2025/05/16 by Yuri Soga, Soga, Yuri
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations #Thermoelastic and Magnetoelastic Phenomena
paper · pdf · doi:10.48550/arxiv.2505.10912
openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing: \begincases ut = Δ(e-vu) amp;in Ω× (0,∞), vt = Δv -v + u amp;in Ω× (0,∞), \endcases 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 Llog 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 ∫Ωev 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.