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Article

Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

Karel Van Bockstal

Research Group NaM2, Department of Electronics and Information systems, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium; karel.vanbockstal@ugent.be

Received: 2 July 2020; Accepted: 30 July 2020 ; Published: 3 August 2020 Abstract: We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative.

Keywords: time-fractional wave equation; distributed-order; non-autonomous; time discretization;

existence; uniqueness

MSC:35A15; 35R11; 47G20; 65M12

1. Introduction

We consider a bounded domainΩ ⊂ Rd(d ∈ N) with a Lipschitz continuous boundary ∂Ω.

We denote the final time byT, and we defineQT :=×(0,T]andΣT:=∂Ω×(0,T]. The objective of this paper is to show the existence of a uniqueufor given f, ˜u0and ˜v0to the fractional wave equation of time distributed-order (DO) with nonlinear source term given by

















D(µ)

t u

(x,t) +L(x,t)u(x,t) = f(x,t) +F(u(x,t)) (x,t)QT,

u(x,t) =0 (x,t)ΣT,

u(x, 0) =u˜0(x) x∈Ω,

tu(x, 0) =v˜0(x) x∈Ω.

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D(µ)

t ustands for the time DO fractional derivative defined by

D(µ)

t u

(x,t) = Z 2

1

βtu

(x,t)µ(β)dβ, (x,t)QT, (2) with weight functionµ:[1, 2]Rsatisfying

µ∈L1(1, 2), µ>0, µ.0, and with∂βtuthe Caputo derivative of orderβ∈(1, 2)defined by [1,2]

βtu

(x,t) = Z t

0

(t−s)1β

Γ(2−β) ssu(x,s) ds, (x,t)QT,

Mathematics2020,8, 1283; doi:10.3390/math8081283 www.mdpi.com/journal/mathematics

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whereΓdenotes the Gamma function. In this paper, we rewrite the DO fractional derivative defined in (2) as

D(µ)

t u

(x,t) = (k∗∂ttu) (x,t), (x,t)QT, where

k(t) = Z 2

1

t1β

Γ(2−β)µ(β)dβ,

and the symbol ‘∗’ stands for the convolution product defined by (k∗z) (t) = R0tk(t−s)z(s) ds.

The second-order linear differential operatorLis defined as follows

L(x,t)u(x,t) =−∇ ·(A(x,t)u(x,t) +b(x,t)u(x,t)) +c(x,t)u(x,t), (3) where ((x,t)QT)

A(x,t) =ai,j(x,t)

i,j=1,...,d, b(x,t) = (b1(x,t),b2(x,t),. . .,bd(x,t)).

The system (1) can be used to model the propagation of mechanical waves in viscoelastic materials [3–9]. We mention the most important results available in literature related to the fractional wave equations, which are also connected to the subject of this paper. The existence and uniqueness of a solution to autonomous (time-independent elliptic part) constant-order fractional wave equations is studied in References [10,11]. A fundamental solution to the fractional wave equation of constant-order is determined in Reference [12] and to the Cauchy problem for the 1D DO diffusion-wave equation in Reference [13]. In Reference [14], a Cauchy problem for a time-fractional DO multi-dimensional diffusion-wave equation is investigated in the space of tempered distributions. The unique solvability of the Cauchy problem for inhomogeneous DO differential equations in a Banach space with a linear bounded operator in the right-hand side is studied in Reference [15]. In Reference [16], the authors give existence and uniqueness of weak solutions results together with energy estimates for problem (1) (see Theorem 3.2) considering instead ofLthe fractional powers of orders∈ (0, 1)of a self-adjoint (AT=Aandb=0) and uniformly elliptic second order operator with time-independent coefficients (the domainΩis an open, bounded, and convex subset ofRd). Their approach is based on a Galerkin technique (the solution has a representation formula separated in the independent variables), spectral theory and the Mittag-Leffler analysis. However, this type of analysis is not appropriate for analyzing problem (1) as the coefficients in the governing operator are allowed to be time dependent. To the best of our knowledge, no paper deals with the existence of a unique weak solution to problem (1), which is the main goal of this paper. Now, we discuss numerical methods for solving the (DO) fractional wave equations as the method that will be applied in this article also includes a time-discrete scheme for computations. A fully discrete difference Crank-Nicolson scheme is derived for a diffusion-wave system in Reference [17]. A difference scheme for a one-dimensional DO time-fractional wave equation is derived and analyzed [18]. Two alternating direction implicit difference schemes for solving the two-dimensional time DO wave equations were developed in Reference [19]. The element-free Galerkin method is used in Reference [20] to solve the two-dimensional DO time-fractional diffusion-wave equation. Finite difference schemes for a multidimensional time-fractional wave equation of DO with a nonlinear source term are studied in Reference [21]. Two temporal second-order schemes are derived and analyzed for the time multi-term fractional diffusion-wave equation based on the order reduction technique in Reference [22]. A fast and linearized finite difference method to solve a nonlinear time-fractional wave equation with multi fractional orders is studied in Reference [23].

