Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 40, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.40 EXISTENCE FOR A NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEM WITH INERTIAL TERM SHUNSUKE KURIMA Abstract. This article presents a nonlocal Penrose-Fife type phase field sys- tem with inertial term. We do not know whether we can prove the existence of solutions to the problem as in Colli-Grasselli-Ito [3] or not. In this article we introduce a time discretization scheme, then pass to the limit as the time step h approaches 0, and obtain an error estimate for the difference between the continuous solution and the discrete solution. 1. Introduction Colli-Grasselli-Ito [3] derived the existence of solutions to the parabolic hyper- bolic Penrose-Fife phase field system ( − 1 u ) t + (λ(ϕ))t −∆u = f in Ω× (0, T ), ϕtt + ϕt −∆ϕ+ β(ϕ) + π(ϕ) = λ′(ϕ)u in Ω× (0, T ), ∂νu+ u = g on ∂Ω× (0, T ), (− 1 u )(0) = − 1 u0 , ϕ(0) = ϕ0, ϕt(0) = v0 in Ω, (1.1) where Ω ⊂ Rd (d = 1, 2, 3) is a bounded domain with smooth boundary ∂Ω, T > 0, λ : R→ R is a smooth function which may have quadratic growth, β : R→ R is a maximal monotone function, π : R → R is an anti-monotone function, ∂ν denotes differentiation with respect to the outward normal of ∂Ω, u0 : Ω→ R, ϕ0 : Ω→ R and v0 : Ω→ R are given functions. Moreover, in the case that λ(ϕ) = ϕ, they have proved the uniqueness of solutions to (1.1). Assuming that |β(r)| ≤ c1|r|3 + c2 for all r ∈ R, where c1, c2 > 0 are some constants, we can obtain an estimate for β(ϕ) by establishing the L∞(0, T ;H1(Ω))-estimate for ϕ and by using the continuity of the embedding H1(Ω) ↪→ L6(Ω). 2020 Mathematics Subject Classification. 35G30, 80A22, 35A40. Key words and phrases. Nonlocal Penrose-Fife type phase field systems; inertial terms; existence; approximation and time discretization. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 29, 2022. Published June 23, 2023. 1 2 S. KURIMA EJDE-2023/40 The existence of solutions to the singular nonlocal phase field system with inertial term (lnu)t + ϕt −∆u = f in Ω× (0, T ), ϕtt + ϕt + a(·)ϕ− J ∗ ϕ+ β(ϕ) + π(ϕ) = u in Ω× (0, T ), ∂νu = 0 on ∂Ω× (0, T ), (lnu)(0) = lnu0, ϕ(0) = ϕ0, ϕt(0) = v0 in Ω (1.2) has been studied in [7], where J : Rd → R is an interaction kernel, a(x) := ∫ Ω J(x− y) dy and (J∗ϕ)(x) := ∫ Ω J(x−y)ϕ(y) dy for x ∈ Ω. To derive the L∞(0, T ;H2(Ω))- estimate for ∫ t 0 u(s) ds is a key to establish an estimate for β(ϕ). Indeed, it holds that 1 2 |ϕ(x, t)|2 = 1 2 |ϕ0(x)|2 + ∫ t 0 ϕt(x, s)ϕ(x, s) ds and 1 2 |ϕt(x, t)|2 + ∫ t 0 |ϕt(x, s)|2 ds+ β̂(ϕ(x, t)) = ∫ t 0 u(x, s)ϕt(x, s) ds+ 1 2 |v0(x)|2 + β̂(ϕ0(x)) − ∫ t 0 (a(x)ϕ(x, s)− (J ∗ ϕ(s))(x))ϕt(x, s) ds, where β̂(r) = ∫ r 0 β(s) ds. Moreover, since u > 0 in Ω× (0, T ), we see that∫ t 0 u(x, s)ϕt(x, s) ds ≤ ‖ϕt‖L∞(Ω×(0,T )) ∫ t 0 u(x, s) ds. Thus, deriving the L∞(0, T ;H2(Ω))-estimate for ∫ t 0 u(x, s) ds from the first equation in (1.2), using the continuity of the embedding H2(Ω) ↪→ L∞(Ω), applying the Young inequality and the Gronwall lemma, we can establish the L∞(Ω × (0, T ))- estimates for ϕt and ϕ, whence we can obtain the L∞(Ω× (0, T ))-estimate for β(ϕ) by assuming that β is continuous. It seems that this is the first study of nonlocal Penrose-Fife type phase field systems with inertial term. So we verify the existence of solutions to the problem( − 1 u ) t + ϕt −∆u = f in Ω× (0, T ), ϕtt + ϕt + a(·)ϕ− J ∗ ϕ+ β(ϕ) + π(ϕ) = u in Ω× (0, T ), ∂νu+ u = g on ∂Ω× (0, T ), (− 1 u )(0) = − 1 u0 , ϕ(0) = ϕ0, ϕt(0) = v0 in Ω, (1.3) where Ω ⊂ Rd (d = 1, 2, 3) is a bounded domain with smooth boundary ∂Ω. More- over, we assume the following conditions: (A1) J(−x) = J(x) for all x ∈ Rd and supx∈Ω ∫ Ω |J(x− y)| dy < +∞. (A2) β : R → R is a single-valued maximal monotone function such that there exists a proper lower semicontinuous convex function β̂ : R → [0,+∞) satisfying that β̂(0) = 0 and β = ∂β̂, where ∂β̂ is the subdifferential of β̂. Moreover, β : R→ R is local Lipschitz continuous. (A3) π : R→ R is a Lipschitz continuous function. EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 3 (A4) f ∈ L2(Ω × (0, T )), g ∈ L2(0, T ;H1/2(∂Ω)), g ≤ 0 a.e. on ∂Ω × (0, T ), θ0 := − 1 u0 ∈ L2(Ω), θ0 > 0 a.e. in Ω, ln θ0 ∈ L1(Ω), ϕ0, v0 ∈ L∞(Ω). Definition 1.1. A pair (u, ϕ) with u ∈ L2(0, T ;H1(Ω)), − 1 u ∈ H1(0, T ; (H1(Ω)) ∗ ) ∩ L∞(0, T ;L2(Ω)), ϕ ∈W 2,2(0, T ;L2(Ω)) ∩W 1,∞(0, T ;L∞(Ω)) is called a weak solution of (1.3) if (u, ϕ) satisfies〈 (− 1 u )t, w 〉 (H1(Ω))∗,H1(Ω) + (ϕt, w)L2(Ω) + ∫ Ω ∇u · ∇w + ∫ ∂Ω (u− g)w = (f, w)L2(Ω) a.e. in (0, T ) for all w ∈ H1(Ω), ϕtt + ϕt + a(·)ϕ− J ∗ ϕ+ β(ϕ) + π(ϕ) = u a.e. in Ω× (0, T ), (− 1 u )(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0 a.e. in Ω. Theorem 1.2. Assume that (A1)–(A4) hold. Then there exists a unique weak solution (u, ϕ) of (1.3). This article is organized as follows. In Section 2 we introduce a time discretiza- tion of (1.3) and set precisely the approximate problem. In Section 3 we prove the existence for the discrete problem. In Section 4 we establish some uniform estimates for the approximate problem. Section 5 obtains Cauchy’s criterion for solutions of the approximate problem and is devoted to the proofs of the existence and unique- ness of weak solutions to (1.3) and an error estimate between the solution of (1.3) and the solution of the approximate problem. 2. Time discretization To prove the existence of weak solutions to (1.3) we deal with the discrete prob- lem θn+1 − θn h + ϕn+1 − ϕn h −∆un+1 = fn+1 in Ω, zn+1 + vn+1 + a(·)ϕn − J ∗ ϕn + β(ϕn+1) + π(ϕn+1) = un+1 in Ω, zn+1 = vn+1 − vn h , vn+1 = ϕn+1 − ϕn h in Ω, ∂νun+1 + un+1 = gn+1 on ∂Ω (2.1) for n = 0, . . . , N − 1, where h = T N , N ∈ N, θj := − 1 uj for j = 0, 1, . . . , N , and fk := 1 h ∫ kh (k−1)h f(s) ds, gk := 1 h ∫ kh (k−1)h g(s) ds for k = 1, . . . , N . Indeed, we can show the existence for (2.1). Theorem 2.1. Assume that (A1)-( A4) hold. Then there exists h0 ∈ (0, 1] such that for all h ∈ (0, h0) there exists a unique solution of (2.1) satisfying un+1 ∈ H2(Ω), ϕn+1 ∈ L∞(Ω) for n = 0, . . . , N − 1. 4 S. KURIMA EJDE-2023/40 Putting θ̂h(t) := θn + θn+1 − θn h (t− nh), (2.2) ϕ̂h(t) := ϕn + ϕn+1 − ϕn h (t− nh), (2.3) v̂h(t) := vn + vn+1 − vn h (t− nh) (2.4) for t ∈ [nh, (n+ 1)h], n = 0, . . . , N − 1, and uh(t) := un+1, θh(t) := θn+1, ϕh(t) := ϕn+1, ϕ h (t) := ϕn, (2.5) vh(t) := vn+1, zh(t) := zn+1, fh(t) := fn+1 (2.6) for t ∈ (nh, (n+ 1)h], n = 0, . . . , N − 1, we can rewrite (2.1) as (θ̂h)t + (ϕ̂h)t −∆uh = fh in Ω× (0, T ), zh + vh + a(·)ϕ h − J ∗ ϕ h + β(ϕh) + π(ϕh) = uh in Ω× (0, T ), zh = (v̂h)t, vh = (ϕ̂h)t in Ω× (0, T ), θh = − 1 uh in Ω× (0, T ), ∂νuh + uh = gh on ∂Ω× (0, T ), θ̂h(0) = θ0, ϕ̂h(0) = ϕ0, v̂h(0) = v0 in Ω. (2.7) Here we can check directly the following identities by (2.2)-(2.6): ‖θ̂h‖L∞(0,T ;L2(Ω)) = max{‖θ0‖L2(Ω), ‖θh‖L∞(0,T ;L2(Ω))}, (2.8) ‖ϕ̂h‖L∞(0,T ;L∞(Ω)) = max{‖ϕ0‖L∞(Ω), ‖ϕh‖L∞(0,T ;L∞(Ω))}, (2.9) ‖v̂h‖L∞(0,T ;L∞(Ω)) = max{‖v0‖L∞(Ω), ‖vh‖L∞(0,T ;L∞(Ω))}, (2.10) ‖θh − θ̂h‖2L2(0,T ;(H1(Ω))∗) = h2 3 ‖(θ̂h)t‖2L2(0,T ;(H1(Ω))∗), (2.11) ‖ϕh − ϕ̂h‖L∞(0,T ;L∞(Ω)) = h‖(ϕ̂h)t‖L∞(0,T ;L∞(Ω)) = h‖vh‖L∞(0,T ;L∞(Ω)), (2.12) ‖vh − v̂h‖2L2(0,T ;L2(Ω)) = h2 3 ‖(v̂h)t‖2L2(0,T ;L2(Ω)) = h2 3 ‖zh‖2L2(0,T ;L2(Ω)), (2.13) ϕ h = ϕh − h(ϕ̂h)t. (2.14) We can prove Theorem 1.2 by passing to the limit in (2.7) as h↘ 0. Moreover, we can obtain the following theorem which asserts an error estimate between the solution of (1.3) and the solution of (2.7). Theorem 2.2. Let h0 be as in Theorem 1.2. Assume that (A1)-(A4) hold. Assume further that f ∈ W 1,1(0, T ;L2(Ω)) and g ∈ W 1,1(0, T ;L2(∂Ω)). Then there exist constants h00 ∈ (0, h0) and M > 0 depending on the data such that ‖1 ? (uh − u)‖C([0,T ];H1(Ω)) + ‖ϕ̂h − ϕ‖C([0,T ];L2(Ω)) + ‖v̂h − ϕt‖C([0,T ];L2(Ω)) ≤Mh1/2 for all h ∈ (0, h00), where (1 ? w)(t) := ∫ t 0 w(s) ds for vector-valued functions w summable in (0, T ). EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 5 3. Existence for the discrete problem In this section we will show Theorem 2.1. Lemma 3.1. For all h > 0, G ∈ L2(Ω), G∂Ω ∈ H1/2(∂Ω), if G∂Ω ≤ 0 a.e. on ∂Ω, then there exists a unique function u ∈ H2(Ω) satisfying u < 0 a.e. in Ω, − 1 u − h∆u = G a.e. in Ω, ∂νu+ u = G∂Ω a.e. on ∂Ω. Proof. We set the operator A : D(A) ⊂ L2(Ω)→ L2(Ω) as Au := −∆u− cu for u ∈ D(A) := {u ∈ H2(Ω) : ∂νu+ u = G∂Ω a.e. on ∂Ω}. Then this operator is maximal monotone for some constant c > 0. Also, we define the operator B : D(B) ⊂ L2(Ω)→ L2(Ω) as Bu := −h −1 u for u ∈ D(B) := {u ∈ L2(Ω) : u < 0 a.e. in Ω}. Then this operator is maximal monotone. Now we set the function b : D(b) ⊂ R→ R as b(r) := −h −1 r for r ∈ D(b) := {r ∈ R : r < 0}. Let λ > 0, let Bλ be the Yosida approximation of B and let bλ be the Yosida approximation of b on R. Then, noting that bλ is monotone, u = λbλ(u) + (1 + λb)−1(u), bλ(u) = − h−1 (1+λb)−1(u) > 0, and G∂Ω ≤ 0 a.e. on ∂Ω, we can confirm that (Au,Bλu)L2(Ω) = ∫ Ω b′λ(u)|∇u|2 + ∫ ∂Ω ubλ(u)− ∫ ∂Ω G∂Ωbλ(u)− c ∫ Ω ubλ(u) ≥ λ‖bλ(u)‖2L2(∂Ω) − h −1|∂Ω| − cλ‖bλ(u)‖2L2(Ω) + ch−1|Ω| ≥ −max{c, h−1|∂Ω|}(λ‖Bλ(u)‖2L2(Ω) + 1) for all u ∈ D(A) and all λ > 0. Therefore we can conclude that the operator A+B is maximal monotone (see e.g., Barbu [2, Theorem 2.7]). � Lemma 3.2. For all G ∈ L2(Ω) and all h ∈ (0,min{1, 1/‖π′‖L∞(R)}) there exists a unique solution ϕ ∈ L2(Ω) of the equation ϕ+ hϕ+ h2β(ϕ) + h2π(ϕ) = G a.e. in Ω. The above lemma can be proved as in [6, Lemma 2.1]. Proof of Theorem 2.1. We can rewrite (2.1) as − 1 un+1 − h∆un+1 = −ϕn+1 + hfn+1 + ϕn + θn, ∂νun+1 + un+1 = gn+1, ϕn+1 + hϕn+1 + h2β(ϕn+1) + h2π(ϕn+1) = h2un+1 + ϕn + hvn + hϕn − h2a(·)ϕn + h2J ∗ ϕn. (3.1) To prove Theorem 2.1 it suffices to establish the existence and uniqueness of so- lutions to (3.1) in the case that n = 0. Let h ∈ (0,min{1, 1/‖π′‖L∞(R)}). Then, owing to Lemma 3.1, for all ϕ ∈ L2(Ω) there exists a unique function u ∈ H2(Ω) such that − 1 u − h∆u = −ϕ+ hf1 + ϕ0 + θ0, ∂νu+ u = g1. (3.2) 6 S. KURIMA EJDE-2023/40 Also, we see from Lemma 3.2 that for all u ∈ L2(Ω) there exists a unique function ϕ ∈ L2(Ω) such that ϕ+ hϕ+ h2β(ϕ) + h2π(ϕ) = h2u+ ϕ0 + hv0 + hϕ0 − h2a(·)ϕ0 + h2J ∗ ϕ0. (3.3) Thus we can set Φ : L2(Ω) → L2(Ω), Ψ : L2(Ω) → L2(Ω) and B : L2(Ω) → L2(Ω) as Φϕ = u, Ψu = ϕ for ϕ, u ∈ L2(Ω), B = Ψ ◦ Φ. Moreover, we can obtain that for all ϕ, ϕ̃ ∈ L2(Ω), ‖Bϕ−Bϕ̃‖L2(Ω) ≤ C1h 1 + h− ‖π′‖L∞(R)h2 ‖ϕ− ϕ̃‖L2(Ω) (cf. [7, Proof of Theorem 1.2]). Then there exists h01 ∈ (0,min{1, 1/‖π′‖L∞(R)}) such that C1h 1 + h− ‖π′‖L∞(R)h2 ∈ (0, 1) for all h ∈ (0, h01). Hence B : L2(Ω)→ L2(Ω) is a contraction mapping in L2(Ω) for all h ∈ (0, h01) and then it follows from the Banach fixed-point theorem that for all h ∈ (0, h01) there exists a unique function ϕ1 ∈ L2(Ω) such that ϕ1 = Bϕ1 ∈ L2(Ω). Thus, for all h ∈ (0, h01), putting u1 := Φϕ1 ∈ H2(Ω) implies that there exists a unique pair (u1, ϕ1) ∈ (L2(Ω))2 satisfying (3.1) in the case that n = 0. Moreover, we can prove that there exists h0 ∈ (0, h01) such that for all h ∈ (0, h0) there exists a constant C1 = C1(h) > 0 such that |ϕ1(x)| ≤ C1 for a.a. x ∈ Ω (cf. [7, Proof of Theorem 1.2]). � 4. Uniform estimates for the discrete problem In this section we derive a priori estimates for (2.7). Lemma 4.1. Let h0 be as in Theorem 2.1. Then there exist constants h1 ∈ (0, h0) and C > 0 depending on the data such that ‖ϕh‖2L∞(0,T ;L2(Ω)) + ‖vh‖2L∞(0,T ;L2(Ω)) + ‖uh‖2L2(0,T ;H1(Ω)) + ‖θh‖L∞(0,T ;L1(Ω)) + ‖ ln θh‖L∞(0,T ;L1(Ω)) ≤ C for all h ∈ (0, h1). Proof. Multiplying the identity vn+1 = ϕn+1−ϕn h by hϕn+1 we obtain 1 2 ‖ϕn+1‖2L2(Ω) − 1 2 ‖ϕn‖2L2(Ω) + 1 2 ‖ϕn+1 − ϕn‖2L2(Ω) = h(ϕn+1, vn+1)L2(Ω). (4.1) We