Global boundedness to a chemotaxis system with singular sensitivity and logistic source

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We consider the parabolic-parabolic Keller–Segel system with singular sensitivity and logistic source: \( u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v) +ru-\mu u^2\), \(v_t=\Delta v-v+u\) under the homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^2\), \(\chi ,\mu >0\) and \(r\in \mathbb {R}\). It is proved that the system exists globally bounded classical solutions if \(r>\frac{\chi ^2}{4}\) for \(0 \chi -1\) for \(\chi >2\).