Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production. (June 2021) - Wang, Wanwan; Li, Yuxiang
Abstract: The purpose of this paper is to study radially symmetric solutions of a parabolic–elliptic chemotaxis system (0.1) u t = Δ u − ∇ ⋅ ( f ( u ) ∇ v ) in B R × ( 0, + ∞ ), 0 = Δ v − μ ( t ) + g ( u ) in B R × ( 0, + ∞ ) subject to homogeneous Neumann boundary conditions, where B R = B R ( 0 ) ⊂ R n with n ⩾ 1, R > 0 and μ ( t ) = 1 | B R | ∫ B R g ( u ( x, t ) ) d x denotes the time-dependent spatial mean of g ( u ) . The chemosensitivity f and nonlinear signal production g are suitably regular functions. We show that the blow-up phenomenon of radially symmetric solution ( u, v ) of (0.1) in finite time for some initial datum u 0, when f ( ξ ) ⩾ K ξ k and g ( ξ ) ⩾ L ξ l for all ξ ⩾ 1 with K, k, L, l > 0 and k + l − 1 > 2 n . However, system (0.1) admits a global bounded classical solution for arbitrary initial datum when f ( ξ ) ⩽ K ξ k and g ( ξ ) ⩽ L ξ l for all ξ ⩾ 1 with k + l − 1 < 2 n .