Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 96, pp. 1–26.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

TIME DISCRETIZATION OF AN ABSTRACT PROBLEM FROM

LINEARIZED EQUATIONS OF A COUPLED SOUND AND

HEAT FLOW

SHUNSUKE KURIMA

Abstract. Recently, a time discretization of simultaneous abstract evolution

equations applied to parabolic-hyperbolic phase-field systems has been stud-
ied. This article focuses on a time discretization of an abstract problem that

has application to linearized equations of coupled sound and heat flow. As

examples, we also study some parabolic-hyperbolic phase-field systems.

1. Introduction

Matsubara-Yokota [10] established the existence, uniqueness, and regularity of
solutions to the initial-boundary value problem for the linearized equations of cou-
pled sound and heat flow

θt + (γ − 1)ϕt − σ∆θ = 0 in Ω× (0,∞),

ϕtt − c2∆ϕ−m2ϕ = −c2∆θ in Ω× (0,∞),

θ = ϕ = 0 on ∂Ω× (0,∞),

θ(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0 in Ω ,

by applying the Hille-Yosida theorem, where c > 0, σ > 0, m ∈ R and γ > 1 are
constants, Ω ⊂ Rd (d ∈ N) is a domain with smooth bounded boundary ∂Ω, and
θ0, ϕ0, v0 are given functions.

Reference [9] presents the existence of solutions to the initial valued problem for
the simultaneous abstract evolution equation

dθ

dt
+
dϕ

dt
+A1θ = f in (0, T ),

L
d2ϕ

dt2
+B

dϕ

dt
+A2ϕ+ Φϕ+ Lϕ = θ in (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0,
dϕ

dt
(0) = v0

where T > 0, L : H → H is a bounded linear positive selfadjoint operator, B :
D(B) ⊂ H → H, Aj : D(Aj) ⊂ H → H (j = 1, 2) are linear maximal monotone
selfadjoint operators, H and V are real Hilbert spaces satisfying V ⊂ H, Vj (j =

2010 Mathematics Subject Classification. 35A35, 47N20, 35G30, 35L70.
Key words and phrases. Simultaneous evolution equations; linearized equations;

coupled sound and heat flow; time discretization; error estimate.
c©2020 Texas State University.

Submitted January 20, 2020. Published September 19, 2020.

1



2 S. KURIMA EJDE-2020/96

1, 2) are linear subspaces of V satisfying D(Aj) ⊂ Vj (j = 1, 2), Φ : D(Φ) ⊂ H → H
is a maximal monotone operator, L : H → H is a Lipschitz continuous operator, f :
(0, T )→ H and θ0 ∈ V1, ϕ0, v0 ∈ V2 are given. By employing a time discretization
scheme in [3, 4], an error estimate for the difference between continuous and discrete
solutions was presented.

Moreover in [9], assuming conditions from [3, Section 2] and [4, 5, 6, 7, 12,
13], some parabolic-hyperbolic phase-field systems are contained as examples un-
der homogeneous Dirichlet–Dirichlet boundary conditions, homogeneous Dirichlet-
Neumann boundary conditions, homogeneous Neumann-Dirichlet boundary condi-
tions, or homogeneous Neumann-Neumann boundary conditions.

In this article we consider the existence and uniqueness of solutions of the ab-
stract problem

dθ

dt
+ η

dϕ

dt
+A1θ = 0 in (0, T ),

L
d2ϕ

dt2
+B1

dϕ

dt
+A2ϕ+ Φϕ+ Lϕ = B2θ in (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0,
dϕ

dt
(0) = v0,

(1.1)

where T > 0, η > 0, L : H → H is a bounded linear positive selfadjoint operator,
Bj : D(Bj) ⊂ H → H, Aj : D(Aj) ⊂ H → H (j = 1, 2) are linear maximal
monotone selfadjoint operators, D(Aj) ⊂ V (j = 1, 2), Φ : D(Φ) ⊂ H → H is
a maximal monotone operator, L : H → H is a Lipschitz continuous operator,
θ0, ϕ0, v0 ∈ V are given. Moreover, we study the problem

δhθn + ηδhϕn +A1θn+1 = 0,

Lzn+1 +B1vn+1 +A2ϕn+1 + Φϕn+1 + Lϕn+1 = B2θn+1,

z0 = z1, zn+1 = δhvn,

vn+1 = δhϕn

(1.2)

for n = 0, . . . , N − 1, where h = T
N , N ∈ N,

δhθn :=
θn+1 − θn

h
, δhϕn :=

ϕn+1 − ϕn
h

, δhvn :=
vn+1 − vn

h
. (1.3)

Putting

θ̂h(0) := θ0,
dθ̂h
dt

(t) := δhθn, ϕ̂h(0) := ϕ0,
dϕ̂h
dt

(t) := δhϕn, (1.4)

v̂h(0) := v0,
dv̂h
dt

(t) := δhvn, (1.5)

θh(t) := θn+1, zh(t) := zn+1, ϕh(t) := ϕn+1, vh(t) := vn+1 (1.6)

for a.a. t ∈ (nh, (n+ 1)h), n = 0, . . . , N − 1, we can rewrite (1.2) as

dθ̂h
dt

+ η
dϕ̂h
dt

+A1θh = 0 in (0, T ),

Lzh +B1vh +A2ϕh + Φϕh + Lϕh = B2θh in (0, T ),

zh =
dv̂h
dt

, vh =
dϕ̂h
dt

in (0, T ),

θ̂h(0) = θ0, ϕ̂h(0) = ϕ0, v̂h(0) = v0.

(1.7)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 3

We will use the following assumptions:

(A1) V and H are real Hilbert spaces satisfying V ⊂ H with dense, continuous
and compact embedding. Moreover, the inclusions V ⊂ H ⊂ V ∗ hold by
identifying H with its dual space H∗, where V ∗ is the dual space of V .

(A2) L : H → H is a bounded linear operator fulfilling

(Lw, z)H = (w,Lz)H for all w, z ∈ H, (Lw,w)H ≥ cL‖w‖2H
for all w ∈ H, where cL > 0 is a constant.

(A3) Aj : D(Aj) ⊂ H → H (j = 1, 2) are linear maximal monotone selfadjoint
operators, where D(Aj) (j = 1, 2) are linear subspaces of H and D(Aj) ⊂
V (j = 1, 2). Moreover, there exist bounded linear monotone operators
A∗j : V → V ∗ (j = 1, 2) such that

〈A∗jw, z〉V ∗,V = 〈A∗jz, w〉V ∗,V for all w, z ∈ V,
A∗jw = Ajw for all w ∈ D(Aj).

Moreover, for all α > 0 and for j = 1, 2 there exists ωj,α > 0 such that

〈A∗jw,w〉V ∗,V + α‖w‖2H ≥ ωj,α‖w‖2V for all w ∈ V.
(A4) Bj : D(Bj) ⊂ H → H (j = 1, 2) are linear maximal monotone selfadjoint

operators, where D(Bj) (j = 1, 2) are linear subspaces of H, satisfying
D(A1) ⊂ D(B2) and

D(B1) ∩D(A2) 6= ∅,
(B1w,A2w)H ≥ 0 for all w ∈ D(B1) ∩D(A2),

(B2w,A1w)H ≥ 0 for all w ∈ D(A1),

(B1w,A2z)H = (B1z,A2w)H for all w, z ∈ D(B1) ∩D(A2).

(A5) There exists a constant CA1,B2
> 0 such that

‖B2θ‖H ≤ CA1,B2
(‖A1θ‖H + ‖θ‖H) for all θ ∈ D(A1).

(A6) Φ : D(Φ) ⊂ H → H is a maximal monotone operator satisfying Φ(0) = 0
and V ⊂ D(Φ). Moreover, there exist constants p, q, CΦ > 0 such that

‖Φw − Φz‖H ≤ CΦ(1 + ‖w‖pV + ‖z‖qV )‖w − z‖V for all w, z ∈ V.
(A7) There exists a lower semicontinuous convex function i : V → {x ∈ R | x ≥

0} such that (Φw,w − z)H ≥ i(w)− i(z) for all w, z ∈ V .
(A8) Φλ(0) = 0, (Φλw,B1w)H ≥ 0 for all w ∈ D(B1), (Φλw,A2w)H ≥ 0 for all

w ∈ D(A2), where λ > 0 and Φλ : H → H is the Yosida approximation of
Φ.

(A9) B∗j : V → V ∗ (j = 1, 2) are bounded linear monotone operators fulfilling

〈B∗jw, z〉V ∗,V = 〈B∗j z, w〉V ∗,V for all w, z ∈ V,
B∗jw = Bjw for all w ∈ D(Bj) ∩ V.

(A10) For all g ∈ H, a, b, c, d, d′ > 0, λ > 0, if there exists ϕλ ∈ V such that

Lϕλ + aB∗1ϕλ + bA∗2ϕλ + cΦλϕλ + dLϕλ + d′B2(I + hA1)−1ϕλ = g

in V ∗, then it follows that ϕλ ∈ D(B1) ∩D(A2) and

Lϕλ + aB1ϕλ + bA2ϕλ + cΦλϕλ + dLϕλ + d′B2(I + hA1)−1ϕλ = g

in H.



4 S. KURIMA EJDE-2020/96

(A11) L : H → H is a Lipschitz continuous operator with Lipschitz constant
CL > 0.

(A12) θ0 ∈ D(A1), A1θ0 ∈ V , ϕ0 ∈ D(B1) ∩D(A2), v0 ∈ D(B1) ∩ V .

We set conditions (A2) and (A3) as in [3, Section 2]. Condition (A10) is equiv-
alent to the elliptic regularity in some cases (see Section 2). We set conditions
(A6)–(A8) and (A11) keeping mind typical examples of not only linearized equa-
tions of coupled sound and heat flow, but also of parabolic-hyperbolic phase-field
systems; see Section 2 and and assumptions in [4, 5, 6, 7, 12, 13].

