On a parabolic-parabolic system with gradient dependent chemotactic coefficient and consumption

Abstract

This paper investigates a parabolic-parabolic system with a gradient dependent chemotactic coefficient and consumption of chemoattractant under homogeneous boundary conditions of Neumann type, in a bounded domain Ω⊂Rn (n≥2) with a smooth boundary, 1 < p < 2. It is proved that if initial data satisfy u0∈C0(Ω¯), v0∈W1,q(Ω)∩(W2,n+2n(Ω),Ln+2n(Ω))nn+2,n+2n for some q > n and 0<‖v0‖L∞(Ω)<14K, then the model admits at least one global weak solution for n<8−2(p−1)p−1 and possesses at least one global renormalized solution for n≥8−2(p−1)p−1. Here, K≔supξ≥0ξ(1+ξ)2⁡ln(1+ξ) is positive and finite.