Large-Data Global Generalized Solutions in a Chemotaxis System with Tensor-Valued Sensitivities

The chemotaxis system uₜ=Δ u - ∇ · (uS(x,u,v)·∇ v);\ vₜ=Δ v - uf(v) (referred to as () in this abstract), for the density $u=u(x,t)$ of a cell population and the concentration $v=v(x,t)$ of an attractive chemical consumed by the former, is considered under no-flux boundary conditions in a bounded domain Ω⊂Rⁿ, n≥ 1, with smooth boundary, where f ∈ C¹([0,∞);[0,∞)) and S ∈ C²(Ω× [0,∞)²;Rn× n) are given functions such that f(0)=0. In contrast to related Keller--Segel-type problems with scalar sensitivities, in the presence of such matrix-valued S the system () in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal L² estimate for ∇ ln (u+1) as a starting point, we derive a series of compactness properties of solutions to suitably regularized versions of (). Motivated by these, we develop a generalized solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component u; in particular, the chemotactic flux uS(x,u,v)·∇ v need not be integrable in this context. On the basis of the above compactness properties, it is finally shown that within this framework, under a mild growth assumption on S and for all sufficiently regular nonnegative initial data, the corresponding initial-boundary value problem for () possesses at least one global generalized solution. This extends known results which in the case of such general matrix-valued S provide statements on global existence only in the two-dimensional setting and under the additional restriction that \|v₀\|L^∞(Ω) be small.