Remark 1.1. Owing to (1.4)-(1.6), the reader can check directly the following
identities:

‖ϕ̂h‖L∞(0,T ;V ) = max{‖ϕ0‖V , ‖ϕh‖L∞(0,T ;V )}, (1.8)

‖v̂h‖L∞(0,T ;V ) = max{‖v0‖V , ‖vh‖L∞(0,T ;V )}, (1.9)

‖θ̂h‖L∞(0,T ;V ) = max{‖θ0‖V , ‖θh‖L∞(0,T ;V )}, (1.10)

‖ϕh − ϕ̂h‖L∞(0,T ;V ) = h
∥∥dϕ̂h
dt

∥∥
L∞(0,T ;V )

= h‖vh‖L∞(0,T ;V ), (1.11)

‖vh − v̂h‖L∞(0,T ;H) = h
∥∥dv̂h
dt

∥∥
L∞(0,T ;H)

= h‖zh‖L∞(0,T ;H), (1.12)

‖θh − θ̂h‖2L2(0,T ;V ) =
h2

3

∥∥dθ̂h
dt

∥∥2

L2(0,T ;V )
. (1.13)

Definition 1.2. A pair (θ, ϕ) with

θ ∈ H1(0, T ;V ) ∩ L∞(0, T ;V ) ∩ L∞(0, T ;D(A1)),

ϕ ∈W 2,∞(0, T ;H) ∩W 1,∞(0, T ;V ) ∩ L2(0, T ;D(A2)),

dϕ

dt
∈ L2(0, T ;D(B1)), Φϕ ∈ L∞(0, T ;H)

is called a solution of (1.1) if (θ, ϕ) satisfies

dθ

dt
+ η

dϕ

dt
+A1θ = 0 in H a.e. on (0, T ), (1.14)

L
d2ϕ

dt2
+B1

dϕ

dt
+A2ϕ+ Φϕ+ Lϕ = B2θ in H a.e. on (0, T ), (1.15)

θ(0) = θ0, ϕ(0) = ϕ0,
dϕ

dt
(0) = v0 in H. (1.16)

Our main results read as follows.

Theorem 1.3. Assume that (A1)–(A12) hold. Then there exists h0 ∈ (0, 1) such
that for all h ∈ (0, h0) there exists a unique solution (θn+1, ϕn+1) of (1.2) satisfying

θn+1 ∈ D(A1), ϕn+1 ∈ D(B1) ∩D(A2) for n = 0, . . . , N − 1.

Theorem 1.4. Assume that (A1)–(A12) hold. Then there exists a unique solution
(θ, ϕ) of (1.1).

Theorem 1.5. Let h0 be as in Theorem 1.3, and assume that (A1)–(A12) hold.
Then there exist constants h00 ∈ (0, h0) and M = M(T ) > 0 such that

‖L1/2(v̂h − v)‖L∞(0,T ;H) + ‖B1/2
1 (vh − v)‖L2(0,T ;H) + ‖ϕ̂h − ϕ‖L∞(0,T ;V )

+ ‖θ̂h − θ‖L∞(0,T ;H) + ‖θh − θ‖L2(0,T ;V )



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 5

+ ‖B1/2
2 (θ̂h − θ)‖L∞(0,T ;H) +

∫ T

0

(B2(θh(t)− θ(t)), A1(θh(t)− θ(t)))H dt

≤Mh1/2

for all h ∈ (0, h00), where v = dϕ
dt .

This article is organized as follows. In Section 2 we give the linearized equations
of coupled sound and heat flow and some parabolic-hyperbolic phase-field systems
as examples. In Section 3 we derive existence of solutions to (1.2). In Section 4
we prove that there exists a solution of (1.1). In Section 5 we establish uniqueness
for (1.1). In Section 6 we obtain error estimates between solutions of (1.1) and
solutions of (1.7).

2. Examples

Example 2.1. We have the problem

θt + (γ − 1)ϕt − σ∆θ = 0 in Ω× (0, T ),

ϕtt − c2∆ϕ−m2ϕ = −c2∆θ in Ω× (0, T ),

θ = ϕ = 0 on ∂Ω× (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0 in Ω,

(2.1)

where c > 0, σ > 0, m ∈ R, γ > 1, T > 0 are constants and Ω ⊂ R3 is a bounded
domain with smooth boundary ∂Ω, under the assumption that

θ0 ∈ H2(Ω) ∩H1
0 (Ω),−∆θ0 ∈ H1

0 (Ω), ϕ0 ∈ H2(Ω) ∩H1
0 (Ω), v0 ∈ H1

0 (Ω).

Indeed, putting

V := H1
0 (Ω), H := L2(Ω), L := I : H → H,

A1 := −σ∆ : D(A1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B1 := 0 : D(B1) := H → H,

A2 := −c2∆ : D(A2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B2 := −c2∆ : D(B2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

and defining the operators A∗1 : V → V ∗, B∗1 : V → V ∗, A∗2 : V → V ∗, Φ : D(Φ) ⊂
H → H, L : H → H, B∗2 : V → V ∗ as

〈A∗1w, z〉V ∗,V := σ

∫
Ω

∇w · ∇z for w, z ∈ V,

〈B∗1w, z〉V ∗,V := 0 for w, z ∈ V,

〈A∗2w, z〉V ∗,V := c2
∫

Ω

∇w · ∇z for w, z ∈ V,

Φz := 0 for z ∈ D(Φ) := H,

Lz := −m2z for z ∈ H,

〈B∗2w, z〉V ∗,V := c2
∫

Ω

∇w · ∇z for w, z ∈ V,

we can check that (A1)–(A12) hold. Similarly, we can confirm that the homogeneous
Neumann-Neumann problem is an example.



6 S. KURIMA EJDE-2020/96

Example 2.2. Now we have the problem

θt + (γ − 1)ϕt − σ∆θ = 0 in Ω× (0, T ),

ϕtt + εϕt − c2∆ϕ+ β(ϕ) + π(ϕ) = −c2∆θ in Ω× (0, T ),

θ = ϕ = 0 on ∂Ω× (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0 in Ω,

(2.2)

where c > 0, σ > 0, ε ≥ 0, γ > 1, T > 0 are constants and Ω ⊂ R3 is a bounded
domain with smooth boundary ∂Ω, under the following conditions:

(A13) β : R → R is a single-valued maximal monotone function and there exists

a proper differentiable (lower semicontinuous) convex function β̂ : R →
[0,+∞) such that β̂(0) = 0 and β(r) = β̂ ′(r) = ∂β̂(r) for all r ∈ R, where

β̂ ′ and ∂β̂, respectively, are the differential and subdifferential of β̂.

(A14) β ∈ C2(R). Moreover, there exists a constant Cβ > 0 such that |β′′(r)| ≤
Cβ(1 + |r|) for all r ∈ R.

(A15) π : R→ R is a Lipschitz continuous function.

(A16) θ0 ∈ H2(Ω) ∩H1
0 (Ω), −∆θ0 ∈ H1

0 (Ω), ϕ0 ∈ H2(Ω) ∩H1
0 (Ω), v0 ∈ H1

0 (Ω).

Indeed, putting

V := H1
0 (Ω), H := L2(Ω), L := I : H → H,

A1 := −σ∆, D(A1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B1 := εI, D(B1) := H → H,

A2 := −c2∆, D(A2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B2 := −c2∆, D(B2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H

and defining the operators A∗1 : V → V ∗, B∗1 : V → V ∗, A∗2 : V → V ∗, Φ : D(Φ) ⊂
H → H, L : H → H, B∗2 : V → V ∗ as

〈A∗1w, z〉V ∗,V := σ

∫
Ω

∇w · ∇z for w, z ∈ V,

〈B∗1w, z〉V ∗,V := ε(w, z)H for w, z ∈ V,

〈A∗2w, z〉V ∗,V := c2
∫

Ω

∇w · ∇z for w, z ∈ V,

Φz := β(z) for z ∈ D(Φ) := {z ∈ H | β(z) ∈ H},
Lz := π(z) for z ∈ H,

〈B∗2w, z〉V ∗,V := c2
∫

Ω

∇w · ∇z for w, z ∈ V,

we can confirm that (A1)–(A12) hold, see [9]. Similarly, we can verify that the
homogeneous Neumann–Neumann problem is an example.

Example 2.3. We have the problem

θt + (γ − 1)ϕt − σ∆θ = 0 in Ω× (0, T ),

ϕtt − ε∆ϕt − c2∆ϕ+ β(ϕ) + π(ϕ) = −c2∆θ in Ω× (0, T ),

θ = ϕ = 0 on ∂Ω× (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0in Ω,

(2.3)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 7

where c > 0, σ > 0, ε ≥ 0, γ > 1, T > 0 are constants and Ω ⊂ R3 is a bounded
domain with smooth boundary ∂Ω, under the three conditions (A13)–(A15) and
the condition

(A17) θ0 ∈ H2(Ω) ∩H1
0 (Ω), −∆θ0 ∈ H1

0 (Ω), ϕ0 ∈ H2(Ω) ∩H1
0 (Ω), v0 ∈ H2(Ω) ∩

H1
0 (Ω).

Indeed, putting

V := H1
0 (Ω), H := L2(Ω), L := I : H → H,

A1 := −σ∆, D(A1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B1 := −ε∆, D(B1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

A2 := −c2∆, D(A2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B2 := −c2∆, D(B2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H

and defining the operators A∗1 : V → V ∗, B∗1 : V → V ∗, A∗2 : V → V ∗, Φ : D(Φ) ⊂
H → H, L : H → H, B∗2 : V → V ∗ as

〈A∗1w, z〉V ∗,V := σ

∫
Ω

∇w · ∇z for w, z ∈ V,

〈B∗1w, z〉V ∗,V := ε

∫
Ω

∇w · ∇z for w, z ∈ V,

〈A∗2w, z〉V ∗,V := c2
∫

Ω

∇w · ∇z for w, z ∈ V,

Φz := β(z) for z ∈ D(Φ) := {z ∈ H | β(z) ∈ H},
Lz := π(z) for z ∈ H,

〈B∗2w, z〉V ∗,V := c2
∫

Ω

∇w · ∇z for w, z ∈ V,

we can verify that (A1)–(A12) hold, see [9]. Similarly, we can check that the
homogeneous Neumann-Neumann problem is an example.

