arXiv subjects

Zulaihat Hassan

Publications and source records attributed to Zulaihat Hassan.

4 recordsLinked to original sources

arXiv2026

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

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 $1 2-\frac{2}{N}$, 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.

math.AP↗

arXiv2025

The spreading of global solutions of chemotaxis systems with logistic source and consumption on $\mathbb{R}^{N}$

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=Δu - χ\nabla\cdot(u\nabla v)+ u(a-bu),\quad &(t,x)\in [0,\infty)\times\mathbb{R}^N, \\ &{τ\partial_tv}=Δv-uv,\quad & (t,x)\in [0,\infty)\times\mathbb{R}^N, \end{aligned} \right. \] where $u(t,x)$ represents the population density of a biological species, and $v(t,x)$ denotes the density of a chemical substance. Key findings of this study include: (i) the species spreads at least at the speed $c^*=2\sqrt a$ (equalling the speed when $v\equiv 0$), suggesting that the chemical substance does not hinder the spreading; (ii) the chemical substance does not induce infinitely fast spreading of $u$; (iii) the spreading speed remains unaffected under conditions that $v(0,\cdot)$ decays spatially or $0<-χ\ll 1$ and $τ=1$. Additionally, our numerical simulations reveal a noteworthy phase transition in $χ$: for $v(0, \cdot)$ uniformly distributed across space, the spreading speed accelerates only when $χ$ surpasses a critical positive value.

math.AP↗

arXiv2025

Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on $\mathbb{R}^N$: I. Global Solvability and Boundedness

This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption: \begin{equation*} \begin{cases} u_t = m\nabla\cdot \left((\eps+u)^{m-1}\nabla u\right) - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{ in } (0,\infty)\times\mathbb{R}^N, \\ v_t = Δv - uv, & \text{ in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where $m > 1$ and $\eps \geq 0$. The present paper focuses on the global solvability and boundedness of weak solutions. For general bounded initial data, which may be non-integrable, we prove the existence of global weak solutions that remain uniformly bounded for all times. The proof relies on deriving local $L^p$ estimates that are uniform in time via a new continuity-type argument and obtaining $L^\infty$ bounds using Moser's iteration; all of these estimates are uniform as $\eps\to0$. In part II, we will study the regularity and uniqueness of weak solutions.

math.AP↗

arXiv2023

Global existence of classical solutions of chemotaxis systems with logistic source and consumption or linear signal production on $\mathbb{R}^{n}$

While much literature on chemotaxis systems focuses on bounded domains, this paper emphasizes the global existence of classical solutions for three primary chemotaxis systems with a logistic source on $\mathbb{R}^n$. We present a unified proof demonstrating global existence of solutions can be deduced from their locally uniform boundedness in $L^p(\mathbb{R}^n)$ for some $p>\max\{1,\frac{n}{2}\}$. We then provide sufficient conditions for the global existence and boundedness of classical solutions. Notably, our findings even improve several existing results for bounded domains.

math.AP↗