test the second equation in (2.1) by hvn+1 to infer that 1 2 ‖vn+1‖2L2(Ω) − 1 2 ‖vn‖2L2(Ω) + 1 2 ‖vn+1 − vn‖2L2(Ω) + h‖vn+1‖2L2(Ω) + (β(ϕn+1), ϕn+1 − ϕn)L2(Ω) = h(un+1, vn+1)L2(Ω) − h(π(ϕn+1), vn+1)L2(Ω) − h(a(·)ϕn − J ∗ ϕn, vn+1)L2(Ω). (4.2) Here the condition (A2) leads to the inequality (β(ϕn+1), ϕn+1 − ϕn)L2(Ω) ≥ ‖β̂(ϕn+1)‖L1(Ω) − ‖β̂(ϕn)‖L1(Ω). (4.3) EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 7 Thus we deduce from (4.1)-(4.3), the Young inequality, (A1), and (A3) that there exists a constant C1 > 0 such that 1 2 ‖ϕn+1‖2L2(Ω) − 1 2 ‖ϕn‖2L2(Ω) + 1 2 ‖ϕn+1 − ϕn‖2L2(Ω) + 1 2 ‖vn+1‖2L2(Ω) − 1 2 ‖vn‖2L2(Ω) + 1 2 ‖vn+1 − vn‖2L2(Ω) + h‖vn+1‖2L2(Ω) + ‖β̂(ϕn+1)‖L1(Ω) − ‖β̂(ϕn)‖L1(Ω) ≤ h(un+1, vn+1)L2(Ω) + C1h+ C1‖ϕn+1‖2L2(Ω) + C1‖ϕn‖2L2(Ω) + C1‖vn+1‖2L2(Ω) (4.4) for all h ∈ (0, h0). Next we multiply the first equation in (2.1) by h(1 + un+1) to obtain that (θn+1 − θn, 1 + un+1)L2(Ω) + h ∫ Ω |∇un+1|2 + h ∫ ∂Ω |un+1|2 = h(fn+1, 1 + un+1)L2(Ω) − h(un+1, vn+1)L2(Ω) − h(vn+1, 1)L2(Ω) − h ∫ ∂Ω un+1 + h ∫ ∂Ω gn+1(1 + un+1). (4.5) Here, noting that un+1 = − 1 θn+1 and r − 1 ≥ ln r for all r > 0, we have that (θn+1 − θn, 1 + un+1)L2(Ω) = ‖θn+1‖L1(Ω) − ‖θn‖L1(Ω) + (θn+1 − θn, un+1)L2(Ω) = ‖θn+1‖L1(Ω) − ‖θn‖L1(Ω) + ∫ Ω ( θn θn+1 − 1) ≥ ‖θn+1‖L1(Ω) − ‖θn‖L1(Ω) + ∫ Ω ln θn θn+1 = ‖θn+1‖L1(Ω) − ‖θn‖L1(Ω) + ∫ Ω (− ln θn+1 + ln θn). (4.6) There exist constants C∗, C ∗ > 0 such that C∗(‖∇w‖2L2(Ω) + ‖w‖2L2(∂Ω)) ≤ ‖w‖ 2 H1(Ω) ≤ C ∗(‖∇w‖2L2(Ω) + ‖w‖2L2(∂Ω)) (4.7) for all w ∈ H1(Ω). Therefore we see from (4.5)-(4.7) and the Young inequality that there exists a constant C2 > 0 such that ‖θn+1‖L1(Ω) − ‖θn‖L1(Ω) + ∫ Ω (− ln θn+1 + ln θn) + 1 2C∗ h‖un+1‖2H1(Ω) ≤ −h(un+1, vn+1)L2(Ω) + C2h+ C2h‖fn+1‖2L2(Ω) + C2h‖gn+1‖2L2(∂Ω) + C2h‖vn+1‖2L2(Ω) (4.8) 8 S. KURIMA EJDE-2023/40 for all h ∈ (0, h0). Therefore we add (4.4) to (4.8) and sum over n = 0, . . . ,m − 1 with 1 ≤ m ≤ N to derive that 1 2 ‖ϕm‖2L2(Ω) + 1 2 ‖vm‖2L2(Ω) + ‖β̂(ϕm)‖L1(Ω) + ‖θm‖L1(Ω) − ∫ Ω ln θm + 1 2C∗ h m−1∑ n=0 ‖un+1‖2H1(Ω) ≤ 1 2 ‖ϕ0‖2L2(Ω) + 1 2 ‖v0‖2L2(Ω) + ‖β̂(ϕ0)‖L1(Ω) + ‖θ0‖L1(Ω) − ∫ Ω ln θ0 + (C1 + C2)T + C2h m−1∑ n=0 ‖fn+1‖2L2(Ω) + C2h m−1∑ n=0 ‖gn+1‖2L2(∂Ω) + 2C1h m−1∑ n=0 ‖ϕn+1‖2L2(Ω) + (C1 + C2)h m−1∑ n=0 ‖vn+1‖2L2(Ω). (4.9) On the other hand, ‖θm‖L1(Ω) − ∫ Ω ln θm = ∫ Ω (θm − ln θm) ≥ 1 3 ∫ Ω (θm + | ln θm|). (4.10) Thus it follows from (4.9) and (4.10) that ( 1 2 − 2C1h)‖ϕm‖2L2(Ω) + ( 1 2 − (C1 + C2)h)‖vm‖2L2(Ω) + ‖β̂(ϕm)‖L1(Ω) + 1 3 ‖θm‖L1(Ω) + 1 3 ‖ ln θm‖L1(Ω) + 1 2C∗ h m−1∑ n=0 ‖un+1‖2H1(Ω) ≤ 1 2 ‖ϕ0‖2L2(Ω) + 1 2 ‖v0‖2L2(Ω) + ‖β̂(ϕ0)‖L1(Ω) + ‖θ0‖L1(Ω) + ‖ ln θ0‖L1(Ω) + (C1 + C2)T + C2h m−1∑ n=0 ‖fn+1‖2L2(Ω) + C2h m−1∑ n=0 ‖gn+1‖2L2(∂Ω) + 2C1h m−1∑ j=0 ‖ϕj‖2L2(Ω) + (C1 + C2)h m−1∑ j=0 ‖vj‖2L2(Ω) and then there exist constants C3 > 0 and h1 ∈ (0, h0) such that ‖ϕm‖2L2(Ω) + ‖vm‖2L2(Ω) + ‖β̂(ϕm)‖L1(Ω) + ‖θm‖L1(Ω) + ‖ ln θm‖L1(Ω) + h m−1∑ n=0 ‖un+1‖2H1(Ω) ≤ C3 + C3h m−1∑ j=0 ‖ϕj‖2L2(Ω) + C3h m−1∑ j=0 ‖vj‖2L2(Ω) for all h ∈ (0, h1) and m = 1, . . . , N . Therefore, owing to the discrete Gronwall lemma (see e.g., [5, Prop. 2.2.1]), there exists a constant C4 > 0 such that ‖ϕm‖2L2(Ω) + ‖vm‖2L2(Ω) + ‖β̂(ϕm)‖L1(Ω) + ‖θm‖L1(Ω) + ‖ ln θm‖L1(Ω) + h m−1∑ n=0 ‖un+1‖2H1(Ω) ≤ C4 EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 9 for all h ∈ (0, h1) and m = 1, . . . , N . � Lemma 4.2. Let h1 be as in Lemma 4.1. Then there exist constants h2 ∈ (0, h1) and C > 0 depending on the data such that ‖θh‖2L∞(0,T ;L2(Ω)) + ‖ ln θh‖2L2(0,T ;H1(Ω)) ≤ C for all h ∈ (0, h2). Proof. Testing the first equation in (2.1) by hθn+1 leads to the identity 1 2 ‖θn+1‖2L2(Ω) − 1 2 ‖θn‖2L2(Ω) + 1 2 ‖θn+1 − θn‖2L2(Ω) + h(−∆un+1, θn+1)L2(Ω) = h(fn+1, θn+1)L2(Ω) − h(vn+1, θn+1)L2(Ω). (4.11) Here, since un+1 = − 1 θn+1 , θn+1 > 0, and gn+1 ≤ 0, we have that h(−∆un+1, θn+1)L2(Ω) = h ∫ Ω ∇un+1 · ∇θn+1 + h ∫ ∂Ω un+1θn+1 − h ∫ ∂Ω gn+1θn+1 ≥ h ∫ Ω |∇ ln θn+1|2 − h|∂Ω|. (4.12) Therefore