Example 2.4. We have the problem

θt + ϕt −∆θ = 0 in Ω× (0, T ),

ϕtt + ϕt −∆ϕ+ β(ϕ) + π(ϕ) = θ in Ω× (0, T ),

θ = ϕ = 0 on ∂Ω× (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0 in Ω,

(2.4)

where Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω, T > 0, under the
four conditions (A13)–(A16). Indeed, putting

V := H1
0 (Ω), H := L2(Ω), L := I : H → H,

A1 := −∆, D(A1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B1 := I, D(B1) := H → H,

A2 := −∆, D(A2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B2 := I, D(B2) := H → H



8 S. KURIMA EJDE-2020/96

and defining the operators A∗1 : V → V ∗, B∗1 : V → V ∗, A∗2 : V → V ∗, Φ : D(Φ) ⊂
H → H, L : H → H, B∗2 : V → V ∗ as

〈A∗1w, z〉V ∗,V :=

∫
Ω

∇w · ∇z for w, z ∈ V,

〈B∗1w, z〉V ∗,V := (w, z)H for w, z ∈ V,

〈A∗2w, z〉V ∗,V :=

∫
Ω

∇w · ∇z for w, z ∈ V,

Φz := β(z) for z ∈ D(Φ) := {z ∈ H | β(z) ∈ H},
Lz := π(z) for z ∈ H,

〈B∗2w, z〉V ∗,V := (w, z)H for w, z ∈ V,

we can confirm that (A1)–(A12) hold, see [9]. Similarly, we can show that the
homogeneous Neumann–Neumann problem is an example.

Example 2.5. We have the problem

θt + ϕt −∆θ = 0 in Ω× (0, T ),

ϕtt −∆ϕt −∆ϕ+ β(ϕ) + π(ϕ) = θ in Ω× (0, T ),

θ = ϕ = 0 on ∂Ω× (0, T ),

θ(0) = θ0, ϕ(0) = ϕ0, ϕt(0) = v0 in Ω

(2.5)

where Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω, T > 0, under the
four conditions (A13)-(A15), (A17). Indeed, putting

V := H1
0 (Ω), H := L2(Ω), L := I : H → H,

A1 := −∆, D(A1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B1 := −∆, D(B1) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

A2 := −∆, D(A2) := H2(Ω) ∩H1
0 (Ω) ⊂ H → H,

B2 := I, D(B2) := H → H

and defining the operators A∗1 : V → V ∗, B∗1 : V → V ∗, A∗2 : V → V ∗, Φ : D(Φ) ⊂
H → H, L : H → H, B∗2 : V → V ∗ as

〈A∗1w, z〉V ∗,V :=

∫
Ω

∇w · ∇z for w, z ∈ V,

〈B∗1w, z〉V ∗,V :=

∫
Ω

∇w · ∇z for w, z ∈ V,

〈A∗2w, z〉V ∗,V :=

∫
Ω

∇w · ∇z for w, z ∈ V,

Φz := β(z) for z ∈ D(Φ) := {z ∈ H | β(z) ∈ H},
Lz := π(z) for z ∈ H,

〈B∗2w, z〉V ∗,V := (w, z)H for w, z ∈ V,

we can check that (A1)–(A12) hold, see [9]. Similarly, we can show that the homo-
geneous Neumann-Neumann problem is an example.



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 9

3. Existence of discrete solutions

In this section we prove Theorem 1.3.

Lemma 3.1. There exists h1 ∈ (0, 1) such that 0 < h1 < h̃, where

h̃ :=
( cL

1 + CL + ηCA1,B2

+
η2C2

A1,B2

4(1 + CL + ηCA1,B2
)

)1/2

− ηCA1,B2

2(1 + CL + ηCA1,B2
)

and for all g ∈ H and all h ∈ (0, h1) there exists a unique solution ϕ ∈ D(B1) ∩
D(A2) of the equation

Lϕ+ hB1ϕ+ h2A2ϕ+ h2Φϕ+ h2Lϕ+ ηh2B2(I + hA1)−1ϕ = g in H.

Proof. We define the operator Ψ : V → V ∗ as

〈Ψϕ,w〉V ∗,V :=(Lϕ,w)H + h〈B∗1ϕ,w〉V ∗,V + h2〈A∗2ϕ,w〉V ∗,V + h2(Φλϕ,w)H

+ h2(Lϕ,w)H + ηh2(B2(I + hA1)−1ϕ,w)H for ϕ,w ∈ V.

Then the operator Ψ : V → V ∗ is monotone, continuous and coercive for all

h ∈ (0, h̃). Indeed, since the condition (A5) yields that

‖B2(I + hA1)−1ϕ‖H ≤ CA1,B2
(‖(I + hA1)−1ϕ‖H + ‖A1(I + hA1)−1ϕ‖H)

≤ CA1,B2(1 + h−1)‖ϕ‖H
(3.1)

for all ϕ ∈ H, we derive, from (A2), (A3), (A11), the monotonicity of B∗1 and Φλ,
and (3.1) that

〈Ψϕ−Ψϕ,ϕ− ϕ〉V ∗,V
= (L(ϕ− ϕ), ϕ− ϕ)H + h〈B∗1(ϕ− ϕ), ϕ− ϕ〉V ∗,V + h2〈A∗2(ϕ− ϕ), ϕ− ϕ〉V ∗,V

+ h2(Φλϕ− Φλϕ,ϕ− ϕ)H + h2(Lϕ− Lϕ,ϕ− ϕ)H

+ ηh2(B2(I + hA1)−1(ϕ− ϕ), (ϕ− ϕ))H

≥ cL‖ϕ− ϕ‖2H + ω2,1h
2‖ϕ− ϕ‖2V − h2‖ϕ− ϕ‖2H − CLh2‖ϕ− ϕ‖2H

− ηCA1,B2
(h+ h2)‖ϕ− ϕ‖2H

≥ ω2,1h
2‖ϕ− ϕ‖2V

for all ϕ,ϕ ∈ V and all h ∈ (0, h̃). It follows from the boundedness of the operators
L : H → H, B∗1 : V → V ∗, A∗2 : V → V ∗, the Lipschitz continuity of Φλ : H → H,
the condition (A11), (3.1) and the continuity of the embedding V ↪→ H that there
exists a constant C1 = C1(λ) > 0 such that

|〈Ψϕ−Ψϕ,w〉V ∗,V |
≤ |(L(ϕ− ϕ), w)H |+ h|〈B∗1(ϕ− ϕ), w〉V ∗,V |+ h2|〈A∗2(ϕ− ϕ), w〉V ∗,V |

+ h2|(Φλϕ− Φλϕ,w)H |+ h2|(Lϕ− Lϕ,w)H |
+ ηh2|(B2(I + hA1)−1(ϕ− ϕ), w)H |
≤ C1(1 + h+ h2)‖ϕ− ϕ‖V ‖w‖V

for all ϕ,ϕ ∈ V and all h > 0. Moreover, the inequality 〈Ψϕ − L0, ϕ〉V ∗,V ≥
ω2,1h

2‖ϕ‖2V holds for all ϕ ∈ V and all h ∈ (0, h̃). Therefore the operator Ψ : V →
V ∗ is surjective for all h ∈ (0, h̃) (see e.g., [2, p. 37]) and then we see from (A10) that



10 S. KURIMA EJDE-2020/96

for all g ∈ H and all h ∈ (0, h̃) there exists a unique solution ϕλ ∈ D(B1) ∩D(A2)
of the equation

Lϕλ + hB1ϕλ + h2A2ϕλ + h2Φλϕλ + h2Lϕλ + ηh2B2(I + hA1)−1ϕλ = g (3.2)

in H. Here, multiplying (3.2) by ϕλ and using the Young inequality, (A11), (3.1),
we infer that

(Lϕλ, ϕλ)H + h(B1ϕλ, ϕλ)H + h2〈A∗2ϕλ, ϕλ〉V ∗,V + h2(Φλϕλ, ϕλ)H

= (g, ϕλ)H − h2(Lϕλ − L0, ϕλ)H − h2(L0, ϕλ)H − ηh2(B2(I + hA1)−1ϕλ, ϕλ)H

≤ cL
2
‖ϕλ‖2H +

1

2cL
‖g‖2H + CLh

2‖ϕλ‖2H +
‖L0‖2H

2
h2 +

1

2
h2‖ϕλ‖2H

+ ηCA1,B2
(h+ h2)‖ϕλ‖2H ,

whence the conditions (A2) and (A3), the monotonicity of B1 and Φλ imply that

there exists h1 ∈ (0,min{1, h̃}) such that for all h ∈ (0, h1) there exists a constant
C2 = C2(h) > 0 satisfying

‖ϕλ‖2V ≤ C2 (3.3)

for all λ > 0. We have from (3.2), (A8), (3.1) and the Young inequality that

h2‖Φλϕλ‖2H
= (g,Φλϕλ)H − (Lϕλ,Φλϕλ)H − h(B1ϕλ,Φλϕλ)H − h2(A2ϕλ,Φλϕλ)H

− h2(Lϕλ,Φλϕλ)H − ηh2(B2(I + hA1)−1ϕλ,Φλϕλ)H

≤ 2

h2
‖g‖2H +

2

h2
‖Lϕλ‖2H + 2h2‖Lϕλ‖2H + 2η2C2

A1,B2
(1 + h)2‖ϕλ‖2H

+
1

2
h2‖Φλϕλ‖2H .

