BLOW-UP SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLINEAR MEMORY

Mohammed Medkour (), Belgacem Rebiai (), Salem Alkhalaf and Asma Alharbi ()
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Mohammed Medkour: Department of Mathematics and Computer Sciences, University of Larbi Tebessi, Tebessa 12002, Algeria
Belgacem Rebiai: Department of Mathematics and Computer Sciences, University of Larbi Tebessi, Tebessa 12002, Algeria
Salem Alkhalaf: Department of Computer Engineering, College of Computer, Qassim University, Buraydah 51452, Saudi Arabia
Asma Alharbi: Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

FRACTALS (fractals), 2025, vol. 33, issue 04, 1-9

Abstract: In this paper, we consider the nonlinear cauchy problem ð œ—Ï + (−Δ)𠜃 2𠜗 = λȷ0/Ï 1−αe𠜗 + μȷ0/Ï 1−α|𠜗|p−1𠜗,x ∈ â„ N,Ï > 0,𠜗(x, 0) = 𠜗0(x), x ∈ â„ N, where J0/Ï Î±f(x) := − 1 Γ(α)∫0Ï f(s) (Ï âˆ’s)1−αds and 0 < 𠜃 ≤ 2, 0 < α < 1. We will study the equation for λ = μ = 1. We demonstrate the local existence and uniqueness of the solution to this problem by the Banach fixed-point principle. Then, it is shown that local solutions experience blow-up. Finally, we present the blow-up rate when 𠜃 = 2.

Keywords: Fractional Derivatives; Fractional Partial Differential Equations; Local Existence; Blow-up (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25400924

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