A finite element approach to the Keller–Segel Chemotaxis system with impact of nonlocal diffusion

Abstract This paper presents the investigation of a boundedness solution for a Keller–Segel chemotaxis (KSC) system with nonlocal diffusion. In order to achieve boundedness, we derive the necessary and sufficient condition based on the assumption of ensuring a unique classical global bounded solution for the proposed KSC system. In this work, the proposed system’s boundedness condition is determined based on three particular situations: when $\chi &lt;\chi ^{*}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>χ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> , $\chi &lt;\kappa $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>κ</mml:mi> </mml:math> , and when $\chi =\kappa $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>κ</mml:mi> </mml:math> . Finally, the effectiveness of the nonlinear and nonlocal diffusion terms in the KSC system is demonstrated through numerical simulations under consideration of the values of system parameters. From the simulation results, we can confirm the effectiveness of the proposed method.