ1, τ2 are two real numbers satisfying 0 ≤ τ1 < τ2, and f1ð:, :Þ, f2ð:,:Þ: R2 ⟶ R are defined functions later. 0, where Q = ð0, LÞ × ð0, TÞ, L < ∞, T < ∞, g1ð:Þ, g2ð:Þ: R+ ⟶ R+, μ1, μ3 > 0, the second integral represents the distributed delay and μ2, μ4 : 1⁄2τ1, τ2 ⟶ R are bounded functions, where τ1, τ2 are two real numbers satisfying 0 ≤ τ1 < τ2, and f1ð:, :Þ, f2ð:,:Þ: R2 ⟶ R are defined functions later Three decades ago, these problems that arise in oneð1Þ dimensional elasticity have been studied and developed with regard to viscosity with long-term memory. The distributed ð2Þ delay in many works has been studied and many authors have taken care of it, for example, [5, 9, 27, 28] Based on all this and the results of the research papers [14, 15, 17, 28–30, 31], the introduction of the term distributed delay as Journal of Function Spaces a damping mechanism in problem (1) makes it a new problem from what has been previously studied. In the final section, the general decay is obtained by applying the energy method and the function of Lyapunov
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