Boundedness of classical solutions for a chemotaxis model with rotational flux terms

Abstract

In this paper, we study the following chemotaxis system with rotational flux terms:where 0 is some non-decreasing function.We prove that the classical solutions to the above system are uniformly in-timebounded if there exists a smooth function ( ) with ′ ( ) ≥ 0 such that for some > 2 the matrix-valued function:be a negative semi-definite matrix. Here, denotes the transpose of and I is an × identity matrix. We show that the preceding matrix-valued function is a negative semi-definite matrix provided thatThese results extend the recent results obtained by Li et al. (Math. Models Methods Appl. Sci.) (2015) and Zhang (Math. Nachr.) (2016).We also study the special case = I with > 0. The above matrix in this case is written as:[ 1 4 ( − 1) 2 + 1 ( − 1) ( ′ ( )) 2 − 1 ′′ ( ) ] I ∶= ( )I . 1864

Boundedness and stabilization in a two-species chemotaxis system with signal absorption

“…Recently, some blow-up properties of (1.1), including blow-up criteria for the local classical solution, lower global blow-up estimate on u L ∞ (Ω) and local non-degeneracy property for the blow-up points, have been obtained in [6]. Furthermore, global solutions and the stabilization for the corresponding variants of (1.3), such as chemotaxis-consumption systems with tensor-valued sensitivities [9,11,22,25] and singular sensitivities [24], system (1.3) with logistic source [1,10] or coupled chemotaxis-fluid system [12,20,21,23] have also been investigated. For more results on the model variations of (1.3), we refer to the recent survey [2] and the references therein.…”