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This paper is concerned with a chemotaxis system ut=Δum−∇⋅(χ1(w)u∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=Δvn−∇⋅(χ2(w)v∇w)+μ2v(1−a2u−v),x∈Ω,t>0,wt=Δw−(αu+βv)w,x∈Ω,t>0,\left\{\begin{array}{ll}{u}_{t}=\Delta {u}^{m}-\nabla \cdot \left({\chi }_{1}\left(w)u\nabla w)+{\mu }_{1}u\left(1-u-{a}_{1}v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta {v}^{n}-\nabla \cdot \left({\chi }_{2}\left(w)v\nabla w)+{\mu }_{2}v\left(1-{a}_{2}u-v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {w}_{t}=\Delta w-\left(\alpha u+\beta v)w,& x\in \Omega ,\hspace{0.33em}t\gt 0,\end{array}\right. under homogeneous Neumann boundary conditions in a bounded domain Ω⊂R3\Omega \subset {{\mathbb{R}}}^{3} with smooth boundary, where μ1,μ2>0{\mu }_{1},{\mu }_{2}\gt 0, a1,a2>0{a}_{1},{a}_{2}\gt 0, α,β>0\alpha ,\beta \gt 0, and the chemotactic sensitivity function χi∈C1([0,∞)){\chi }_{i}\in {C}^{1}({[}0,\infty )), χi′≥0{\chi }_{i}^{^{\prime} }\ge 0. It is proved that for any large initial data, for any m,n>1m,n\gt 1, the system admits a global weak solution, which is uniformly bounded.

In this paper, we define the entropy number in probabilistic setting and determine the exact order of entropy number of finite-dimensional space in probabilistic setting. Moreover, we also estimate the sharp order of entropy number of univariate Sobolev space in probabilistic setting by discretization method.

Patellofemoral pain syndrome (PFPS) is a common injury among runners, and it is thought that abnormal lower extremity biomechanics contribute to its development. However, the relationship between biomechanical changes after a marathon and PFPS injury remains limited. This study aims to investigate whether differences in knee and hip kinematics and lower extremity muscle activities exist in recreational runners before and after a marathon. Additionally, it aims to explore the relationship between these biomechanical changes and the development of PFPS injury. 12 recreational runners participated in the study. Kinematics and muscle activities of the lower extremity were recorded during walking (5 km/h) and running (10 km/h) tasks within 24 hours before and within 5 hours after a marathon. After the marathon, there was a significant decrease in peak knee flexion (walking: p = 0.006; running: p = 0.006) and an increase in peak hip internal rotation (walking: p = 0.026; running: p = 0.015) during the stance phase of both walking and running compared to before the marathon. The study demonstrates a decrease in knee flexion and an increase in hip internal rotation during the stance phase of gait tasks after completing a marathon, which may increase the risk of developing PFPS injury. [ABSTRACT FROM AUTHOR]