Boundedness criteria for a class of indirect (and direct) chemotaxis-consumption models in high dimensions

Frassu S.;Viglialoro G.

2022-01-01

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Abstract

In a bounded and smooth domain Ω of Rn, n ≥ 5, we, mainly, consider for some ξ,χ,δ positive and Tmax ∈ (0,∞] the zero-flux chemotaxis model with indirect signal absorption ut = ξ∆u−χ∇·(u∇v), vt =∆v−wv, wt =−δw+u, in Ω×(0,Tmax), equipped with sufficiently regular initial data u(x,0) = u0(x) ≥ 0, v(x,0) = v0(x) ≥ 0 and w(x,0) = w0(x) ≥ 0. We establish the existence of ξ∗ = ξ∗(n) > 1 such that whenever χv0 L∞(Ω) obeys certain constraints, functions of n and ξ (0 < ξ < ξ∗), the initial-boundary value problem has a unique classical solution in Ω × (0,∞), which is bounded. In the frame of both direct and indirect chemotaxis models, our work (partially) improves and generalizes known results.