Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source

Abstract

We consider the fully parabolic Keller–Segel system with singular sensitivity and logistic-type source: ut=Δu−χ∇⋅(uv∇v)+ru−μuk, vt=Δv−v+u under the non-flux boundary conditions in a smooth bounded convex domain Ω⊂Rn, χ,r,μ>0, k>1. A global very weak solution for the system with n≥2 is obtained under one of the following conditions: (i) r>χ24 for 0<χ≤2, or r>max⁡{χ24(1−p02),χ−1} for χ>2 with p0=4(k−1)4+(2−k)kχ2 if k∈(2−1n,2]; (ii) χ2<min⁡{2(r+r2)k,4k(k−1)(k−2)} if k>2. Furthermore, this global very weak solution should be globally bounded in fact provided rμ and the initial data ‖u0‖L2(Ω),‖∇v‖L4(Ω) suitably small for n=2,3. In addition, if k>3(n+2)n+4 replaces k>2 in the condition (ii), the system admits globally bounded classical solutions. All these describe the influence of the exponent k>1 in the logistic-type source ru−μuk to the behavior of solutions for the considered fully parabolic Keller–Segel system with singular sensitivity.