Boundedness in a Keller–Segel system with external signal production

We study the Neumann initial-boundary problem for the chemotaxis system{ut=Δu−∇⋅(u∇v),x∈Ω,t>0,vt=Δv−v+u+f(x,t),x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω in a smooth, bounded domain Ω⊂RnΩ⊂Rn with n≥2n≥2 and f∈L∞([0,∞);Ln2+δ0(Ω))∩Cα(Ω×(0,∞)) with some α>0α>0 and δ0∈(0,1)δ0∈(0,1). First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of n=2n=2 and f   being constant in time, requiring the nonnegative initial data u0u0 to fulfill the property ∫Ωu0dx<4π ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller–Segel model (coinciding with f≡0f≡0 in the system above) to the case f≢0f≢0. Under certain smallness conditions imposed on the initial data and f   we furthermore show that for more general space dimension n≥2n≥2 and f not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller–Segel system to the present situation.