Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping

Abstract

This paper is concerned with a class of signal-dependent motility Keller-Segel systems{ut=∇⋅(D(v)∇u−S(v)u∇v)+f(u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under no-flux boundary conditions in a smooth bounded domain Ω⊂Rn. This model describes the aggregation phase of Dictyostelium discoideum cells in response to the secreted chemical signal cyclic adenosine monophosphate. The parameter functions D and S satisfyD,S∈C3((0,+∞)), and there exist constants k,K,x0>0 such thatk≤D(s)≤K,|S(s)|≤Kfor alls∈(x0,+∞). The generalized logistic source f∈C1([0,+∞)) satisfies f(0)=0 and there exists l>max⁡{2(n+2)n+4,2(n−1)n} such thatlims→+∞⁡f(s)sl=−∞. The main result of the paper is to show that the existence of the global-in-time solution in an appropriately generalized sense to the model for all sufficiently smooth initial data which indicates that superlinear damping can rule out blow-up for such kind of models.