Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions

This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=\Delta (\gamma (v) u ) &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] v_t=\Delta v - v + u &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] \displaystyle \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 &\mathrm{on}\ \partial\Omega \times (0,\infty), \\[1mm] u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 &\mathrm{in}\ \Omega, \end{cases} \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^n$($n \geq 3$), $\gamma (v)=v^{-k}$ ($k>0$) and the initial data $(u_0,v_0)$ is positive and regular. This system has striking features similar to those of the logarithmic Keller--Segel system. It is established that classical solutions of the system exist globally in time and remain uniformly bounded in time if $k \in (0,n/(n-2))$, independently the magnitude of mass. This constant $n/(n-2)$ is conjectured as the optimal range guaranteeing global existence and boundedness in the corresponding logarithmic Keller--Segel system. We will derive sufficient estimates for solutions through some single evolution equation that some auxiliary function satisfies. The cornerstone of the analysis is the refined comparison estimate for solutions, which enables us to control the nonlinearity of the auxiliary equation

Weak Solutions to a Parabolic-Elliptic System of Chemotaxis

AbstractWe study a parabolic-elliptic system of partial differential equations, which describes the chemotactic feature of slime molds. It is known that the blowup solution forms singularities such as delta functions, referred to as the collapses. Here, we study the case that the domain is a flat torus and show that the post-blowup continuation of the solution is possible only when those collapses are quantized with the mass 8π

Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type

A Cauchy problem for a parabolic-elliptic system of drift-di usion type is considered. The problem is formally of the form Ut = r (rU Ur( ) 1U): This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I in the sense that its blowup rate is the same as the corresponding ordinary di erential equation yt = y2 (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three. 2000 Mathematics Subject Classi cation. 35K55, 35K57, 92C17.

Blowup of solutions to an indirect chemotaxis system (Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations)

"Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations". May 29-31, 2019. edited by Takayoshi Ogawa, Keiichi Kato, Mishio Kawashita and Masashi Misawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This manuscript summarizes results by Fujie and Senba (2017, 2019). In this manuscript, we describe properties of solutions to an indirect chemotaxis system. The system is one of chemotaxis systems, and has three unknown functions. These three functions correspond to density of living thing and concentrations of two kinds of chemical substances, respectively. When the dimension of the domain is less than four, our system does not have blowup solutions. In four dimensional case, our system has blowup solutions. In this manuscript, I will describe details of these results and sketch of these proofs