Application of an Adams type inequality to a two-chemical substances chemotaxis system

Abstract

This paper deals with positive solutions of the fully parabolic system,{ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0).In the lower dimensional case (n≤3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded.In the case n=4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if ‖u0‖L1(Ω)<(8π)2/χ; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if ‖u0‖L1(Ω)<(8π)2/χ without radial symmetry.The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8π)2/χ is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8π problem in the Keller–Segel system to higher dimensions.