In all these articles it is usually assumed that the solutionu(t)(ignoring the space variable) lies in C2([0,T])or C3([0,T]), but it is well-known that∂ttublows up ast→0+in some practical situations [16,24,25]. An overview of numerical methods for time-fractional evolution equations with nonsmooth data is given in Reference [26].

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However, in this contribution, we follow a standard procedure for showing the existence of a weak solution (time-discretization method), which is recently applied in Reference [27] for a non-autonomous time fractional diffusion equation of DO. We first state this result below, see (Reference [27] Theorem 3.1).

Theorem 1 (Fractional diffusion equation (of DO)). Consider

βtu

(t) = (k∗∂tu) (t) = Rt

0 (ts)β

Γ(1β)su(s) ds withβ∈(0, 1)or

D(µ)

t u

(t) = (k∗∂tu) (t) =R01βtu

(t)µ(β)dβ(i.e., DO) in problem (1)with F≡0. Assume that

• µ∈L1(0, 1),µ>0,µ.0(if DO);

• u˜0∈H10();

• f ∈H1

(0,T), H10(), or f ∈L2

(0,T), H10()withk∂tf(t)kH1

0() 6Ctαfor all t∈ (0,T]and some constantα∈(0, 1);

A∈ L

QT

d×d

is uniformly elliptic withAT =Aand∂tA∈ L

QT

d×d

;

b=0;

• c∈L QT

with c>0in QT, and∂tc∈L QT

.

Then, there exists a unique weak solution u to the problem (1) with u ∈ C

[0,T], H10(Ω) L

(0,T), H10()and k∗∂tu∈L2

(0,T), H10().

We will show that for the fractional wave equation (of DO) the analysis goes through without information about∂tf and whenb , 0(i.e., non-symmetric L). In Section2, we define the weak formulation of (1). Afterwards, in Section 3, we establish the existence of a unique solution to the variational problem by applying Rothe’s method, which is the most general result in literature concerning the well-posedness of the non-autonomous time-fractional wave equation. The obtained regularity onutakes into account the possible singularity att=0 in case of non-smooth solutions.

We finally note that when showing the existence and uniqueness of a solution, we will use Proposition 6 and 10 from Reference [28], which can be used independently of the type of operator (diffusion/wave).

Remark 1(Additional notations). We denote by(·,·)the standard inner product inL2()and byk·kits induced norm. The spaceL2(∂Ω)on the boundary is defined analogously as the spaceL2(). The multi-indexα is a d-dimensional vector withα= (α1,. . .,αd),αi≥0withαi∈Z. The length ofαis given by|α|1=Pdi=1αi. The function u∈L2()is an element ofHk()if all generalized (weak) derivatives Dαu of u up to order k exist and belong toL2(), that is,

Hk() =nu∈L2():Dαu∈L2(),|α|16ko . The spaceHk(Ω)is called a Sobolev space of order k and is equipped with the norm

kuk

Hk()=







 X

|α|16k

Dαu

2 L2(Ω)







12

.

The function spaceH10(Ω)is defined as

H10():={u∈H1():γ(u) =0}={u∈H1():u=0on∂Ω}, wereγdenotes the trace map fromH1()toL2(∂Ω).

Next, we consider an abstract Banach space X with normk·kXand p∈[1,∞). The spaceLp((0,T),X) consists of measurable functions u:(0,T)X such that

kukLp((0,T),X)= Z T

0

u(t)pXdt

!1/p

<∞.

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The spaceC([0,T],X)consists of continuous functions u:[0,T]X satisfying kukC([0,T],X)= max

t[0,T]

u(t)X<.

The spaceL((0,T),X)consists of all measurable functions u:(0,T)X that are essentially bounded.

