Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions

The paper proves the Lq-boundedness (Theorem 1.1) by deriving an upper differential inequality for ∫Ω u^q that crucially requires an upper bound on the chemotactic cross term; this is achieved via Proposition 3.1 and related lemmas, culminating in inequality (3.3) and the absorbing ODE estimate for ∫Ω u^q . By contrast, the candidate solution replaces the chemotaxis contribution I2 with a lower bound obtained from the identity −Δ ln v = νu/v + |∇ ln v|^2 − μ. Because I2 appears with a plus sign in the energy balance, an upper bound is needed to control Yp' from above; a lower bound is insufficient and reverses the required inequality direction, so the key step leading to Yp'(t) ≤ c0 Yp − c1 Yp^{1+1/p} is invalid. The paper’s assumptions (H) and the structure of (1.1) are correctly used in the paper’s proof, and the result matches Theorem 1.1 as stated .