Thus, owing to the boundedness of the operator L : H → H, (A11) and (3.3), it
holds that for all h ∈ (0, h1) there exists a constant C3 = C3(h) > 0 such that

‖Φλϕλ‖2H ≤ C3 (3.4)

for all λ > 0. Then equation (3.2) yields

h‖B1ϕλ‖2H
= (g,B1ϕλ)H − (Lϕλ, B1ϕλ)H − h2(A2ϕλ, B1ϕλ)H − h2(Φλϕλ, B1ϕλ)H

− h2(Lϕλ, B1ϕλ)H − ηh2(B2(I + hA1)−1ϕλ, B1ϕλ)H ,

and hence we deduce from the boundedness of the operator L : H → H, (A4), (A8),
(A11), (3.1), the Young inequality and (3.3) that for all h ∈ (0, h1) there exists a
constant C4 = C4(h) > 0 satisfying

‖B1ϕλ‖2H ≤ C4(h) (3.5)

for all λ > 0. We derive from (3.1)-(3.5) that for all h ∈ (0, h1) there exists a
constant C5 = C5(h) > 0 such that

‖A2ϕλ‖2H ≤ C5(h) (3.6)

for all λ > 0. Hence the inequalities (3.3)-(3.6) mean that there exist ϕ ∈ D(B1)∩
D(A2) and ξ ∈ H such that

ϕλ → ϕ weakly in V, (3.7)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 11

Lϕλ → Lϕ weakly in H, (3.8)

Φλ(ϕλ)→ ξ weakly in H, (3.9)

B1ϕλ → B1ϕ weakly in H, (3.10)

A2ϕλ → A2ϕ weakly in H (3.11)

as λ = λj → +0. Here it follows from (3.3), (3.7), the compact of the embedding
V ↪→ H that

ϕλ → ϕ strongly in H (3.12)

as λ = λj → +0. Also, we see from (3.9) and (3.12) that (Φλϕλ, ϕλ)H → (ξ, ϕ)H
as λ = λj → +0. Thus the inclusion and the identity

ϕ ∈ D(Φ), ξ = Φϕ (3.13)

hold (see e.g., [1, Lemma 1.3, p. 42]).
Thanks to (3.2), (3.8)-(3.13) and (A11), we can verify that there exists a solution

ϕ ∈ D(B1) ∩D(A2) of the equation

Lϕ+ hB1ϕ+ h2A2ϕ+ h2Φϕ+ h2Lϕ+ ηh2B2(I + hA1)−1ϕ = g in H.

Moreover, the solution ϕ of this problem is unique by (A2), (A3), the monotonicity
of B1 and Φ, (A11) and (3.1). �

Proof of Theorem 1.3. Let h1 be as in Lemma 3.1 and let h ∈ (0, h1). Then we
infer from (1.3), the linearity of the operators A1, L, B1, B2 and A2 that problem
(1.2) can be written as

θn+1 + hA1θn+1 = θn + η(ϕn − ϕn+1),

Lϕn+1 + hB1ϕn+1 + h2A2ϕn+1 + h2Φϕn+1 + h2Lϕn+1

+ ηh2B2(I + hA1)−1ϕn+1

= Lϕn + hLvn + hB1ϕn + h2B2(I + hA1)−1(ηϕn + θn)

(3.14)

and then proving Theorem 1.3 is equivalent to show existence and uniqueness of
solutions to (3.14) for n = 0, . . . , N − 1. It suffices to consider the case that n = 0.
Owing to Lemma 3.1, there exists a unique solution ϕ1 ∈ D(B1) ∩ D(A2) of the
equation

Lϕ1 + hB1ϕ1 + h2A2ϕ1 + h2Φϕ1 + h2Lϕ1 + ηh2B2(I + hA1)−1ϕ1

= Lϕ0 + hLv0 + hB1ϕ0 + h2B2(I + hA1)−1(ηϕ0 + θ0).

Therefore, putting θ1 := (I + hA1)−1(θ0 + η(ϕ0 −ϕ1)), we can conclude that there
exists a unique solution (θ1, ϕ1) of (3.14) in the case that n = 0. �

4. Uniform estimates for (1.7) and passage to the limit

In this section we will derive a priori estimates for (1.7) and will show Theorem
1.4 by passing to the limit in (1.7) as h→ +0.

Lemma 4.1. Let h0 be as in Theorem 1.3. Then there exist constants h2 ∈ (0, h0)
and C = C(T ) > 0 such that

‖vh‖2L∞(0,T ;H) + h‖zh‖2L2(0,T ;H) + ‖B1/2
1 vh‖2L2(0,T ;H) + ‖ϕh‖2L∞(0,T ;V )

+ h‖vh‖2L2(0,T ;V ) + ‖B1/2
2 θh‖2L∞(0,T ;H) + h

∥∥B1/2
2

dθ̂h
dt

∥∥2

L2(0,T ;H)
≤ C



12 S. KURIMA EJDE-2020/96

for all h ∈ (0, h2).

Proof. We test the second equation in (1.2) by hvn+1 (= ϕn+1 − ϕn) and recall
(1.3) to obtain that

(L(vn+1 − vn), vn+1)H + h‖B1/2
1 vn+1‖2H + 〈A∗2ϕn+1, ϕn+1 − ϕn〉V ∗,V

+ (ϕn+1, ϕn+1 − ϕn)H + (Φϕn+1, ϕn+1 − ϕn)H

= h(B2θn+1, vn+1)H − h(Lϕn+1, vn+1)H + h(ϕn+1, vn+1)H .

(4.1)

Here it holds

(L(vn+1 − vn), vn+1)H

= (L1/2(vn+1 − vn), L1/2vn+1)H

=
1

2
‖L1/2vn+1‖2H −

1

2
‖L1/2vn‖2H +

1

2
‖L1/2(vn+1 − vn)‖2H

(4.2)

and

〈A∗2ϕn+1, ϕn+1 − ϕn〉V ∗,V + (ϕn+1, ϕn+1 − ϕn)H

=
1

2
〈A∗2ϕn+1, ϕn+1〉V ∗,V −

1

2
〈A∗2ϕn, ϕn〉V ∗,V

+
1

2
〈A∗2(ϕn+1 − ϕn), ϕn+1 − ϕn〉V ∗,V

+
1

2
‖ϕn+1‖2H −

1

2
‖ϕn‖2H +

1

2
‖ϕn+1 − ϕn‖2H .

(4.3)

The first equation in (1.2) yields

h(B2θn+1, vn+1)H

=
h

η

(
B2θn+1,−

θn+1 − θn
h

−A1θn+1

)
H

= − 1

2η

(
‖B1/2

2 θn+1‖2H − ‖B
1/2
2 θn‖2H + ‖B1/2

2 (θn+1 − θn)‖2H
)

− h

η
(B2θn+1, A1θn+1)H .

(4.4)

From (4.1)–(4.4), (A4), (A7), (A11), the continuity of the embedding V ↪→ H, and
Young’s inequality, we have that there exist constants C1, C2 > 0 such that

1

2
‖L1/2vn+1‖2H −

1

2
‖L1/2vn‖2H +

1

2
‖L1/2(vn+1 − vn)‖2H + h‖B1/2

1 vn+1‖2H

+
1

2
〈A∗2ϕn+1, ϕn+1〉V ∗,V −

1

2
〈A∗2ϕn, ϕn〉V ∗,V

+
1

2
〈A∗2(ϕn+1 − ϕn), ϕn+1 − ϕn〉V ∗,V +

1

2
‖ϕn+1‖2H −

1

2
‖ϕn‖2H

+
1

2
‖ϕn+1 − ϕn‖2H + i(ϕn+1)− i(ϕn)

+
1

2η
‖B1/2

2 θn+1‖2H −
1

2η
‖B1/2

2 θn‖2H +
1

2η
‖B1/2

2 (θn+1 − θn)‖2H

≤ h‖vn+1‖2H + C1h‖ϕn+1‖2V + C2h

(4.5)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 13

for all h ∈ (0, h0). Moreover, summing (4.5) over n = 0, . . . ,m−1 with 1 ≤ m ≤ N
leads to the inequality

1

2
‖L1/2vm‖2H +

1

2

m−1∑
n=0

‖L1/2(vn+1 − vn)‖2H + h

m−1∑
n=0

‖B1/2
1 vn+1‖2H

+
1

2
〈A∗2ϕm, ϕm〉V ∗,V +

1

2
‖ϕm‖2H +

1

2

m−1∑
n=0

〈A∗2(ϕn+1 − ϕn), ϕn+1 − ϕn〉V ∗,V

+
1

2

m−1∑
n=0

‖ϕn+1 − ϕn‖2H + i(ϕm) +
1

2η
‖B1/2

2 θm‖2H

+
1

2η

m−1∑
n=0

‖B1/2
2 (θn+1 − θn)‖2H

≤ 1

2
‖L1/2v0‖2H +

1

2
〈A∗2ϕ0, ϕ0〉V ∗,V +

1

2
‖ϕ0‖2H + i(ϕ0) +

1

2η
‖B1/2

2 θ0‖2H

+ h

m−1∑
n=0

‖vn+1‖2H + C1h

m−1∑
n=0

‖ϕn+1‖2V + C2T

(4.6)

for all h ∈ (0, h0). We see from (A3) that

1

2
〈A∗2ϕm, ϕm〉V ∗,V +

1

2
‖ϕm‖2H ≥

ω2,1

2
‖ϕm‖2V (4.7)

and

1

2

m−1∑
n=0

〈A∗2(ϕn+1 − ϕn), ϕn+1 − ϕn〉V ∗,V +
1

2

m−1∑
n=0

‖ϕn+1 − ϕn‖2H

≥ ω2,1

2

m−1∑
n=0

‖ϕn+1 − ϕn‖2V =
ω2,1

2
h2

m−1∑
n=0

‖vn+1‖2V .

