Running summarizing tools on a new article

Abstract: This paper deals with the Neumann problem for the coupled chemotaxis–haptotaxis model of cancer invasion given by for x ∈ Ω and t > 0 with, respectively, given nonnegative initial data u 0 and w 0 , where χ , ξ and μ are positive parameters, Ω is a bounded domain in , and n ⩾ 1, with smooth boundary. The goal of this work is to identify two parallels between the solution behaviour in (⋆) and that in the corresponding two-component chemotaxis system obtained when w ≡ 0: • For the latter, it has been known that for any choice of , solutions are global and remain bounded under the condition The first result of this paper says that the above statement remains true for arbitrarily large haptotactic ingredients: If (⋆⋆) holds, then for any ξ > 0 and all reasonably smooth w 0 , (⋆) possesses a globally defined solution which is bounded in each of its components. • With regard to the qualitative solution behaviour, this work identifies an explicit smallness condition on w 0 which under the assumption (⋆⋆) asserts exponential decay of w in the large time limit, whereas both u and v persist in a certain sense.