Asymptotic analysis for Cahn–Hilliard type phase‐field systems related to tumor growth in general domains

Abstract

This article considers a limit system by passing to the limit in the following Cahn–Hilliard type phase‐field system related to tumor growth as β↘0: urn:x-wiley:mma:media:mma5520:mma5520-math-0001 in a bounded or an unbounded domain with smooth‐bounded boundary. Here, , T > 0, α > 0, β > 0, p ≥ 0, B is a maximal monotone graph, and π is a Lipschitz continuous function. In the case that Ω is a bounded domain, p and −Δ + 1 are replaced with p(φβ) and −Δ, respectively, and p is a Lipschitz continuous function; Colli, Gilardi, Rocca, and Sprekels (Discrete Contin Dyn Syst Ser S 2017; 10:37–54) have proved existence of solutions to the limit problem with this approach by applying the Aubin–Lions lemma for the compact embedding H1(Ω)↪L2(Ω) and the continuous embedding L2(Ω)↪(H1(Ω))∗. However, the Aubin–Lions lemma cannot be applied directly when Ω is an unbounded domain. The present work establishes existence of weak solutions to the limit problem along with uniqueness and error estimates in terms of the parameter β↘0. To this end, we construct an applicable theory by noting that the embedding H1(Ω)↪L2(Ω) is not compact in the case that Ω is an unbounded domain.