Global existence of weak solutions to chemotaxis models with porous medium diffusion, linear production, and logistic source on $\mathbb{R}^N$

Abstract

This paper investigates the global solvability, boundedness, and uniqueness of weak solutions to the chemotaxis system \begin{equation*} \begin{cases} u_t = Δu^m - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{in } (0,\infty)\times\mathbb{R}^N, \\ τv_t = Δv - λv + μu, & \text{in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where \(m>1\), \(τ\in\{0,1\}\), \(λ,μ,a,b>0\), and \(χ\in\mathbb{R}\). For every \(m>1\), we establish the existence of weak solutions for initial data that are not necessarily integrable, although such solutions need not be bounded in general. We then show that globally bounded weak solutions exist in the parabolic-parabolic case \((τ=1)\) when \(m>\frac{2N}{N+2}\), and also when \(1<m\le \frac{2N}{N+2}\) provided that the logistic damping coefficient \(b\) is sufficiently large. In the parabolic-elliptic case \((τ=0)\), we prove the existence of globally bounded weak solutions when \(m>2-\frac{2}{N}\), and also when \(1<m\le 2-\frac{2}{N}\) provided that \(b\) is sufficiently large. Finally, for \(1<m\le 3\), we prove uniqueness of weak solutions that are Hölder continuous up to the initial time.