The spaceH1((0,T),X)consists of functions u:(0,T)X such that the weak derivative u0exists and

kuk

H1((0,T),X)=

"Z T 0

u(t)2

X+u0(t)2

X

dt

#12

<∞.

The values C,εand Cεare considered to be generic and positive constants (independent of the discretization parameter) throughout the paper, whereεis arbitrarily small and Cεarbitrarily large, that is, Cε=C

1+ε+1ε. The same notation for different constants is used, but the meaning should be clear from the context.

2. Weak Formulation

First, we define the variational formulation of problem (1) as follows: searchu∈L2

(0,T), H10(Ω) with k∗∂ttu ∈ L2

(0,T), H10() such that for almost all (a.a.) t ∈ (0,T) and for all ϕ ∈ H10() it holds that

h(kttu) (t),ϕi

H10()×H10()+L(t) (u(t),ϕ) = (f,ϕ) + (F(u(t)),ϕ), (4) where a bilinear formLis associated with the differential operatorLdefined in (3) as follows

L(t) (u(t),ϕ):= (L(t)u(t),ϕ)

= (A(t)u(t) +b(t)u(t),∇ϕ) + (c(t)u(t),ϕ), withu(t),ϕ∈H10(). For completeness, we summarize the properties of the singular kernelkbelow

• kis strongly positive definite sincek(t)>0 for allt>0,∂tk(t)60 for allt>0 and∂ttk(t)>0 for allt>0 [29];

• k∈L1(0,T)since Z T

0

k(t)dt= Z 2

1

µ(β) Γ(2−β)

Z T 0

t1β

!

dtdβ= Z 2

1

µ(β) Γ(3−β)T

2βdβ62µmax{1,T}, (5)

using thatΓ(z)> 12forz∈(1, 2)andµ:=R12µ(β)dβ;

• ∂tk∈L1loc(0,T)for any[t1,t2](0,T)since Z t2

t1

tk(t)dt= Z 2

1

µ(β) Γ(2β)

t11β−t12β

dβ62µmax{1,t11},

using thatΓ(z)>Γ(1) =1 forz∈(0, 1).

We state the following assumptions on the data functions in (3), which will be used throughout the paper. The matrixA=ai j(x,t)is ad×dmatrix-valued function such thatA

L QTd×d

is uniformly elliptic, that is, there exists a strict positive constantαsuch that

Xd i,j=1

ai j(x,t)ξiξj >α|ξ|2, for a.a. (x,t)QTand for allξ∈Rd.

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We suppose thatbL QT

andc∈L QT

such that

c(x,t) kbk2

L(QT)

2α >0, (x,t)QT. Moreover, we suppose that

AT=A;

• ∂tA∈ L

QT

d×d

;

• (∇ ·b)(t)L()for allt∈(0,T);

• ∂tb∈ L

QT

d

. Therefore, we have that

L(t) (u,ϕ)6Ckuk

H10()

ϕ

H1

0(), ∀u,ϕ∈H10(); L(t) (ϕ,ϕ)> α

2 ∇ϕ

2, ∀ϕ∈H10(Ω). (6)

It follows from the Friedrichs inequality that the bilinear formLis H10()-elliptic. In the next section, we study the existence and uniqueness of a solution.

3. Existence of a Solution

We employ a semidiscretization in time based on Rothe’s method to address the existence of a weak solution to (1). First, we discretize the time interval[0,T]inton∈Nequidistant subintervals[ti1,ti] with uniform time stepτ= Tn <1, that is,ti =iτ,i=0,. . .,n. We denote the approximation ofuat timet=ti(06i6n) byui, and we approximate the first and second order time derivative at time t=tiby the backward Euler finite-difference formulas

tu(ti)δui= uiui1

τ , ∂ttu(ti)δ2ui = δuiδui1

τ = uiui1

τ2 −δui1

τ , 16i6n.

These notations are also used for any functionz,u. Next, we define the time discrete convolution as follows

(k∗z)(ti)(k∗z)i:= Xi l=1

ki+1lzlτ, (7)

with

(k∗z)0:=0.

