A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivity. (16th April 2015)

• Record Type: Journal Article
• Authors: Lankeit, Johannes
• Is Part Of: Mathematical methods in the applied sciences. Volume 39:Number 3(2016:Feb. 15)
• Page Start: 394
• Page End: 404
• Publication Date: 2015-04-16
• Subjects: chemotaxis -- singular sensitivity -- global existence -- boundedness; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3489
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We consider the parabolic chemotaxis model u t = Δu − χ ∇ · u v ∇ v v t = Δv − v + u in a smooth, bounded, convex two‐dimensional domain and show global existence and boundedness of solutions for χ ∈(0, χ 0 ) for some χ 0 >1, thereby proving that the value χ = 1 is not critical in this regard. Our main tool is consideration of the energy functional F a, b ( u, v ) = ∫ Ω u ln u − a ∫ Ω u ln v + b ∫ Ω ∇ v 2 for a > 0, b ≥0, where using nonzero values of b appears to be new in this context. Copyright © 2015 John Wiley & Sons, Ltd.