(4.8)

Thus from (4.6)-(4.8) and (A2) we obtain

(cL
2
− h
)
‖vm‖2H +

cL
2
h2

m−1∑
n=0

‖zn+1‖2H + h

m−1∑
n=0

‖B1/2
1 vn+1‖2H

+
(ω2,1

2
− C1h

)
‖ϕm‖2V +

ω2,1

2
h2

m−1∑
n=0

‖vn+1‖2V +
1

2η
‖B1/2

2 θm‖2H

+
1

2η
h2

m−1∑
n=0

‖B1/2
2 δhθn‖2H

≤ 1

2
‖L1/2v0‖2H +

1

2
〈A∗2ϕ0, ϕ0〉V ∗,V +

1

2
‖ϕ0‖2H + i(ϕ0) +

1

2η
‖B1/2

2 θ0‖2H

+ h

m−1∑
j=0

‖vj‖2H + C1h

m−1∑
j=0

‖ϕj‖2V + C2T,



14 S. KURIMA EJDE-2020/96

whence there exist constants h2 ∈ (0, h0) and C3 = C3(T ) > 0 such that

‖vm‖2H + h2
m−1∑
n=0

‖zn+1‖2H + h

m−1∑
n=0

‖B1/2
1 vn+1‖2H

+ ‖ϕm‖2V + h2
m−1∑
n=0

‖vn+1‖2V + ‖B1/2
2 θm‖2H + h2

m−1∑
n=0

‖B1/2
2 δhθn‖2H

≤ C3h

m−1∑
j=0

‖vj‖2H + C3h

m−1∑
j=0

‖ϕj‖2V + C3

(4.9)

for all h ∈ (0, h2). Therefore from inequality (4.9) and the discrete Gronwall lemma
(see e.g., [8, Prop. 2.2.1]) there exists a constant C4 = C4(T ) > 0 such that

‖vm‖2H + h2
m−1∑
n=0

‖zn+1‖2H + h

m−1∑
n=0

‖B1/2
1 vn+1‖2H

+ ‖ϕm‖2V + h2
m−1∑
n=0

‖vn+1‖2V + ‖B1/2
2 θm‖2H + h2

m−1∑
n=0

‖B1/2
2 δhθn‖2H ≤ C4

for all h ∈ (0, h2) and m = 1, . . . , N . �

Lemma 4.2. Let h2 be as in Lemma 4.1. Then there exists a constant C = C(T ) >
0 such that

‖z1‖2H + h‖B1/2
1 z1‖2H + ‖v1‖2V + h2‖z1‖2V + 〈B∗2(ηv1 +A1θ1), ηv1 +A1θ1〉V ∗,V ≤ C

for all h ∈ (0, h2).

Proof. The second equation in (1.2), the identities v1 = v0 +hz1 and ϕ1 = ϕ0 +hv1

yield that

Lz1 +B1v0 + hB1z1 +A2ϕ0 + hA2v1 + Φϕ1 + Lϕ1 = B2θ1. (4.10)

Then we test (4.10) by z1 to infer that

‖L1/2z1‖2H + (B1v0, z1)H + h(B1z1, z1)H + (A2ϕ0, z1)H + h(A2v1, z1)H

+ (Φϕ1, z1)H + (Lϕ1, z1)H

= (B2θ1, z1)H .

(4.11)

From (A3) we obtain

h(A2v1, z1)H

= (A2v1, v1 − v0)H

= 〈A∗2v1, v1 − v0〉V ∗,V

=
1

2
〈A∗2v1, v1〉V ∗,V −

1

2
〈A∗2v0, v0〉V ∗,V +

1

2
〈A∗2(v1 − v0), v1 − v0〉V ∗,V

≥ ω2,1

2
‖v1‖2V −

1

2
‖v1‖2H −

1

2
〈A∗2v0, v0〉V ∗,V +

ω2,1

2
‖v1 − v0‖2V

− 1

2
‖v1 − v0‖2H .

(4.12)

We see from (A6) and Lemma 4.1 that there exists a constant C1 = C1(T ) > 0
such that

|(Φϕ1, z1)H | ≤ CΦ(1 + ‖ϕ1‖pV )‖ϕ1‖V ‖z1‖H ≤ C1‖z1‖H . (4.13)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 15

Also, the first equation in (1.2) and the identity v1 − v0 = hz1 imply that

1

2η
〈B∗2(ηv1 +A1θ1), ηv1 +A1θ1〉V ∗,V

− 1

2η
〈B∗2(ηv0 +A1θ0), ηv0 +A1θ0〉V ∗,V

+
1

2η
〈B∗2(η(v1 − v0) +A1(θ1 − θ0)), η(v1 − v0) +A1(θ1 − θ0)〉V ∗,V

=
1

η
〈B∗2(ηv1 +A1θ1), η(v1 − v0) +A1(θ1 − θ0)〉V ∗,V

= −(B2θ1, z1)H + (B2θ0, z1)H −
1

ηh
(B2(θ1 − θ0), A1(θ1 − θ0))H .

(4.14)

It follows from (4.11)-(4.14), (A2), (A4) and the monotonicity of B∗2 : V → V ∗ that

cL‖z1‖2H + h‖B1/2
1 z1‖2H +

ω2,1

2
‖v1‖2V +

ω2,1

2
h2‖z1‖2V

+
1

2η
〈B∗2(ηv1 +A1θ1), ηv1 +A1θ1〉V ∗,V

≤ −(B1v0, z1)H − (A2ϕ0, z1)H +
1

2
‖v1‖2H +

1

2
〈A∗2v0, v0〉V ∗,V

+
1

2
‖v1 − v0‖2H + C1‖z1‖H − (Lϕ1, z1)H + (B2θ0, z1)H

+
1

2η
〈B∗2(ηv0 +A1θ0), ηv0 +A1θ0〉V ∗,V .

(4.15)

Thus we deduce from (4.15), (A11), the Young inequality and Lemma 4.1 that
Lemma 4.2 holds. �

Lemma 4.3. Let h2 be as in Lemma 4.1. Then there exist constants h3 ∈ (0, h2)
and C = C(T ) > 0 such that

‖zh‖2L∞(0,T ;H) + ‖B1/2
1 zh‖2L2(0,T ;H) + ‖vh‖2L∞(0,T ;V ) + h‖zh‖2L2(0,T ;V ) ≤ C

for all h ∈ (0, h3).

Proof. Let n ∈ {1, . . . , N − 1}. Then we have from the second equation in (1.2)
that

L(zn+1 − zn) + hB1zn+1 + hA2vn+1 + Φϕn+1 − Φϕn + Lϕn+1 − Lϕn
= B2(θn+1 − θn).

Since

(L(zn+1 − zn), zn+1)H = (L1/2(zn+1 − zn), L1/2zn+1)H

=
1

2
‖L1/2zn+1‖2H −

1

2
‖L1/2zn‖2H +

1

2
‖L1/2(zn+1 − zn)‖2H ,

it follows that
1

2
‖L1/2zn+1‖2H −

1

2
‖L1/2zn‖2H +

1

2
‖L1/2(zn+1 − zn)‖2H + h‖B1/2

1 zn+1‖2H
+ 〈A∗2vn+1, vn+1 − vn〉V ∗,V + (vn+1, vn+1 − vn)H

= −h
(Φϕn+1 − Φϕn

h
, zn+1

)
H
− h
(Lϕn+1 − Lϕn

h
, zn+1

)
H

+ (B2(θn+1 − θn), zn+1)H + h(vn+1, zn+1)H .

(4.16)



16 S. KURIMA EJDE-2020/96

On the other hand,

〈A∗2vn+1, vn+1 − vn〉V ∗,V + (vn+1, vn+1 − vn)H

=
1

2
〈A∗2vn+1, vn+1〉V ∗,V −

1

2
〈A∗2vn, vn〉V ∗,V

+
1

2
〈A∗2(vn+1 − vn), vn+1 − vn〉V ∗,V +

1

2
‖vn+1‖2H −

1

2
‖vn‖2H

+
1

2
‖vn+1 − vn‖2H .

(4.17)

Condition (A6) and Lemma 4.1 mean that there exists a constant C1 = C1(T ) > 0
such that

− h
(Φϕn+1 − Φϕn

h
, zn+1

)
H

≤ CΦh(1 + ‖ϕn+1‖pV + ‖ϕn‖qV )‖vn+1‖V ‖zn+1‖H
≤ C1h‖vn+1‖V ‖zn+1‖H

(4.18)

for all h ∈ (0, h2). Also, the first equation in (1.2) and the identity vn+1 − vn =
hzn+1 yield

1

2η
〈B∗2(ηvn+1 +A1θn+1), ηvn+1 +A1θn+1〉V ∗,V

− 1

2η
〈B∗2(ηvn +A1θn), ηvn +A1θn〉V ∗,V

+
1

2η
〈B∗2(η(vn+1 − vn) +A1(θn+1 − θn)), η(vn+1 − vn)

+A1(θn+1 − θn)〉V ∗,V

=
1

η
〈B∗2(ηvn+1 +A1θn+1), η(vn+1 − vn) +A1(θn+1 − θn)〉V ∗,V

= −(B2(θn+1 − θn), zn+1)H −
1

ηh
(B2(θn+1 − θn), A1(θn+1 − θn))H .