We approximate problem (4) at timet=tiby: Findui∈H10(),i=1, 2,. . .,n, such that h(kδ2u)i,ϕiH1

0()×H10()+Li(ui,ϕ) = (fi,ϕ) + (F(ui1),ϕ), ϕH10(). (8) Employing the time discrete convolution (7), the discrete problem can be equivalently written as ai(ui,ϕ) =hF˜i,ϕi, ∀ϕ∈H10(), (9) with

ai(u,ϕ):= k(τ)

τ (u,ϕ) +Li(u,ϕ), and

hF˜i,ϕi:= (fi,ϕ) + (F(ui1),ϕ) + k(τ)

τ (ui1,ϕ) +k(τ) (δui1,ϕ)

i1

X

l=1

ki+1l

δ2ul,ϕ τ.

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The well-posedness of this problem under appropriate assumptions on the initial data follows inductively from the Lax-Milgram lemma and it is stated in the following lemma.

Lemma 1. Suppose thatu˜0 ∈ L2(),v˜0 ∈ L2() and f ∈ L

(0,T), L2(). Moreover, assume that the nonlinear source term F:RRis Lipschitz continuous, that is,

F(s1)F(s2)6LF|s1−s2|, si∈R,

where LFis a positive constant. Then, for any i=1, 2,. . .,n, there exists a unique ui∈H10()solving(8).

3.1. A Priori Estimates

In this subsection, we derive the a priori estimates that we require to be able to show the existence of a solution. Consider the evolution triple H10() L2() (L2()) H10(). Then, from ([30], Lemma 3.2), it follows for a sequence(zi)iNin H10()that

j

X

i=1

hδ(k∗z)i,zii

H10()×H10()τ=

j

X

i=1

(δ(k∗z)i,zi)τ> 1 2

k∗ kzk2

j+1 2

j

X

i=1

kikzik2τ, j∈N, (10) with

k∗ kzk2

j:=

j

X

l=1

kj+1lkzlk2τ.

Lemma 2. Let the assumptions of Lemma1be fulfilled. Moreover, assume thatu˜0∈H10(Ω). Then, positive constants C andτ0exist such that for anyτ < τ0and for every j=1, 2,. . .,n, the following relation holds

k∗ kδuk2

j+

j

X

i=1

kikδuik2τ+

j

X

i=1

kδuik2τ+uj

2+

j

X

i=1

k∇ui− ∇ui1k26C.

Proof. We putϕ= δuiτin (8) and sum the result up fori =1,. . .,jwith 16 j6n. Employing the following relation for any sequence{zi}i

N⊂L2():

δ(k∗z)i(x) =ki(x)z0(x) + (k∗δz)i(x), for a.ax∈Ω, (11) we obtain that

j

X

i=1

(δ(k∗δu)i,δui)τ+

j

X

i=1

Li(ui,δui)τ=

j

X

i=1

(fi,δui)τ+

j

X

i=1

(F(ui1),δui)τ+

j

X

i=1

(ki0,δui)τ. (12) For the first term on the LHS, we get that

j

X

i=1

(δ(k∗δu)i,δui)τ(10)> 1 2

k∗ kδuk2

j+1 4

j

X

i=1

kikδuik2τ+k(T) 4

j

X

i=1

kδuik2τ.

We use per partes formula for a symmetric bilinear form for the second term on the LHS of (12):

j

X

i=1

r(ti;zi,zi−zi1) =1

2r(tj;zj,zj)1

2r(0;z0,z0) + 1 2

j

X

i=1

r(ti;δzi,δzi)τ2δr(ti;zi1,zi1)τ.

Therefore, by the symmetry ofA, we have that

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j

X

i=1

(Ai∇ui,∇δui)τ= 1 2

Aj∇uj,∇uj

−1

2(A0∇u˜0,∇u˜0)1 2

j

X

i=1

(δAiui1,∇ui1)τ

+1 2

j

X

i=1

(Ai(ui− ∇ui1),∇ui− ∇ui1), and thus

j

X

i=1

(Ai∇ui,∇δui)τ> α 2 ∇uj

2−C−C

j1

X

i=1

k∇uik2τ+α 2

j

X

i=1

k∇ui− ∇ui1k2. Using theε-Young inequality and the Friedrichs inequality, we get that

j

X

i=1

(ciui,δui)τ

6Cε1

j

X

i=1

k∇uik2τ+ε1

j

X

i=1

kδuik2τ,

and

j

X

i=1

(biui,∇δui)τ=

j

X

i=1

(∇ ·(biui),δui)τ

6Cε2

j

X

i=1

k∇uik2τ+ε2

j

X

i=1

kδuik2τ.

