Finite speed of propagation in 1-D degenerate Keller-Segel system

We consider the following Keller-Segel system of degenerate type: (KS): ∂u/∂t = ∂/∂x (∂u ^m/∂x-u ^q-1∂v/∂x), x ∈ R{double-struck}, t > 0,0 = ∂ ²v/∂x ²-γν + u,x ∈ R{double-struck}, t > 0, u(x,0)=u ₀ (x), x ∈ R{double-struck}, where m > 1, γ > 0, q ≥ 2m. We shall first construct a weak solution u(x, t) of (KS) such that u ^{m - 1} is Lipschitz continuous and such that u ^m-1+δ for δ > 0 is of class C ¹ with respect to the space variable x. As a by-product, we prove the property of finite speed of propagation of a weak solution u(x, t) of (KS), i.e., that a weak solution u(x, t) of (KS) has a compact support in x for all t > 0 if the initial data u ₀(x) has a compact support in R{double-struck}. We also give both upper and lower bounds of the interface of the weak solution u of (KS).