arXiv subjects

Yuri Soga

Publications and source records attributed to Yuri Soga.

3 recordsLinked to original sources

arXiv2026

A sharp criterion and complete classification of global-in-time solutions and finite time blow-up of solutions to a chemotaxis system in supercritical dimensions

We consider the chemotaxis system with indirect signal production in the whole space, \begin{equation}\label{abst:p}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + w,\\ w_t = \Delta w + u \end{cases} \end{equation} with emphasis on supercritical dimensions. In contrast to the classical parabolic-elliptic Keller--Segel system, where the analysis can be reduced to a single equation, the above system is essentially parabolic-parabolic and does not admit such a reduction. In this paper, we establish a sharp threshold phenomenon separating global-in-time existence from finite time blow-up in terms of scaling-critical Morrey norms of the initial data. In particular, we prove the existence of singular stationary solutions and show that their Morrey norm values serve as the critical thresholds determining the long-time behavior of solutions. Consequently, we identify new critical exponents at which the long-time behavior of solutions changes. This yields a complete classification of the long-time behavior of solutions, providing the first such results for the essentially parabolic-parabolic chemotaxis system \eqref{abst:p} in supercritical dimensions.

math.AP↗

arXiv2025

A sufficient condition for absence of mass quantization in a chemotaxis system with local sensing

We analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing: \begin{equation*} \begin{cases} u_t = Δ(e^{-v}u)\qquad &\mathrm{in}\ Ω\times (0,\infty), v_t = Δv -v + u\qquad &\mathrm{in}\ Ω\times (0,\infty), \end{cases} \end{equation*} where $Ω$ is a ball in two-dimensional space and with nonnegative radially symmetric initial data. In the case of the Keller--Segel system which has a similar mathematical structure with our system, it was shown that the solutions blow up in finite time if and only if $L\log L$ for the first component $u$ diverges in finite time. On the other hand, focusing on the variational structure induced by a signal-dependent motility function $e^{-v}$, we show that an unboundedness of $\int_Ωe^v dx$ for the second component $v$ gives rise to blowup solutions in infinite time under the assumption of radial symmetry. Moreover we prove mass concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system develop a singularity like a Dirac delta function in infinite time. Here we investigate the weight of this singularity. Consequently it is shown that mass quantization may not occur; that is, the weight of the singularity can exceed $8π$ under the assumption of a uniform-in-time lower bound for the Lyapunov functional. This type of behavior cannot be observed in the Keller--Segel system.

math.AP↗

arXiv2025

Concentration phenomena to a chemotaxis system with indirect signal production

We consider a parabolic-ODE-parabolic chemotaxis system with radially symmetric initial data in a two-dimensional disk under the $0$-Neumann boundary condition. Although our system shares similar mathematical structures as the Keller--Segel system, the remarkable characteristic of the system we consider is that its solutions cannot blow up in finite time. In this paper, focusing on blow-up solutions in infinite time, we confirm concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system have a singularity like a Dirac delta function in infinite time. This means that there exist a time sequence $\{t_k\}$, a weight $m \ge 8π$, and a nonnegative function $f \in L^1(Ω)$ such that \begin{align*} u(\cdot,t_k) \stackrel{*}{\rightharpoonup} m δ(0) + f\ \mathrm{as}\ t_k \to \infty. \end{align*} We highlight this result is obtained by showing uniform-in-time boundedness of some energy functional. Moreover, we study whether $m = 8π$ or $m > 8π$, which is an open problem in the Keller--Segel system. It is proved that the weight $m$ of a delta function singularity is larger than $8π$ under a specific assumption associated with a Lyapunov functional. This finding suggests the relationship between solutions blowing up in infinite time and an unboundedness of a Lyapunov functional, which contrasts with the Keller--Segel system.

math.AP↗