Global solvability of a model for tuberculosis granuloma formation

authored by

Mario Fuest, Johannes Lankeit, Masaaki Mizukami

Abstract

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 u_t=D_uΔu−χ_u∇⋅(u∇v)−γ_uuv−δ_uu+β_u,v_t=D_vΔv+ρ_vv−γ_vuv+μ_vw,w_t=D_wΔw+γ_wuv−α_wwz−μ_ww,z_t=D_zΔz−χ_z∇⋅(z∇w)+α_zf(w)z−δ_zzin 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 utilizing a dissipative term of the (quasi-)energy functional, which may just be discarded in simpler consumption systems.

Organisation(s)

Faculty of Mathematics and Physics

External Organisation(s)

Kyoto University of Education

Type

Article

Journal

Nonlinear Analysis: Real World Applications

Volume

85

ISSN

1468-1218

Publication date

28.03.2025

Publication status

E-pub ahead of print

Peer reviewed

Yes

ASJC Scopus subject areas

Analysis, General Engineering, General Economics,Econometrics and Finance, Computational Mathematics, Applied Mathematics

Sustainable Development Goals

SDG 3 - Good Health and Well-being

Electronic version(s)

https://doi.org/10.1016/j.nonrwa.2025.104369 (Access: Closed)
https://doi.org/ arXiv:2411.00542 (Access: Open)