The terms in the RHS of (12) can be estimated as follows

j

X

i=1

(fi,δui)τ

6Cε3+ε3

j

X

i=1

kδuik2τ,

j

X

i=1

(F(ui1),δui)τ

6Cε4+Cε4

j1

X

i=1

k∇uik2τ+ε4

j

X

i=1

kδuik2τ

and

j

X

i=1

(ki0,δui

6Cε55 j

X

i=1

kikδuik2τ,

sincek∈L1(0,T). Collecting the previous estimates gives 1

2

k∗ kδuk2

j+ 1

4−ε5

Xj

i=1

kikδuik2τ+ k(T)

4 −ε1−ε2−ε3−ε4

! j X

i=1

kδuik2τ +α

2 ∇uj

2+α 2

j

X

i=1

kui−ui1k2

H1(Ω)6Cε3+Cε4+Cε5+ (Cε1+Cε2+Cε4)

j

X

i=1

k∇uik2τ.

We fix(εi)5i=1sufficiently small such thatε5 < 14 andP4

i=1εi < k(T)4 , and afterwards we apply Grönwall’s lemma to conclude the proof.

Remark 2. In this paper, we have supposed that f ∈L

(0,T), L2(). If f ∈L

(0,T), H10(), then the term containing f on the right-hand side of (12) cannot be estimated directly. Instead, we can assume that f ∈ H1

(0,T), H10(Ω), since in that case we can estimate this term by using the following partial summation rule

j

X

i=1

hfi,δuii

H10(Ω)×H10(Ω)τ=hfj,uji

H10(Ω)×H10(Ω)− hf0,u0i

H10(Ω)×H10(Ω)

j

X

i=1

hδfi,ui1i

H10(Ω)×H10(Ω)τ,

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i.e.,

j

X

i=1

hfi,δuii

H10()×H10()τ

6Cε+εuj

2+C

j1

X

i=1

k∇uik2τ.

We note also that the condition on f in Theorem1cannot be relaxed to f ∈L

(0,T), L2()following the approach in (Reference [27] Lemma 3.3 and 3.4). However, it is clear that Theorem1is also satisfied when f is satisfying one of the conditions on f with the spaceH10(Ω)replaced byL2(Ω).

Corollary 1. Let the assumptions of Lemma2be fulfilled. Then, there exist positive constants C such that for every j=1, 2,. . .,n, the following relation holds

(k∗δ2u)j

H10(Ω) 6C.

Proof. The estimate follows from

(k∗δ2u)iH1

0(Ω) = sup

kϕk

H10()=1

h(k∗δ2u)i,ϕi

H10(Ω)×H10(Ω)

= sup

kϕk

H10()=1

(fi,ϕ) + (F(ui1),ϕ)− Li(ui,ϕ) 6

fi

+C+Ckui1k+Ck∇uik, and the result of Lemma2.

3.2. Convergence

In this subsection, the existence of a weak solution is proved using Rothe’s method. We define the following piecewise linear in time functionsun :[0,T]L2()

un:[0,T]L2():t7→





˜

u0 t=0

ui1+ (t−ti1)δui t∈(ti1,ti], 16i6n, and the piecewise constant in time functionsun,vn, ˜un :[0,T]L2(Ω):

un:[0,T]L2(Ω):t7→





˜

u0 t=0

ui t∈(ti1,ti], 16i6n;

vn :[0,T]L2(Ω):t7→





˜

v0 t=0

δui t∈(ti1,ti], 16i6n;

˜

un:[0,T]L2(Ω):t7→





0 t∈[0,τ]

un(t−τ) t∈(ti1,ti], 26i6n.

Similarly, we definekn,Lnand fn. We rewrite Equation (8) on the whole time frame by aid of these Rothe’s functions and Equation (11) as follows

h∂t(k∗δu)n(t)kn(t)v˜0,ϕi

H10(Ω)×H10(Ω)+Ln(t)(un(t),ϕ)

=fn(t),ϕ

+ (F(u˜n(t)),ϕ), ∀ϕ∈H10(). (13)

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Note that

Ln(t) (un(t),ϕ) =an(t)un(t) +bn(t)un(t),∇ϕ

+ (cn(t)un(t),ϕ). We show the existence of a unique weak solution in the following theorem.

Theorem 2(Existence and uniqueness). Suppose that the conditions of Lemma2are fulfilled. Then, there exists a unique weak solution u to the problem (4) with u ∈ C

[0,T], L2(Ω)L

(0,T), H10(Ω)with

tu∈C

[0,T], H10()L2

(0,T), L2()and k∗∂ttu∈L2

(0,T), H10().

