Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on RN: I. Global Solvability and Boundedness

This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption: {equation*} {cases} u_t = m∇· ((+u)ᵐ⁻¹∇ u) - χ∇ · (u ∇ v) + u(a - b u), & in (0,∞)^N, \\ v_t = Δv - uv, & in (0,∞)^N, {cases} {equation*} where $m > 1$ and ≥ 0. The present paper focuses on the global solvability and boundedness of weak solutions. For general bounded initial data, which may be non-integrable, we prove the existence of global weak solutions that remain uniformly bounded for all times. The proof relies on deriving local Lᵖ estimates that are uniform in time via a new continuity-type argument and obtaining L^∞ bounds using Moser's iteration; all of these estimates are uniform as →0. In part II, we will study the regularity and uniqueness of weak solutions.