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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:α   ∂  t    μ  β   +   ∂  t    φ  β   −  Δ   μ  β   =  p  (   σ  β   −   μ  β   )    in    Ω  ×  (  0  ,  T  )  ,μ  β   =  β   ∂  t    φ  β   +  (  −  Δ  +  1  )   φ  β   +   ξ  β   +  π  (   φ  β   )  ,ξ  β   ∈  B  (   φ  β   )    in    Ω  ×  (  0  ,  T  )  ,∂  t    σ  β   −  Δ   σ  β   =  −  p  (   σ  β   −   μ  β   )    in    Ω  ×  (  0  ,  T  )  ,in a bounded or an unbounded domain   Ω  ⊂   R  N    with smooth‐bounded boundary. Here,   N  ∈  N  ,  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 ( φ   β  ) 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  H  1 (Ω)↪ L  2 (Ω) and the continuous embedding  L  2 (Ω)↪( H  1 (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  H  1 (Ω)↪ L  2 (Ω) is not compact in the case that Ω is an unbounded domain.

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