This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function χ(v) and the growth term f(u) under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that 0 < χ(v) ≤ χ0/vk (k ≥ 1, χ0 > 0) and λ1 − µ1u ≤ f(u) ≤ λ2 − µ2u (λ1, λ2, µ1, µ2 > 0). It is shown that if χ0 is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota.
This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation ∂u ∂t − (λ + iα)∆u + (κ + iβ)|u| q−1 u − γu = 0 in R N × (0, ∞) with L p -initial data u0 in the subcritical case (1 6 q < 1 + 2p/N), where u is a complex-valued unknown function, α, β, γ, κ ∈ R, λ > 0, p > 1, i = √ −1 and N ∈ N. The proof is based on the L p -L q estimates of the linear semigroup {exp(t(λ + iα)∆)} and usual fixed-point argument.