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: 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. |