Time: 10:00-11:00, Tuesday, December 16
Venue: E14-212
Speaker: Minh Le
Titile: Boundedness in Chemotaxis Systems with General Weak Singular Sensitivity and Logistic Sources
Abstract: In this talk, we consider the following system in an open, bounded domain \( \Omega \subset \mathbb{R}^n \) with \( n \geq 2 \): \begin{equation*} \begin{cases} u_t &= \Delta u - \nabla \cdot (u \chi(v) \nabla v) + ru - \dfrac{\mu u^2}{\ln^\gamma(u+e)}, \\ \kappa v_t &= \Delta v - \alpha v + \beta u, \end{cases} \end{equation*} where \( \chi(\cdot) \in C^1((0, \infty)) \) is positive, \( r, \mu, \alpha, \beta > 0 \), \( \kappa \in \{0, 1\} \), and \( \gamma \geq 0 \). We introduce the concept of a weak singular sensitivity condition and show that if \( \chi \) satisfies this condition, then solutions are global and bounded. Furthermore, when \( \kappa = 0 \) and \( \chi(v) = \dfrac{\chi_0}{v} \) for some \( \chi_0 > 0 \), {solutions remain globally bounded provided $ {\chi_0} < \min \left\{ \dfrac{1}{2}, \dfrac{2}{n} \right\} $}.
