Classical solutions to Cauchy problems for parabolic-elliptic systems of Keller-Segel type

Abstract

The Cauchy problem in ℝn, n ≥ 2, for [equaction presented] is considered for general matrices S ∈ ℝn×n. A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC (ℝn)Lp(ℝn) with some p ∈ [ 1, n), there exist Tmax ∈ (0, ∞ ] and a uniquely determined u ∈ C0([ 0, Tmax); BUC (ℝn))⊂ C0 ([ 0, Tmax) such that with v:= Γ∗u, and with Γ denoting the Newtonian kernel on ℝn, the pair (u, v) forms a classical solution of (∗) in ℝn × (0, Tmax), which has the property that if [equaction presented]. An exemplary application of this provides a result on global classical solvability in cases when |S + 1| is sufficiently small, where 1 = diag (1, ..., 1).