(4.19)

It follows from (4.16)-(4.19), (A4) and (A11) that there exists a constant C2 =
C2(T ) > 0 such that

1

2
‖L1/2zn+1‖2H −

1

2
‖L1/2zn‖2H +

1

2
‖L1/2(zn+1 − zn)‖2H + h‖B1/2

1 zn+1‖2H

+
1

2
〈A∗2vn+1, vn+1〉V ∗,V −

1

2
〈A∗2vn, vn〉V ∗,V

+
1

2
〈A∗2(vn+1 − vn), vn+1 − vn〉V ∗,V

+
1

2
‖vn+1‖2H −

1

2
‖vn‖2H +

1

2
‖vn+1 − vn‖2H

+
1

2η
〈B∗2(ηvn+1 +A1θn+1), ηvn+1 +A1θn+1〉V ∗,V

− 1

2η
〈B∗2(ηvn +A1θn), ηvn +A1θn〉V ∗,V

+
1

2η

〈
B∗2(η(vn+1 − vn) +A1(θn+1 − θn)), η(vn+1 − vn)

+A1(θn+1 − θn)
〉
V ∗,V

≤ C2h‖vn+1‖V ‖zn+1‖H

(4.20)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 17

for all h ∈ (0, h2). Then we sum (4.20) over n = 1, . . . , ` − 1 with 2 ≤ ` ≤ N to
obtain

1

2
‖L1/2z`‖2H +

1

2

`−1∑
n=1

‖L1/2(zn+1 − zn)‖2H + h

`−1∑
n=1

‖B1/2
1 zn+1‖2H

+
1

2
〈A∗2v`, v`〉V ∗,V +

1

2

`−1∑
n=1

〈A∗2(vn+1 − vn), vn+1 − vn〉V ∗,V

+
1

2
‖v`‖2H +

1

2

`−1∑
n=1

‖vn+1 − vn‖2H +
1

2η
〈B∗2(ηv` +A1θ`), ηv` +A1θ`〉V ∗,V

≤ 1

2
‖L1/2z1‖2H +

1

2
〈A∗2v1, v1〉V ∗,V +

1

2
‖v1‖2H

+
1

2η
〈B∗2(ηv1 +A1θ1), ηv1 +A1θ1〉V ∗,V + C2h

`−1∑
n=0

‖vn+1‖V ‖zn+1‖H .

Thus from (A2) and (A3) we have

cL
2
‖z`‖2H + h

`−1∑
n=1

‖B1/2
1 zn+1‖2H +

ω2,1

2
‖v`‖2V +

ω2,1

2
h2

`−1∑
n=1

‖zn+1‖2V

+
1

2η
〈B∗2(ηv` +A1θ`), ηv` +A1θ`〉V ∗,V

≤ 1

2
‖L1/2z1‖2H +

1

2
〈A∗2v1, v1〉V ∗,V +

1

2
‖v1‖2H

+
1

2η
〈B∗2(ηv1 +A1θ1), ηv1 +A1θ1〉V ∗,V + C2h

`−1∑
n=0

‖vn+1‖V ‖zn+1‖H

(4.21)

for all h ∈ (0, h2) and ` = 2, . . . , N . Therefore we infer from (4.21), the boundedness
of L and A∗2, and Lemma 4.2 that there exists a constant C3 = C3(T ) > 0 such
that

cL
2
‖zm‖2H + h

m−1∑
n=0

‖B1/2
1 zn+1‖2H +

ω2,1

2
‖vm‖2V +

ω2,1

2
h2

m−1∑
n=0

‖zn+1‖2V

≤ C3 + C2h

m−1∑
n=0

‖vn+1‖V ‖zn+1‖H

(4.22)

for all h ∈ (0, h2) and m = 1, . . . , N . Moreover, we see from (4.22) and the Young
inequality that

1

2
(cL − C2h)‖zm‖2H + h

m−1∑
n=0

‖B1/2
1 zn+1‖2H +

1

2
(ω2,1 − C2h)‖vm‖2V

+
ω2,1

2
h2

m−1∑
n=0

‖zn+1‖2V

≤ C3 +
C2

2
h

m−1∑
j=0

‖vj‖2V +
C2

2
h

m−1∑
j=0

‖zj‖2H

(4.23)



18 S. KURIMA EJDE-2020/96

for all h ∈ (0, h2) and m = 1, . . . , N . Hence there exist constants h3 ∈ (0, h2) and
C4 = C4(T ) > 0 such that

‖zm‖2H + h

m−1∑
n=0

‖B1/2
1 zn+1‖2H + ‖vm‖2V + h2

m−1∑
n=0

‖zn+1‖2V

≤ C4 + C4h

m−1∑
j=0

‖vj‖2V + C4h

m−1∑
j=0

‖zj‖2H

for all h ∈ (0, h3) and m = 1, . . . , N . Therefore, owing to the discrete Gronwall
lemma (see e.g., [8, Prop. 2.2.1]), there exists a constant C5 = C5(T ) > 0 satisfying

‖zm‖2H + h

m−1∑
n=0

‖B1/2
1 zn+1‖2H + ‖vm‖2V + h2

m−1∑
n=0

‖zn+1‖2V ≤ C5

for all h ∈ (0, h3) and m = 1, . . . , N . �

Lemma 4.4. Let h2 be as in Lemma 4.1. Then there exists a constant C = C(T ) >
0 such that

‖Φϕh‖L∞(0,T ;H) ≤ C

for all h ∈ (0, h2).

The above lemma follows from (A6) and Lemma 4.1.

Lemma 4.5. Let h3 be as in Lemma 4.3. Then there exist constants h4 ∈ (0, h3)
and C = C(T ) > 0 such that

∥∥dθ̂h
dt

∥∥2

L2(0,T ;H)
+ h
∥∥dθ̂h
dt

∥∥2

L2(0,T ;V )
+ ‖θh‖2L∞(0,T ;V ) ≤ C

for all h ∈ (0, h4).

Proof. We multiply the first equation in (1.2) by θn+1 − θn and by hθn+1, respec-
tively, and use the Young inequality to obtain that

h‖θn+1 − θn
h

‖2H + 〈A∗1θn+1, θn+1 − θn〉V ∗,V + (θn+1 − θn, θn+1)H

+ h(A1θn+1, θn+1)H

= −ηh
(
vn+1,

θn+1 − θn
h

)
H
− ηh(vn+1, θn+1)H

≤ η2h‖vn+1‖2H +
1

2
h‖θn+1 − θn

h
‖2H +

1

2
h‖θn+1‖2H .

(4.24)

Here it holds that

〈A∗1θn+1, θn+1 − θn〉V ∗,V + (θn+1 − θn, θn+1)H

=
1

2
〈A∗1θn+1, θn+1〉V ∗,V −

1

2
〈A∗1θn, θn〉V ∗,V

+
1

2
〈A∗1(θn+1 − θn), θn+1 − θn〉V ∗,V

+
1

2
‖θn+1‖2H −

1

2
‖θn‖2H +

1

2
‖θn+1 − θn‖2H .

(4.25)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 19

From (4.24), (4.25) and the continuity of the embedding V ↪→ H, there exists a
constant C1 > 0 such that

1

2
h‖θn+1 − θn

h
‖2H +

1

2
〈A∗1θn+1, θn+1〉V ∗,V −

1

2
〈A∗1θn, θn〉V ∗,V

+
1

2
〈A∗1(θn+1 − θn), θn+1 − θn〉V ∗,V +

1

2
‖θn+1‖2H −

1

2
‖θn‖2H

+
1

2
‖θn+1 − θn‖2H

≤ η2h‖vn+1‖2H + C1h‖θn+1‖2V

(4.26)

for all h ∈ (0, h3). Therefore we can prove Lemma 4.5 by summing (4.26) over
n = 0, . . . ,m− 1 with 1 ≤ m ≤ N , the condition (A3), Lemma 4.1 and the discrete
Gronwall lemma (see e.g., [8, Prop. 2.2.1]). �

Lemma 4.6. Let h4 be as in Lemma 4.5. Then there exists a constant C = C(T ) >
0 such that ∥∥dθ̂h

dt

∥∥2

L2(0,T ;V )
+ ‖A1θh‖2L∞(0,T ;H) ≤ C

for all h ∈ (0, h4).

Proof. It follows from the first equation in (1.2) that

h
〈
A∗1

θn+1 − θn
h

,
θn+1 − θn

h

〉
V ∗,V

+ h
∥∥θn+1 − θn

h

∥∥2

H

+
1

2
‖A1θn+1‖2H −

1

2
‖A1θn‖2H +

1

2
‖A1(θn+1 − θn)‖2H

= −ηh
〈
A∗1

θn+1 − θn
h

, vn+1

〉
V ∗,V

+ h
∥∥θn+1 − θn

h

∥∥2

H

and then we can prove this lemma by (A3), the boundedness of the operator A∗1 :
V → V ∗, the Young inequality, Lemma 4.3, summing over n = 0, . . . ,m − 1 with
1 ≤ m ≤ N and Lemma 4.5. �

Lemma 4.7. Let h4 be as in Lemma 4.5. Then there exists a constant C = C(T ) >
0 such that

‖B2θh‖2L∞(0,T ;H) + ‖B1vh‖2L2(0,T ;H) + ‖A2ϕh‖2L2(0,T ;H) ≤ C

for all h ∈ (0, h4).

Proof. By (A5) and Lemmas 4.5 and 4.6, there exists a constant C1 = C1(T ) > 0
such that

‖B2θh‖2L∞(0,T ;H) ≤ C1 (4.27)

for all h ∈ (0, h4). The second equation in (1.2) yields

h‖B1vn+1‖2H = h(B1vn+1, B1vn+1)H

= −h(Lzn+1, B1vn+1)H − h(A2ϕn+1, B1vn+1)H − h(Φϕn+1, B1vn+1)H

− h(Lϕn+1, B1vn+1)H + h(B2θn+1, B1vn+1)H

and then by Young’s inequality, the boundedness of the operator L : H → H, (A11)
and Lemma 4.1, there exists a constant C2 = C2(T ) > 0 satisfying

h‖B1vn+1‖2H ≤ C2h‖zn+1‖2H − h(A2ϕn+1, B1vn+1)H + C2h‖Φϕn+1‖2H
+ C2h‖B2θn+1‖2H + C2h

(4.28)



20 S. KURIMA EJDE-2020/96

for all h ∈ (0, h4). From (A4) we have

−h(A2ϕn+1, B1vn+1)H = −(A2ϕn+1, B1ϕn+1 −B1ϕn)H

= −1

2
(A2ϕn+1, B1ϕn+1)H +

1

2
(A2ϕn, B1ϕn)H

− 1

2
(A2(ϕn+1 − ϕn), B1(ϕn+1 − ϕn))H .

