Boundedness and asymptotic stability in a model for tuberculosis granuloma formation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910041277202432 |
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| author | Mizukami, Masaaki Tanaka, Yuya |
| author_facet | Mizukami, Masaaki Tanaka, Yuya |
| contents | This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - uv - u + β, &x \in Ω,\ t>0, \\ v_t = Δv + v -uv + μw, &x \in Ω,\ t>0, \\ w_t = Δw + uv - wz - w, &x \in Ω,\ t>0, \\ z_t = Δz - \nabla \cdot (z \nabla w) + f(w)z -z, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ ($n\ge 2$) is a smooth bounded domain, $β,μ>0$ and $f$ is some function, and shows that if initial data are small in some sense then the solution $(u,v,w,z)$ of the problem exists globally and convergences to $(β,0,0,0)$ exponentially when $β>1$ and the reproduction number $R_0 := \frac{μβ+ 1}β$ satisfies $R_0<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundedness and asymptotic stability in a model for tuberculosis granuloma formation Mizukami, Masaaki Tanaka, Yuya Analysis of PDEs This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - uv - u + β, &x \in Ω,\ t>0, \\ v_t = Δv + v -uv + μw, &x \in Ω,\ t>0, \\ w_t = Δw + uv - wz - w, &x \in Ω,\ t>0, \\ z_t = Δz - \nabla \cdot (z \nabla w) + f(w)z -z, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ ($n\ge 2$) is a smooth bounded domain, $β,μ>0$ and $f$ is some function, and shows that if initial data are small in some sense then the solution $(u,v,w,z)$ of the problem exists globally and convergences to $(β,0,0,0)$ exponentially when $β>1$ and the reproduction number $R_0 := \frac{μβ+ 1}β$ satisfies $R_0<1$. |
| title | Boundedness and asymptotic stability in a model for tuberculosis granuloma formation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.16752 |