Masaaki Mizukami

We consider the fully parabolic, spatially heterogeneous chemotaxis-growth system $ \begin{align*} \begin{cases} u_t = \Delta u - \nabla\cdot(u\nabla v) + \kappa(x)u-\mu(x)u^2, \\ v_t = \Delta v - v + u \end{cases} \end{align*} $ in bounded domains $ \Omega\subset \mathbb{R}^2 $ and show that the blow-up set is contained in the set of zeroes of $ \mu $.

We discuss a nonlinear system of partial differential equations modelling the formation of granuloma during tuberculosis infections and prove the global solvability of the homogeneous Neumann problem for \begin{align*} \begin{cases} u_t = D_u Δu - χ_u \nabla \cdot (u \nabla v) - γ_u uv - δ_u u + β_u, \\ v_t = D_v Δv + ρ_v v - γ_v uv + μ_v w,\\ w_t = D_w Δw + γ_w uv - α_w wz - μ_w w,\\ z_t = D_z Δz - χ_z \nabla \cdot (z \nabla w) + α_z f(w)z - δ_z z \end{cases} \end{align*} in bounded domains in the classical and weak sense in the two- and three-dimensional setting, respectively. In order to derive suitable a~priori estimates, we study the evolution of the well-known energy functional for the chemotaxis-consumption system both for the $(u, v)$- and the $(z, w)$-subsystem. A key challenge compared to "pure" consumption systems consists of overcoming the difficulties raised by the additional, in part positive, terms in the second and third equations. This is inter alia achieved by utilising a dissipative term of the (quasi-)energy functional, which may just be discarded in simpler consumption systems.

This paper deals with the fully parabolic chemotaxis-convection model with sensitivity functions for tumor angiogenesis, \begin{align*} \begin{cases} u_t=Δu-\nabla \cdot (uχ_1(v)\nabla v) +\nabla \cdot (uχ_2(w)\nabla w), &x \in Ω,\ t>0, \\[1.05mm] v_t=Δv+\nabla \cdot (vξ(w)\nabla w)+αu-βv, &x \in Ω,\ t>0, \\[1.05mm] w_t=Δw+γu-δw, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \le 3)$ is a bounded domain with smooth boundary, $χ_1, χ_2, ξ$ are functions satisfying some conditions and $α, β, γ, δ>0$ are constants. The purpose of this paper is to establish global existence and boundedness in this system.

This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=\Delta u-\nabla \cdot (u\chi(v)\nabla v) +\nabla \cdot (u\xi(w)\nabla w), &x \in \Omega,\ t>0,\\[1.05mm] v_t=\Delta v-v+u, &x \in \Omega,\ t>0,\\[1.05mm] w_t=\Delta w-w+u, &x \in \Omega,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with smooth boundary, $\chi, \xi$ are functions satisfying some conditions. Global existence and boundedness of classical solutions to the system with logistic source have already been obtained by taking advantage of the effect of logistic dampening (J. Math. Anal. Appl.; 2020;489;124153). This paper establishes existence of global bounded classical solutions despite the loss of logistic dampening.

Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= \Delta n - \nabla \cdot (n \nabla c), \\ 0 &= \Delta c - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions \[ \partial_{\nu} c = (\gamma-c) g \] for $c$ (with $\gamma>0$ and $C^{1+\beta}(\partial\Omega) \ni g > 0$ for some $\beta\in(0,1)$) are established in bounded domains $\Omega\subset\mathbb{R}^{N}$, $N\ge 1$. Under a smallness condition on $\gamma$, moreover, we show convergence to the stationary solution.