we can verify that Lemma 4.2 holds by combining (4.11), (4.12), by summing over n = 0, . . . ,m−1 with 1 ≤ m ≤ N , by applying the discrete Gronwall lemma, Lemma 4.1, the Poincaré-Wirtinger inequality. � Lemma 4.3. Let h2 be as in Lemma 4.2. Then there exists a constant C > 0 depending on the data such that ‖(θ̂h)t‖L2(0,T ;(H1(Ω))∗) ≤ C for all h ∈ (0, h2). Proof. We can obtain this lemma by the first equation in (2.7) and Lemma 4.1. � Lemma 4.4. Let h2 be as in Lemma 4.2. Then there exists a constant C > 0 depending on the data such that h max 1≤m≤N ∥∥m−1∑ n=0 (−un+1) ∥∥ H2(Ω) ≤ C for all h ∈ (0, h2). Proof. We can prove this lemma by Lemmas 4.1, 4.2 and the elliptic regularity theory (cf. [7, Proof of Lemma 4.5]). � Lemma 4.5. Let h2 be as in Lemma 4.2. Then there exist constants h3 ∈ (0, h2) and C > 0 depending on the data such that ‖ϕh‖2L∞(Ω×(0,T )) + ‖vh‖2L∞(Ω×(0,T )) ≤ C for all h ∈ (0, h3). 10 S. KURIMA EJDE-2023/40 Proof. From [7, Proof of Lemma 4.6], we can confirm that there exists a constant C1 > 0 such that 1 2 |ϕm(x)|2 + 1 2 |vm(x)|2 ≤ h m−1∑ n=0 un+1(x)vn+1(x) + C1h m−1∑ n=0 ‖ϕn+1‖2L∞(Ω) + C1h m−1∑ n=0 ‖vn+1‖2L∞(Ω) + C1 (4.13) for all h ∈ (0, h2) and for a.a. x ∈ Ω, m = 1, . . . , N . Here, noting that −uj > 0 a.e. in Ω for j = 0, 1, . . . , N , we deduce from Lemma 4.4 and the continuity of the embedding H2(Ω) ↪→ L∞(Ω) that there exists a constant C2 > 0 such that h m−1∑ n=0 un+1(x)vn+1(x) = h m−1∑ n=0 (−un+1(x))(−vn+1(x)) ≤ ( max 1≤m≤N ‖ − vm‖L∞(Ω) ) h m−1∑ n=0 (−un+1(x)) ≤ ( max 1≤m≤N ‖vm‖L∞(Ω) ) h ∥∥m−1∑ n=0 (−un+1) ∥∥ L∞(Ω) ≤ C2 max 1≤m≤N ‖vm‖L∞(Ω) (4.14) for all h ∈ (0, h2) and for a.a. x ∈ Ω, m = 1, . . . , N . Thus we see from (4.13) and (4.14) that 1 2 |ϕm(x)|2 + 1 2 |vm(x)|2 ≤ C2 max 1≤m≤N ‖vm‖L∞(Ω) + C1h m−1∑ n=0 ‖ϕn+1‖2L∞(Ω) + C1h m−1∑ n=0 ‖vn+1‖2L∞(Ω) + C1 for a.a. x ∈ Ω and for all h ∈ (0, h2), m = 1, . . . , N , whence the inequality 1 2 ‖ϕm‖2L∞(Ω) + 1 2 ‖vm‖2L∞(Ω) ≤ C2 max 1≤m≤N ‖vm‖L∞(Ω) + C1h m−1∑ n=0 ‖ϕn+1‖2L∞(Ω) + C1h m−1∑ n=0 ‖vn+1‖2L∞(Ω) + C1 holds. Then there exist constants h3 ∈ (0, h2) and C3 > 0 such that ‖ϕm‖2L∞(Ω) + ‖vm‖2L∞(Ω) ≤ C3 max 1≤m≤N ‖vm‖L∞(Ω) + C3h m−1∑ j=0 ‖ϕj‖2L∞(Ω) + C3h m−1∑ j=0 ‖vj‖2L∞(Ω) + C3 for all h ∈ (0, h3) and m = 1, . . . , N . Hence by the discrete Gronwall lemma there exists a constant C4 > 0 such that ‖ϕm‖2L∞(Ω) + ‖vm‖2L∞(Ω) ≤ C4 + C4 max 1≤m≤N ‖vm‖L∞(Ω) for all h ∈ (0, h3) and m = 1, . . . , N . Therefore it holds that max 1≤m≤N ‖ϕm‖2L∞(Ω) + max 1≤m≤N ‖vm‖2L∞(Ω) ≤ C4 + C4 max 1≤m≤N ‖vm‖L∞(Ω) EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 11 ≤ C4 + 1 2 max 1≤m≤N ‖vm‖2L∞(Ω) + C2 4 2 , which leads to Lemma 4.5. � Lemma 4.6. Let h3 be as in Lemma 4.5. Then there exists a constant C > 0 depending on the data such that ‖ϕ h ‖L∞(Ω×(0,T )) ≤ C for all h ∈ (0, h3). Proof. This lemma can be obtained by (A4) and Lemma 4.5. � Lemma 4.7. Let h3 be as in Lemma 4.5. Then there exists a constant C > 0 depending on the data such that ‖β(ϕh)‖L∞(Ω×(0,T )) ≤ C for all h ∈ (0, h3). Proof. We can prove this lemma by the continuity of β and Lemma 4.5. � Lemma 4.8. Let h3 be as in Lemma 4.5. Then there exists a constant C > 0 depending on the data such that ‖zh‖L2(0,T ;L2(Ω)) ≤ C for all h ∈ (0, h3). Proof. We can verify that this lemma holds by the second equation in (2.7), Lemmas 4.1, 4.5-4.7, and the conditions (A1), (A3). � Lemma 4.9. Let h3 be as in Lemma 4.5. Then there exists a constant C > 0 depending on the data such that ‖θ̂h‖H1(0,T ;(H1(Ω))∗)∩L∞(0,T ;L2(Ω)) + ‖v̂h‖H1(0,T ;L2(Ω))∩L∞(Ω×(0,T )) + ‖ϕ̂h‖W 1,∞(0,T ;L∞(Ω)) ≤ C for all h ∈ (0, h3). Proof. Lemmas 4.2, 4.3, 4.5, 4.8, along with (2.8)-(2.10), lead to Lemma 4.9. � 5. Existence for (1.3) and error estimate In this section we will derive the existence and uniqueness of solutions to (1.3) by passing to the limit in (2.7) as h↘ 0 and will establish an error estimate between the solution of (1.3) and the solution of (2.7). Lemma 5.1. Let h3 be as in Lemma 4.5. Then there exists a constant M1 > 0 depending on the data such that ‖(1 ? (uh − uτ ))(t)‖2H1(Ω) ≤M1(h+ τ) +M1 ∫ t 0 ‖(1 ? (uh − uτ ))(s)‖2H1(Ω) ds +M1 ∫ t 0 ‖v̂h(s)− v̂τ (s)‖2L2(Ω) ds+M1‖fh − fτ‖2L2(0,T ;L2(Ω)) +M1‖gh − gτ‖2L2(0,T ;L2(∂Ω)) (5.1) for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. 12 S. KURIMA EJDE-2023/40 Proof. We have that (θ̂h − θ̂τ , w)L2(Ω) + (ϕ̂h − ϕ̂τ , w)L2(Ω) + ∫ Ω ∇(1 ? (uh − uτ )) · ∇w + ∫ ∂Ω (1 ? (uh − uτ ))w = ((1 ? (fh − fτ )), w)L2(Ω) + ∫ ∂Ω (1 ? (gh − gτ ))w (5.2) a.e. in (0, T ) for all w ∈ H1(Ω). Taking w = uh − uτ in (5.2) and integrating over (0, t), where t ∈ [0, T ], we have∫ t 0 (θ̂h(s)− θ̂τ (s), uh(s)− uτ (s))L2(Ω) ds + ∫ t 0 (ϕ̂h(s)− ϕ̂τ (s), uh(s)− uτ (s))L2(Ω) ds+ 1 2 ‖∇(1 ? (uh − uτ ))(s)‖2L2(Ω) + 1 2 ‖(1 ? (uh − uτ ))(s)‖2L2(∂Ω) = ∫ t 0 ((1 ? (fh − fτ ))(s), (1 ? (uh − uτ ))(s))L2(Ω) ds + ∫ t 0 (∫ ∂Ω (1 ? (gh − gτ ))(s)(1 ? (uh − uτ ))(s) ) ds. (5.3) Here we see from the identity θh = − 1 uh that∫ t 0 (θ̂h(s)− θ̂τ (s), uh(s)− uτ (s))L2(Ω) ds = ∫ t 0 〈 θ̂h(s)− θh(s), uh(s)− uτ (s) 〉 (H1(Ω))∗,H1(Ω) ds + ∫ t 0 〈 θτ (s)− θ̂τ (s), uh(s)− uτ (s) 〉 (H1(Ω))∗,H1(Ω) ds + ∫ t 0 (α(uh(s))− α(uτ (s)), uh(s)− uτ (s))L2(Ω) ds ≥ ∫ t 0 〈 θ̂h(s)− θh(s), uh(s)− uτ (s) 〉 (H1(Ω))∗,H1(Ω) ds + ∫ t 0 〈 θτ (s)− θ̂τ (s), uh(s)− uτ (s) 〉 (H1(Ω))∗,H1(Ω) ds, (5.4) where α(r) := −1/r for r ∈ D(α) := {r ∈ R | r < 0} and the monotonicity of α was used. Integrating by parts with respect to time yields that∫ t 0 (ϕ̂h(s)− ϕ̂τ (s), uh(s)− uτ (s))L2(Ω) ds = ∫ t 0 ((1 ? (vh − vτ ))(s), (1 ? (uh − uτ ))′(s))L2(Ω) ds = ((1 ? (vh − vτ ))(t), (1 ? (uh − uτ ))(t))L2(Ω) − ∫ t 0 (vh(s)− vτ (s), (1 ? (uh − uτ ))(s))L2(Ω) ds. (5.5) EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 13 Also, it holds that∫ t 0 ((1 ? (fh − fτ ))(s), uh(s)− uτ (s))L2(Ω) ds = ∫ t 0 ((1 ? (fh − fτ ))(s), (1 ? (uh − uτ ))′(s))L2(Ω) ds = ((1 ? (fh − fτ ))(t), (1 ? (uh − uτ ))(t))L2(Ω) − ∫ t 0 (fh(s)− fτ (s), (1 ? (uh − uτ ))(s))L2(Ω) ds (5.6) and ∫ t 0 (∫ ∂Ω (1 ? (gh − gτ ))(s)(uh(s)− uτ (s)) ) ds = ∫ t 0 (∫ ∂Ω (1 ? (gh − gτ ))(s)(1 ? (uh − uτ ))′(s) ) ds = ∫ ∂Ω (1 ? (gh − gτ ))(t)(1 ? (uh − uτ ))(t) − ∫ t 0 (∫ ∂Ω (gh − gτ )(s)(1 ? (uh − uτ ))(s) ) ds. (5.7) Therefore, since vh − vτ = vh − v̂h + v̂τ − vτ + v̂h − v̂τ , we can prove Lemma 5.1 by (5.3)-(5.7), the Schwarz inequality, the Young inequality, (2.11), (2.13), Lemmas 4.1, 4.3, 4.8. � Lemma 5.2. Let h3 be as in Lemma 4.5. Then there exists a constant M2 > 0 depending on the data such that ‖ϕ̂h(t)− ϕ̂τ (t)‖2L2(Ω) + ‖v̂h(t)− v̂τ (t)‖2L2(Ω) ≤M2(h+ τ) +M2 ∫ t 0 ‖ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds +M2 ∫ t 0 ‖v̂h(s)− v̂τ (s)‖2L2(Ω) ds+M2‖(1 ? (uh − uτ ))(t)‖2H1(Ω) (5.8) for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. Proof. We see from (2.14) and Lemma 4.1 that there exists a constant C1 > 0 such that ∫ t 0 ‖ϕ h (s)− ϕ τ (s)‖2L2(Ω) ds ≤ 3 ∫ t 0 ‖ϕh(s)− ϕτ (s)‖2L2(Ω) ds+ 3h2 ∫ t 0 ‖(ϕ̂h)s(s)‖2L2(Ω) ds + 3τ2 ∫ t 0 ‖(ϕ̂τ )s(s)‖2L2(Ω) ds ≤ 3 ∫ t 0 ‖ϕh(s)− ϕτ (s)‖2L2(Ω) ds+ C1h 2 + C1τ 2 (5.9) 14 S. KURIMA EJDE-2023/40 for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. Here, owing to (2.12) and Lemma 4.5, it holds that there exists a constant C2 > 0 such that 3 ∫ t 0 ‖ϕh(s)− ϕτ (s)‖2L2(Ω) ds = 3 ∫ t 0 ‖ϕh(s)− ϕ̂h(s) + ϕ̂τ (s)− ϕτ (s) + ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds ≤ 9 ∫ t 0 ‖ϕh(s)− ϕ̂h(s)‖2L2(Ω) ds+ 9 ∫ t 0 ‖ϕ̂τ (s)− ϕτ (s)‖2L2(Ω) ds + 9 ∫ t 0 ‖ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds ≤ C2h 2 + C2τ 2 + 9 ∫ t 0 ‖ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds (5.10) for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. We derive from the identity vh(s) = (ϕ̂h)s(s), (2.13) and Lemma 4.8 that there exists a constant C3 > 0 such that ‖ϕ̂h(t)− ϕ̂τ (t)‖2L2(Ω) = ∥∥∫ t 0 (vh(s)− vτ (s)) ds ∥∥2 L2(Ω) = ∥∥∫ t 0 (vh(s)− v̂h(s) + v̂τ (s)− vτ (s) + v̂h(s)− v̂τ (s)) ds ∥∥2 L2(Ω) ≤ C3h 2 + C3τ 2 + C3 ∫ t 0 ‖v̂h(s)− v̂τ (s)‖2L2(Ω) ds (5.11) for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. Thus, since v̂h − v̂τ + ϕ̂h − ϕ̂τ + a(·)(1 ? (ϕ h − ϕ τ ))− J ∗ (1 ? (ϕ h − ϕ τ )) + 1 ? (β(ϕh)− β(ϕτ )) + 1 ? (π(ϕh)− π(ϕτ )) = 1 ? (uh − uτ ), we deduce from (A1), Lemma 4.5, the local Lipschitz continuity of β, (A3), and (5.9)-(5.11) that there exists a constant C4 > 0 such that ‖v̂h(t)− v̂τ (t)‖2L2(Ω) ≤ C4(h2 + τ2) + C4 ∫ t 0 ‖ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds + C4 ∫ t 0 ‖v̂h(s)− v̂τ (s)‖2L2(Ω) ds+ C4‖(1 ? (uh − uτ ))(t)‖2H1(Ω) (5.12) for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. On the other hand, it follows from the identity vh(s) = (ϕ̂h)s(s), the Schwarz inequality, the Young inequality, (2.13), Lemmas 4.8 EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 15 and 4.9 that there exists a constant C5 > 0 such that 1 2 ‖ϕ̂h(t)− ϕ̂τ (t)‖2L2(Ω) = ∫ t 0 (vh(s)− vτ (s), ϕ̂h(s)− ϕ̂τ (s))L2(Ω) ds = ∫ t 0 (vh(s)− v̂h(s), ϕ̂h(s)− ϕ̂τ (s))L2(Ω) ds + ∫ t 0 (v̂τ (s)− vτ (s), ϕ̂h(s)− ϕ̂τ (s))L2(Ω) ds + ∫ t 0 (v̂h(s)− v̂τ (s), ϕ̂h(s)− ϕ̂τ (s))L2(Ω) ds ≤ C5h+ C5τ + 1 2 ∫ t 0 ‖v̂h(s)− v̂τ (s)‖2L2(Ω) ds + 1 2 ∫ t 0 ‖ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds (5.13) for all h, τ ∈ (0, h3) and all t ∈ [0, T ]. Therefore we can show Lemma 5.2 by (5.12) and (5.13). � Lemma 5.3. Let h3 be as in Lemma 4.5. Then there exists a constant M > 0 depending on the data such that ‖1 ? (uh − uτ )‖C([0,T ];H1(Ω)) + ‖ϕ̂h − ϕ̂τ‖C([0,T ];L2(Ω)) + ‖v̂h − v̂τ‖C([0,T ];L2(Ω)) ≤M(h1/2 + τ1/2) +M‖fh − fτ‖L2(0,T ;L2(Ω)) +M‖gh − gτ‖L2(0,T ;L2(∂Ω)) for all h, τ ∈ (0, h3). Proof. Combining (5.1) and (5.8) leads to the inequality 1 2 ‖(1 ? (uh − uτ ))(t)‖2H1(Ω) + 1 2M2 ‖ϕ̂h(t)− ϕ̂τ (t)‖2L2(Ω) + 1 2M2 ‖v̂h(t)− v̂τ (t)‖2L2(Ω) ≤ ( M1 + 1 2 ) (h+ τ) +M1 ∫ t 0 ‖(1 ? (uh − uτ ))(s)‖2H1(Ω) ds + 1 2 ∫ t 0 ‖ϕ̂h(s)− ϕ̂τ (s)‖2L2(Ω) ds+ ( M1 + 1 2 ) ∫ t 0 ‖v̂h(s)− v̂τ (s)‖2L2(Ω) ds +M1‖fh − fτ‖2L2(0,T ;L2(Ω)) +M1‖gh − gτ‖2L2(0,T ;L2(∂Ω)). Thus by the Gronwall lemma we can obtain Lemma 5.3. � Proof of Theorem 1.2. From Lemmas 4.1-4.3, 4.5-4.9, 5.3, the Aubin-Lions lemma for the compact embedding L2(Ω) ↪→ (H1(Ω)) ∗ , and properties (2.11)-(2.14), there exist some functions u, θ, ϕ, ξ such that u ∈ L2(0, T ;H1(Ω)), θ ∈ H1(0, T ; (H1(Ω)) ∗ ) ∩ L∞(0, T ;L2(Ω)), ϕ ∈W 2,2(0, T ;L2(Ω)) ∩W 1,∞(0, T ;L∞(Ω)), ξ ∈ L∞(Ω× (0, T )) and θ̂h → θ weakly∗ in H1(0, T ; (H1(Ω)) ∗ ) ∩ L∞(0, T ;L2(Ω)), (5.14) θ̂h → θ strongly in C([0, T ]; (H1(Ω)) ∗ ), (5.15) 16 S. KURIMA EJDE-2023/40 α(uh) = θh → θ weakly∗ in L∞(0, T ;L2(Ω)), (5.16) uh → u weakly in L2(0, T ;H1(Ω)), (5.17) zh → ϕtt weakly in L2(0, T ;L2(Ω)), (5.18) v̂h → ϕt strongly in C([0, T ];L2(Ω)), (5.19) vh → ϕt weakly∗ in L∞(Ω× (0, T )), (5.20) ϕ̂h → ϕ weakly∗ in W 1,∞(0, T ;L∞(Ω)), (5.21) ϕ̂h → ϕ strongly in C([0, T ];L2(Ω)), (5.22) ϕh → ϕ weakly∗ in L∞(Ω× (0, T )), (5.23) ϕ h → ϕ weakly∗ in L∞(Ω× (0, T )), (5.24) β(ϕh)→ ξ weakly∗ in L∞(Ω× (0, T )) (5.25) as h = hj ↘ 0, where α(r) := − 1 r for r ∈ D(α) := {r ∈ R | r < 0}. We see from (2.11), Lemmas 4.1, 4.3, (5.15), and (5.17) that∫ T 0 (α(uh(t)), uh(t))L2(Ω) dt = ∫ T 0 (θh(t), uh(t))L2(Ω) dt = ∫ T 0 〈θh(t)− θ̂h(t), uh(t)〉(H1(Ω))∗,H1(Ω) dt+ ∫ T 0 〈θ̂h(t), uh(t)〉(H1(Ω))∗,H1(Ω) dt → ∫ T 0 〈θ(t), u(t)〉(H1(Ω))∗,H1(Ω) dt = ∫ T 0 (θ(t), u(t))L2(Ω) dt as h = hj ↘ 0. Thus, noting that α : D(α) ⊂ R → R is maximal monotone, we can obtain that θ = α(u) = − 1 u a.e. in Ω× (0, T ) (5.26) (see, e.g., [1, Lemma 1.3, p. 42]). On the other hand, it follows from (2.12), Lemma 4.5 and (5.22) that ‖ϕh − ϕ‖L∞(0,T ;L2(Ω)) ≤ ‖ϕh − ϕ̂h‖L∞(0,T ;L2(Ω)) + ‖ϕ̂h − ϕ‖L∞(0,T ;L2(Ω)) ≤ |Ω|1/2h‖vh‖L∞(Ω×(0,T )) + ‖ϕ̂h − ϕ‖C([0,T ];L2(Ω)) → 0 (5.27) as h = hj ↘ 0. Then combining (5.25) and (5.27) yields that∫ T 0 (β(ϕh(t)), ϕh(t))L2(Ω) dt→ ∫ T 0 (ξ(t), ϕ(t))L2(Ω) dt as h = hj ↘ 0, and hence it holds that ξ = β(ϕ) a.e. in Ω× (0, T ). (5.28) Therefore by (5.14), (5.15), (5.17)-(5.28), (A1), and (A3), and by observing that fh → f strongly in L2(0, T ;L2(Ω)) and gh → g strongly in L2(0, T ;L2(∂Ω)) as h ↘ 0 (see e.g., [4, Section 5]), we can derive the existence of weak solutions to (1.3). Moreover, we can show the uniqueness of weak solutions to (1.3) in a similar way to the proofs of Lemmas 5.1, 5.2 and 5.3. � EJDE-2023/40 NONLOCAL PENROSE-FIFE TYPE PHASE FIELD SYSTEMS 17 Proof of Theorem 2.2. Since we have from f ∈ L2(0, T ;L2(Ω))∩W 1,1(0, T ;L2(Ω)) and g ∈ L2(0, T ;L2(∂Ω)) ∩W 1,1(0, T ;L2(∂Ω)) that there exists a constant C1 > 0 such that ‖fh − f‖L2(0,T ;L2(Ω)) ≤ C1h 1/2, ‖gh − g‖L2(0,T ;L2(∂Ω)) ≤ C1h 1/2 for all h > 0 (see e.g., [4, Section 5]), we can prove Theorem 2.2 by Lemma 5.3. � Remark 5.4. Even if in [3] we consider the approximation (µMuM + ρM (uM ))t + (ϕM )t −∆uM = f in Ω× (0, T ), (ϕM )tt + (ϕM )t + a(·)ϕM − J ∗ ϕM + βM (ϕM ) + π(ϕM ) = −(ρM (uM ))−1 in Ω× (0, T ), ∂νuM + uM = g on ∂Ω× (0, T ), (uM )(0) = −(ρM (u0))−1, ϕM (0) = ϕ0, (ϕM )t(0) = v0 in Ω, (5.29) we do not know whether we can establish a priori estimates for (5.29) or not. Here M ∈ N, µM := 1 1+M2 , the function ρM : R→ R is defined by ρM (r) :=  1 M+1 if r < −(M + 1), − 1 r if − (M + 1) ≤ r ≤ − 1 M+1 , M + 1 if − 1 M+1 < r, and the function βM : R→ R is defined by βM (r) :=  −M if β(r) ≤ −M, β(r) if −M < β(r) < M, M if M ≤ β(r). Although we can obtain that 1 2 |ϕM (x, t)|2 = 1 2 |ϕ0(x)|2 + ∫ t 0 (ϕM )t(x, s)ϕM (x, s) ds and 1 2 |(ϕM )t(x, t)|2 + ∫ t 0 |(ϕM )t(x, s)|2 ds+ β̂M (ϕM (x, t)) = ∫ t 0 (ρM (uM (x, s)))−1(−(ϕM )t(x, s)) ds+ · · · , where β̂M (r) = ∫ r 0 βM (s) ds, we do not know whether the L∞(Ω× (0, T ))-estimate for { ∫ t 0 (ρM (uM (x, s)))−1 ds } M can be derived or not, and then we do not know whether the L∞(Ω× (0, T ))-estimates for {(ϕM )t}M , {ϕM}M and {β(ϕM )}M can be obtained or not. Even if we replace −(ρM (uM ))−1 with uM in (5.29), since the inequality −uM ≥ 0 does not hold, we see that∫ t 0 (−uM (x, s))(−(ϕM )t(x, s)) ds 6≤ ‖ − (ϕM )t‖L∞(Ω×(0,T )) ∫ t 0 (−uM (x, s)) ds, whence we do not know whether the L∞(Ω × (0, T ))-estimates for {(ϕM )t}M , {ϕM}M and {β(ϕM )}M can be established or not. In this paper, we can prove 18 S. KURIMA EJDE-2023/40 the existence of solutions to (1.3) by introducing the time discrete problem (2.1) and obtain an error estimate between the solution of (1.3) and the solution of (2.7). Acknowledgments. The author would like to thank the anonymous referees for their comments and suggestions. References [1] V. Barbu; Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. [2] V. Barbu; Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. [3] P. Colli, M. Grasselli, A. Ito; On a parabolic-hyperbolic Penrose-Fife phase-field system, Electron. J. Differential Equations 2002, No. 100, 30 pp. (Erratum: Electron. J. Differential Equations 2002, No. 100, 32 pp.). [4] P. Colli, S. Kurima; Time discretization of a nonlinear phase field system in general domains, Comm. Pure Appl. Anal., 18 (2019), 3161-3179. [5] J. W. Jerome; Approximations of Nonlinear Evolution Systems, Mathematics in Science and Engineering, 164, Academic Press Inc., Orlando, 1983. [6] S. Kurima; Time discretization of a nonlocal phase-field system with inertial term, Matem- atiche (Catania) 77 (2022), 47-66. [7] S. Kurima; Existence for a singular nonlocal phase field system with inertial term, Acta Appl. Math., 178 (2022), Paper No. 10, 20 pp. Shunsuke Kurima Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku- ku, Tokyo 162-8601, Japan Email address: shunsuke.kurima@gmail.com 1. Introduction 2. Time discretization 3. Existence for the discrete problem 4. Uniform estimates for the discrete problem 5. Existence for (??) and error estimate Acknowledgments References