(4.29)

Thus summing (4.28) over n = 0, . . . ,m − 1 with 1 ≤ m ≤ N and using (4.27),
(4.29), Lemmas 4.3 and 4.4 imply the existence of a constant C3 = C3(T ) > 0 such
that

‖B1vh‖2L2(0,T ;H) ≤ C3 (4.30)

for all h ∈ (0, h4). Moreover, from the second equation in (1.7), (4.27), (4.30),
Lemmas 4.3 and 4.4, (A11) and Lemma 4.1 there exists a constant C4 = C4(T ) > 0
satisfying

‖A2ϕh‖2L2(0,T ;H) ≤ C4

for all h ∈ (0, h4). �

Lemma 4.8. Let h4 be as in Lemma 4.5. Then there exists a constant C = C(T ) >
0 such that

‖ϕ̂h‖W 1,∞(0,T ;V ) + ‖v̂h‖W 1,∞(0,T ;H) + ‖v̂h‖L∞(0,T ;V )

+ ‖θ̂h‖H1(0,T ;V ) + ‖θ̂h‖L∞(0,T ;V ) ≤ C

for all h ∈ (0, h4).

The above lemma follows from (1.8)-(1.10) and Lemmas 4.1, 4.3, 4.5 and 4.6.

Proof of Theroem 1.4 (existence part). By Lemmas 4.1, 4.3-4.8, and (1.11)-(1.13),
there exist functions

θ ∈ H1(0, T ;V ) ∩ L∞(0, T ;V ) ∩ L∞(0, T ;D(A1)),

ϕ ∈ L∞(0, T ;V ) ∩ L2(0, T ;D(A2)),

ξ ∈ L∞(0, T ;H)

such that

dϕ

dt
∈ L∞(0, T ;V ) ∩ L2(0, T ;D(B1)),

d2ϕ

dt2
∈ L∞(0, T ;H)

and

ϕ̂h → ϕ weakly∗ in W 1,∞(0, T ;V ), (4.31)

vh →
dϕ

dt
weakly∗ in L∞(0, T ;V ), (4.32)

v̂h →
dϕ

dt
weakly∗ in W 1,∞(0, T ;H) ∩ L∞(0, T ;V ), (4.33)

zh →
d2ϕ

dt2
weakly∗ in L∞(0, T ;H), (4.34)

Lzh → L
d2ϕ

dt2
weakly∗ in L∞(0, T ;H), (4.35)

θ̂h → θ weakly∗ in H1(0, T ;V ) ∩ L∞(0, T ;V ), (4.36)

ϕh → ϕ weakly∗ in L∞(0, T ;V ), (4.37)



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 21

θh → θ weakly∗ in L∞(0, T ;V ), (4.38)

A1θh → A1θ weakly∗ in L∞(0, T ;H), (4.39)

B1vh → B1
dϕ

dt
weakly in L2(0, T ;H), (4.40)

A2ϕh → A2ϕ weakly in L2(0, T ;H), (4.41)

Φϕh → ξ weakly∗ in L∞(0, T ;H), (4.42)

B2θh → B2θ weakly∗ in L∞(0, T ;H) (4.43)

as h = hj → +0. From Lemma 4.8, the compactness of the embedding V ↪→ H
and the convergence (4.31) we infer that

ϕ̂h → ϕ strongly in C([0, T ];H) (4.44)

as h = hj → +0 (see e.g., [11, Section 8, Corollary 4]). From (1.11) and Lemma
4.3 we have

ϕh → ϕ strongly in L∞(0, T ;H) (4.45)

as h = hj → +0. Hence the convergences (4.42) and (4.45) yield

∫ T

0

(Φϕh(t), ϕh(t))H dt→
∫ T

0

(ξ(t), ϕ(t))H dt

as h = hj → +0 and then

ξ = Φϕ in H a.e. on (0, T ) (4.46)

(see e.g., [1, Lemma 1.3, p. 42]). On the other hand, from Lemma 4.8, the com-
pactness of the embedding V ↪→ H and (4.36) it follows that

θ̂h → θ strongly in C([0, T ];H) (4.47)

as h = hj → +0. Similarly, we derive from (4.33) that

v̂h →
dϕ

dt
strongly in C([0, T ];H) (4.48)

as h = hj → +0. Therefore, combining (4.31), (4.35), (4.36), (4.39)-(4.48) and
(A11), we can verify that there exists a solution of (1.1). �

5. Uniqueness for (1.1)

Proof of Theorem 1.4 (uniqueness part). We let (θ, ϕ), (θ, ϕ) be two solutions of

(1.1) and put θ̃ := θ − θ, ϕ̃ := ϕ − ϕ. Then by (1.15), Young’s inequality, (A6),
(A11), Lemma 4.1, the continuity of the embedding V ↪→ H and (A2), there exists



22 S. KURIMA EJDE-2020/96

a constant C1 = C1(T ) > 0 satisfying

1

2

d

dt
‖L1/2 dϕ̃

dt
(t)‖2H +

(
B1

dϕ̃

dt
(t),

dϕ̃

dt
(t)
)
H

+
1

2

d

dt

∥∥A1/2
2 ϕ̃(t)

∥∥2

H

=
(
B2θ̃(t),

dϕ̃

dt
(t)
)
H
−
(

Φϕ(t)− Φϕ(t),
dϕ̃

dt
(t)
)
H

−
(
Lϕ(t)− Lϕ(t),

dϕ̃

dt
(t)
)
H

≤
(
B2θ̃(t),

dϕ̃

dt
(t)
)
H

+
C2

Φ

2
(1 + ‖ϕ(t)‖pV + ‖ϕ(t)‖qV )2‖ϕ̃(t)‖2V

+
C2
L

2
‖ϕ̃(t)‖2H + ‖dϕ̃

dt
(t)‖2H

≤
(
B2θ̃(t),

dϕ̃

dt
(t)
)
H

+ C1‖ϕ̃(t)‖2V +
1

cL
‖L1/2 dϕ̃

dt
(t)‖2H

(5.1)

for a.a. t ∈ (0, T ). From Young’s inequality, (A2) and the continuity of the embed-
ding V ↪→ H, there exists a constant C2 > 0 such that

1

2

d

dt
‖ϕ̃(t)‖2H =

(dϕ̃
dt

(t), ϕ̃(t)
)
H
≤ 1

2cL
‖L1/2 dϕ̃

dt
(t)‖2H + C2‖ϕ̃(t)‖2V (5.2)

for a.a. t ∈ (0, T ). We see from (A3) that

1

2

∥∥A1/2
2 ϕ̃(t)

∥∥2

H
+

1

2
‖ϕ̃‖2H =

1

2
〈A∗2ϕ̃(t), ϕ̃(t)〉V ∗,V +

1

2
‖ϕ̃‖2H

≥ ω2,1

2
‖ϕ̃(t)‖2V .

(5.3)

Moreover, the identity (1.14) yields that

(
B2θ̃(t),

dϕ̃

dt
(t)
)
H

=
1

η

(
B2θ̃(t),−

dθ̃

dt
(t)−A1θ̃(t)

)
H

= − 1

2η

d

dt

∥∥B1/2
2 θ̃(t)

∥∥2

H
− 1

η
(B2θ̃(t), A1θ̃(t))H .

(5.4)

From (5.1)-(5.4) and (A4), there exists a constant C3 = C3(T ) > 0 such that

1

2
‖L1/2 dϕ̃

dt
(t)‖2H +

ω2,1

2
‖ϕ̃(t)‖2V +

1

2η
‖B1/2

2 θ̃(t)‖2H

≤ C3

∫ t

0

‖L1/2 dϕ̃

dt
(s)‖2H ds+ C3

∫ t

0

‖ϕ̃(s)‖2V ds

for a.a. t ∈ (0, T ), whence we obtain that dϕ̃
dt = ϕ̃ = 0 by the Gronwall lemma and

(A2). Then (1.14) leads to

1

2

d

dt
‖θ̃(t)‖2H + (A1θ̃(t), θ̃(t))H = 0. (5.5)

Thus θ̃ = 0. �



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 23

6. Error estimates

Proof of Theorem 1.5. Let h4 be as in Lemma 4.5. Then, putting z := dv
dt , we

derive from the identity dv̂h
dt = zh, the second equation in (1.7) and (1.15) that

1

2

d

dt
‖L1/2(v̂h(t)− v(t))‖2H

= (L(zh(t)− z(t)), v̂h(t)− vh(t))H + (L(zh(t)− z(t)), vh(t)− v(t))H

= (L(zh(t)− z(t)), v̂h(t)− vh(t))H − (B1(vh(t)− v(t)), vh(t)− v(t))H

− (A2(ϕh(t)− ϕ(t)), vh(t)− v(t))H − (Φϕh(t)− Φϕ(t), vh(t)− v(t))H

− (Lϕh(t)− Lϕ(t), vh(t)− v(t))H + (B2(θh(t)− θ(t)), vh(t)− v(t))H .

