Global solvability and asymptotic behavior of solutions for a fully parabolic nutrient taxis system

Abstract

In this paper, we consider the fully parabolic nu’trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights.