A chemotaxis system with singular sensitivity for burglaries in the higher-dimensional settings: generalized solvability and long-time behavior.

We study the no-flux initial-boundary value problem of a chemotaxis system with singular sensitivity of the following form ⋆ u t = Δ u - χ ∇ · u ∇ ln v - u + g 1 , v t = Δ v - v + u v (1 - v) + g 2 , over a bounded domain Ω ⊂ R n , with chemotaxis coefficient χ > 0 and nonnegative source functions g 1 and g 2 , which was proposed by Pitcher to describe the dynamics of burglaries. From the recent results it is known that, if n = 2 , then such problem admits a global generalized solution for any initial data and any χ > 0 , and possesses a global classical solution for small initial data and small χ . This paper presents that for all reasonably regular initial data and any χ > 0 the corresponding homogeneous Neumann initial-boundary value problem possesses a generalized solution in higher-dimensional settings (i.e., n ≥ 3 ). The asymptotic behavior of generalized solutions is explored as well under some additional assumptions on the source functions. [ABSTRACT FROM AUTHOR]