(6.1)

The boundedness of the operator L : H → H implies the existence of a constant
C1 > 0 such that

(L(zh(t)− z(t)), v̂h(t)− vh(t))H ≤ ‖L(zh(t)− z(t))‖H‖v̂h(t)− vh(t)‖H
≤ C1‖zh(t)− z(t)‖H‖v̂h(t)− vh(t)‖H

(6.2)

for a.a. t ∈ (0, T ) and all h ∈ (0, h4). From the identities vh = dϕ̂h

dt , v = dϕ
dt and

the boundedness of the operator A∗2 : V → V ∗, there exists a constant C2 > 0 such
that

− (A2(ϕh(t)− ϕ(t)), vh(t)− v(t))H

= −〈A∗2(ϕh(t)− ϕ̂h(t)), vh(t)− v(t)〉V ∗,V −
1

2

d

dt
‖A1/2

2 (ϕ̂h(t)− ϕ(t))‖2H

≤ C2‖ϕh(t)− ϕ̂h(t)‖V ‖vh(t)− v(t)‖V −
1

2

d

dt
‖A1/2

2 (ϕ̂h(t)− ϕ(t))‖2H

(6.3)

for a.a. t ∈ (0, T ) and all h ∈ (0, h4). From (A6), Lemma 4.1, Young’s inequality
and (A2), there exists a constant C3 = C3(T ) > 0 such that

− (Φϕh(t)− Φϕ(t), vh(t)− v(t))H

≤ CΦ(1 + ‖ϕh(t)‖pV + ‖ϕ(t)‖qV )‖ϕh(t)− ϕ(t)‖V ‖vh(t)− v(t)‖H
≤ C3‖ϕh(t)− ϕ(t)‖V ‖vh(t)− v(t)‖H

≤ C3

2
‖ϕh(t)− ϕ(t)‖2V +

C3

2
‖vh(t)− v(t)‖2H

≤ C3‖ϕh(t)− ϕ̂h(t)‖2V + C3‖ϕ̂h(t)− ϕ(t)‖2V

+ C3‖vh(t)− v̂h(t)‖2H +
C3

cL
‖L1/2(v̂h(t)− v(t))‖2H

(6.4)

for a.a. t ∈ (0, T ) and all h ∈ (0, h4). By (A11), the continuity of the embedding
V ↪→ H, Young’s inequality and (A2), there exists a constant C4 > 0 such that

− (Lϕh(t)− Lϕ(t), vh(t)− v(t))H

≤ C4‖ϕh(t)− ϕ(t)‖V ‖vh(t)− v(t)‖H

≤ C4

2
‖ϕh(t)− ϕ(t)‖2V +

C4

2
‖vh(t)− v(t)‖2H

≤ C4‖ϕh(t)− ϕ̂h(t)‖2V + C4‖ϕ̂h(t)− ϕ(t)‖2V

+ C4‖vh(t)− v̂h(t)‖2H +
C4

cL
‖L1/2(v̂h(t)− v(t))‖2H

(6.5)



24 S. KURIMA EJDE-2020/96

for a.a. t ∈ (0, T ) and all h ∈ (0, h4). From the first equation in (1.7) and (1.14) it
follows that

(B2(θh(t)− θ(t)), vh(t)− v(t))H

= −1

η

(
B2(θh(t)− θ(t)), dθ̂h

dt
(t)− dθ

dt
(t)
)
H

− 1

η
(B2(θh(t)− θ(t)), A1(θh(t)− θ(t)))H

= −1

η

〈
B∗2(θh(t)− θ̂h(t)),

dθ̂h
dt

(t)− dθ

dt
(t)
〉
V ∗,V

− 1

2η

d

dt
‖B1/2

2 (θ̂h(t)− θ(t))‖2H

− 1

η
(B2(θh(t)− θ(t)), A1(θh(t)− θ(t)))H .

(6.6)

From (6.1)-(6.6), the integration over (0, t), where t ∈ [0, T ], the boundedness of
the operator B∗2 : V → V ∗, (1.11)-(1.13), Lemmas 4.3, 4.6, and the inequalities
0 < h4 < 1, there exists a constant C5 = C5(T ) > 0 such that

1

2
‖L1/2(v̂h(t)− v(t))‖2H +

1

2
‖A1/2

2 (ϕ̂h(t)− ϕ(t))‖2H

+

∫ t

0

‖B1/2
1 (vh(s)− v(s))‖2H ds+

1

2η
‖B1/2

2 (θ̂h(t)− θ(t))‖2H

+
1

η

∫ t

0

(B2(θh(s)− θ(s)), A1(θh(s)− θ(s)))H ds

≤ C5h+ C5

∫ t

0

‖ϕ̂h(s)− ϕ(s)‖2V ds+ C5

∫ t

0

‖L1/2(v̂h(s)− v(s))‖2H ds

(6.7)

for all t ∈ [0, T ] and all h ∈ (0, h4). From dϕ̂h

dt = vh, dϕ
dt = v, Young’s inequality,

(A2) and the continuity of the embedding V ↪→ H, there exists a constant C6 > 0
such that

1

2

d

dt
‖ϕ̂h(t)− ϕ(t)‖2H

= (vh(t)− v(t), ϕ̂h(t)− ϕ(t))H

≤ 1

2
‖vh(t)− v(t)‖2H +

1

2
‖ϕ̂h(t)− ϕ(t)‖2H

≤ ‖vh(t)− v̂h(t)‖2H +
1

cL
‖L1/2(v̂h(t)− v(t))‖2H + C6‖ϕ̂h(t)− ϕ(t)‖2V

(6.8)

for a.a. t ∈ (0, T ) and all h ∈ (0, h4). Thus, integrating (6.8) over (0, t), where t ∈
[0, T ], we deduce from (6.7) and (A3) that there exists a constant C7 = C7(T ) > 0



EJDE-2020/96 TIME DISCRETIZATION OF AN ABSTRACT PROBLEM 25

such that

1

2
‖L1/2(v̂h(t)− v(t))‖2H +

ω2,1

2
‖ϕ̂h(t)− ϕ(t)‖2V

+

∫ t

0

‖B1/2
1 (vh(s)− v(s))‖2H ds+

1

2η
‖B1/2

2 (θ̂h(t)− θ(t))‖2H

+
1

η

∫ t

0

(B2(θh(s)− θ(s)), A1(θh(s)− θ(s)))H ds

≤ C7h+ C7

∫ t

0

‖ϕ̂h(s)− ϕ(s)‖2V ds+ C7

∫ t

0

‖L1/2(v̂h(s)− v(s))‖2H ds

(6.9)

for all t ∈ [0, T ] and all h ∈ (0, h4).
Next the first equation in (1.7) and (1.14) lead to

1

2

d

dt
‖θ̂h(t)− θ(t)‖2H

= −η(vh(t)− v(t), θ̂h(t)− θ(t))H − (A1(θh(t)− θ(t)), θ̂h(t)− θh(t))H

− 〈A∗1(θh(t)− θ(t)), θh(t)− θ(t)〉V ∗,V .

(6.10)

Here we use the Young inequality and (A2) to infer that

− (vh(t)− v(t), θ̂h(t)− θ(t))H

≤ 1

2
‖vh(t)− v(t)‖2H +

1

2
‖θ̂h(t)− θ(t)‖2H

≤ ‖vh(t)− v̂h(t)‖2H + ‖v̂h(t)− v(t)‖2H +
1

2
‖θ̂h(t)− θ(t)‖2H

≤ ‖vh(t)− v̂h(t)‖2H +
1

cL
‖L1/2(v̂h(t)− v(t))‖2H +

1

2
‖θ̂h(t)− θ(t)‖2H .

(6.11)

We have from (A3) that

− 〈A∗1(θh(t)− θ(t)), θh(t)− θ(t)〉V ∗,V
≤ −ω1,1‖θh(t)− θ(t)‖2V + ‖θh(t)− θ(t)‖2H
≤ −ω1,1‖θh(t)− θ(t)‖2V + 2‖θh(t)− θ̂h(t)‖2H + 2‖θ̂h(t)− θ(t)‖2H .

(6.12)

Hence, owing to (6.10)-(6.12), the integration over (0, t), where t ∈ [0, T ], (1.12),
(1.13), Lemmas 4.3 and 4.6, there exists a constant C8 = C8(T ) > 0 such that

1

2
‖θ̂h(t)− θ(t)‖2H + ω1,1

∫ t

0

‖θh(s)− θ(s)‖2V ds

≤ C8h+ C8

∫ t

0

‖L1/2(v̂h(s)− v(s))‖2H ds+ C8

∫ t

0

‖θ̂h(s)− θ(s)‖2H ds
(6.13)

for all t ∈ [0, T ] and all h ∈ (0, h4).
Therefore we can obtain Theorem 1.5 by combining (6.9), (6.13) and by applying

the Gronwall lemma. �

Acknowledgments. The author would like to thank the anonymous referees for
their careful reading, helpful comments and suggestions. The author is supported
by JSPS Research Fellowships for Young Scientists (No. 18J21006).



26 S. KURIMA EJDE-2020/96

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, A. Favini; Time discretization of nonlinear Cauchy problems applying to mixed

hyperbolic-parabolic equations, Internat. J. Math. Math. Sci., 19 (1996), 481–494.

[4] P. Colli, S. Kurima; Time discretization of a nonlinear phase-field system in general domains,
Commun. Pure Appl. Anal., 18 (2019), 3161–3179.

[5] M. Grasselli, V. Pata; Existence of a universal attractor for a parabolic-hyperbolic phase-field
system, Adv. Math. Sci. Appl., 13 (2003), 443-459.

[6] M. Grasselli, V. Pata; Asymptotic behavior of a parabolic-hyperbolic system, Comm. Pure

Appl. Anal., 3 (2004), 849-881.
[7] M. Grasselli, H. Petzeltová, G. Schimperna; Convergence to stationary solutions for a

parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827–838.

[8] J. W. Jerome; Approximations of Nonlinear Evolution Systems, Mathematics in Science and
Engineering 164, Academic Press Inc., Orlando, 1983.

[9] S. Kurima; Time discretization of an initial value problem for a simultaneous abstract evo-

lution equation applying to parabolic-hyperbolic phase-field systems, ESAIM Math. Model.
Numer. Anal., 54 (2020), 977–1002.

[10] A. Matsubara, T. Yokota; Applications of the Hille–Yosida theorem to the linearized equa-

tions of coupled sound and heat flow, AIMS Mathematics 1 (2016), 165–177.
[11] J. Simon; Compact sets in the space Lp(0, T ;B), Ann. Mat. Pura Appl., (4) 146 (1987),

65–96.
[12] H. Wu, M. Grasselli, S. Zheng; Convergence to equilibrium for a parabolic-hyperbolic phase-

field system with Neumann boundary conditions, Math. Models Methods Appl. Sci., 17

(2007), 125–153.
[13] H. Wu, M. Grasselli, S. Zheng; Convergence to equilibrium for a parabolic-hyperbolic phase-

field system with dynamical boundary condition, J. Math. Anal. Appl., 329 (2007), 948-976.

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. Examples
	3. Existence of discrete solutions
	4. Uniform estimates for (1.7) and passage to the limit
	5. Uniqueness for (1.1) 
	6. Error estimates
	Acknowledgments

	References