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

In this paper, we study the following chemotaxis system with rotational flux terms:u t = ∇ · ( ∇ u − u S ( x , u , v ) ∇ v ) ,x ∈ Ω , t > 0 ,v t = Δ v − f ( x , u , v ) ,x ∈ Ω , t > 0 ,under no‐flux boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary. Here, S ∈ C 2 ( Ω ¯ × [ 0 , ∞ ) 2 ; R n × n )is a matrix‐valued function and| S ( x , u , v ) | ≤ S 0 ( v ) , where S 0 is some non‐decreasing function. Also, f ∈ C 1 ( Ω ¯ × [ 0 , ∞ ) 2 ; R )is a non‐negative function withf ( x , u , 0 ) = 0 and f ( x , u , v ) ≤ f 0 ( v ) ( 1 + u ) , where f 0 is some non‐decreasing function. We prove that the classical solutions to the above system are uniformly in‐time‐bounded if there exists a smooth function z ( s ) withz ′ ( s ) ≥ 0 such that for some p > n 2the matrix‐valued function:p − 1 4S t ( x , r , s ) S ( x , r , s ) + 1 p ( p − 1 )( z ′ ( s ) ) 2 − 1 p z ′ ′( s )I nbe a negative semi‐definite matrix. Here, S t denotes the transpose of S and I n is an n × n identity matrix. We show that the preceding matrix‐valued function is a negative semi‐definite matrix provided that∥ v 0∥L ∞ ( Ω )S 0 ( ∥ v 0∥ L ∞ ( Ω )) < π 2 n . These 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 S = χ I nwith χ > 0 . The above matrix in this case is written as:1 4 ( p − 1 ) χ 2 + 1 p ( p − 1 )( z ′ ( s ) ) 2 − 1 p z ′ ′( s )I n : = Z ( s ) I n . For this case, we present a smooth function z ( s ) withz ′ ( s ) ≥ 0 such that the matrix‐valued function Z ( s ) I nis a negative semi‐definite matrix provided that0 < ∥ v 0∥L ∞ ( Ω )< π χ2 n. This result extends the result obtained for this problem which asserts the boundedness of classical solutions under the condition0 < ∥ v 0∥L ∞ ( Ω )< 1 6 χ ( n + 1 ). More precisely, by comparing the two conditions, we can writelim n → ∞π χ2 n1 6 χ ( n + 1 )= + ∞ .