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

Abstract

AbstractIn this paper, we study the following chemotaxis system with rotational flux terms: urn:x-wiley:00442267:media:zamm201700091:zamm201700091-math-0001under no‐flux boundary conditions in a bounded domain , with smooth boundary. Here, is a matrix‐valued function and , where S0 is some non‐decreasing function. Also, is a non‐negative function with urn:x-wiley:00442267:media:zamm201700091:zamm201700091-math-0006where f0 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 with such that for some the matrix‐valued function: urn:x-wiley:00442267:media:zamm201700091:zamm201700091-math-0010be a negative semi‐definite matrix. Here, denotes the transpose of S and is an identity matrix. We show that the preceding matrix‐valued function is a negative semi‐definite matrix provided that urn:x-wiley:00442267:media:zamm201700091:zamm201700091-math-0014These 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 with . The above matrix in this case is written as: urn:x-wiley:00442267:media:zamm201700091:zamm201700091-math-0017For this case, we present a smooth function with such that the matrix‐valued function is a negative semi‐definite matrix provided that . This result extends the result obtained for this problem which asserts the boundedness of classical solutions under the condition . More precisely, by comparing the two conditions, we can write .