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

This paper deals with positive solutions of the fully parabolic system, {u_t=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ₁v_t=Δv−v+winΩ×(0,∞),τ₂w_t=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.