The role of the mixed-quadratic damping in the construction of solutions to chemotaxis system with singular sensitivity

A chemotaxis system with singular sensitivity for criminal activities is considered over a smoothly bounded domain Ω⊂Rn with n ≥ 3: ut=Δu−χ∇⋅uv∇v−κuv+h1, vt = Δv − v + u + h2, where hi, i = 1, 2, are source functions and χ,κ∈R+. For κ = 0, the previous results showed that there exists a generalized solution with integrable first component provided that χ is small suitably. In this work it is shown that, whenever κ > 0, for any χ > 0 the corresponding initial-boundary value problem possesses a globally generalized solution, with the property that its first component is integrable. Our result, in particular, reveals that the mixed-quadratic damping −κuv admits a regularization effect on solutions, which is similar to that of the super-linear damping shown by Winkler, Ann. Sc. Norm. Super. Pisa Cl. Sci. 24(1), 141–172 (2023).