Boundedness of classical solutions for a chemotaxis system with general sensitivity function. Issue 3 (17th February 2019) - Khelghati, Ali; Baghaei, Khadijeh
ABSTRACT: This paper deals with the chemotaxis system with general sensitivity function:u t = ∇ · ( δ ∇ u - u χ ( v ) ∇ v ), x ∈ Ω, t > 0, 0 = Δ v - v + u, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domainΩ ⊂ R n, n ≥ 2, with smooth boundary. Here, δ > 0 and the initial functionu ( x, 0 ) = u 0 and the sensitivity functionχ satisfy: u 0 ∈ C 0 ( Ω ¯ ) with ∫ Ω u 0 d x > 0, χ ( s ) > 0 for s > 0 and χ ∈ C 1 ( [ 0, ∞ ) ) . We prove that the classical solutions to the above system are uniformly in-time-bounded provided that there exists a smooth positive functionφ such that for somep > n 2 and0 < λ < 1, the following differential inequality holdsφ ( s ) + ( p - 1 ) χ ( s ) ≤ - 4 λ δ ( p - 1 ) p φ ′ ( s ), s > 0, whereφ ′ ( s ) < 0 ands φ ( s ) is bounded from above. We also present our results for the special case0 < χ ( s ) ≤ χ 0 s k withχ 0 > 0 andk ≥ 1 . These results coincide with the results obtained by Fujie et al. [Math Methods Appl Sci. 2015] in the case ofk = 1 and extend their results in the case ofk > 1 .