Boundedness and asymptotic stability of classical solutions to a model for tuberculosis granuloma formation

Publication date: Jun 20, 2025

This paper deals with a problem which describes tuberculosis granuloma formation begin{align*} begin{cases} u_t = Delta u – nabla cdot (u nabla v) – uv – u + beta, &x in Omega, t>0, v_t = Delta v + v -uv + mu w, &x in Omega, t>0, w_t = Delta w + uv – wz – w, &x in Omega, t>0, z_t = Delta z – nabla cdot (z nabla w) + f(w)z -z, &x in Omega, t>0 end{cases} end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Omega subset mathbb{R}^n$ ($nge 2$) is a smooth bounded domain, $beta,mu>0$ 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 $(beta,0,0,0)$ exponentially when $beta>1$ and the reproduction number $R_0 := frac{mu beta + 1}{beta}$ satisfies $R_0

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