A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients

Abstract

This manuscript is concerned with the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous issue of key importance in virtually any proof of boundedness features in Keller–Segel type systems. A strategy is proposed which at its core relies on L∞$L^\infty$ bounds for such quantities, conditional in the sense of involving certain Lebesgue norms of solution components that explicitly influence the signal evolution. Applications of this procedure firstly provide apparently novel boundedness results for two particular classes chemotaxis systems, and apart from that are shown to significantly condense proofs for basically well‐known statements on boundedness in two further Keller–Segel type problems.

Global solutions to a two-species chemotaxis system with gradient dependent sensitivity

“…Recently, Winkler [23] presented the results of the initial boundary value problem for (6). When the second equation in ( 6) is replaced by 0 = ∆v − µ + u with µ := 1 |Ω| Ω udx, Winkler [22] found a critical exponent α = n−2 2(n−1) .…”