Proof. The compact embedding of H10() in L2() follows from the Rellich-Kondrachov theorem ([31] Theorem 6.6-3). Lemma 2 gives that the sequences (un)nN is bounded in L2

(0,T), H10(Ω). Thus we have the existence of an elementuin L2

(0,T), L2(Ω)and a subsequence (unl)lNof(un)nNsuch that

unl →uin L2

(0,T), L2(Ω)asl→ ∞. Moreover, the reflexivity of the space L2

(0,T), H10(Ω)implies the existence of a subsequence (indexed bynlagain) such that

unl *uin L2

(0,T), H10() asl→ ∞. Lemma2gives also thatu∈L

(0,T), H10()and that(tunl =vnl)lNis bounded in the reflexive space L2

(0,T), L2(Ω). Therefore,

vnl =tunl * ∂tuin L2

(0,T), L2() asl→ ∞. (14) Hence,u∈C

[0,T], L2(), see ([32], Lemma 7.3). Finally, from Lemma2, it follows that Z T

0

unl(t)unl(t)2H1

0()dt+ Z T

0

unl(t)nl(t)2H1

0()dt62τnl nl

X

i=1

k∇ui− ∇ui1k26Cτnl, that is, we have that

unl, ˜unl →uin L2

(0,T), L2() asl→ ∞. and

unl *uin L2

(0,T), H10() asl→ ∞.

Next, we integrate Equation (13) in time over(0,η) (0,T)for the resulting subsequence to obtain that

h(k∗δu)nl(η),ϕi

H10(Ω)×H10(Ω)− Z η

0

hv˜0knl(t),ϕi

H10(Ω)×H10(Ω)dt +

Z η

0

Ln

l(t)(unl(t),ϕ)dt=

Z η

0

fnl(t),ϕ dt+

Z η

0

F(u˜nl(t)),ϕ

dt. (15) We have that

AnA

→0, bnb

→0 and cn−c

→0 almost everywhere (a.e.) inQTas n→ ∞. Using the results above, we obtain forη∈(0,T)that

Z η

0

Ln

l(t)(unl(t),ϕ)dt−

Z η

0

L(t)(u(t),ϕ)dt

→0 asn→ ∞,

(10)

Z η

0

fn

l(t),ϕdt−

Z η

0

(f,ϕ)dt

→0 asn→ ∞,

Z η

0

F(u˜nl(t)),ϕ dt−

Z η

0

(F(u(t)),ϕ)dt

→0 asn→ ∞.

Now, we explain the limit transition in the first term of (15) by proving in two steps that

llim→∞

Z T 0

h(k∗δu)nl(t),ϕi

H10()×H10()dt− Z T

0

h(k∗vnl)(t),ϕi

H10()×H10()dt

=0. (16)

These two steps are given by (i) lim

l→∞

Z T 0

h(k∗δu)nl(t)dt− Z T

0

(k∗δu)n

l(t),ϕi

H10(Ω)×H10(Ω)dt

=0;

(ii) lim

l→∞

Z T 0

h(k∗δu)n

l(t)dt−

Z T 0

(k∗vnl)(t),ϕi

H10()×H10()dt

=0.

From Corollary1and the Lebesgue dominated theorem, we obtain the limit transition (i) as follows Z T

0

h(k∗δu)nl(t)(k∗δu)n

l(t),ϕi

H10(Ω)×H10(Ω)

dt

=

nl

X

i=1

Z ti ti1

h(t−ti)δ(k∗δu)i,ϕi

H10()×H10()

dt

(11)

6

nl

X

i=1

τ2nl

h(k∗δ2u)iki0,ϕi

H10(Ω)×H10(Ω)

6Cτnl +C Z 2

1

τ2nlβµ(β)dβl−→→∞0.

For the limit transition (ii), we see that (dteτ=tiwhent∈(ti1,ti]) Z T

0

h(k∗δu)n

l(t)(k∗vnl)(t),ϕi

H10(Ω)×H10(Ω)

dt

= Z T

0

h(knl∗vnl)(dteτ)(k∗vnl)(t),ϕi

H10(Ω)×H10(Ω)

dt 6

Z T 0

h

Z t 0

knl(dteτ−s)k(t−s)vnl(s) ds,ϕi

H10(Ω)×H10(Ω)

dt +

Z T 0

h

Z dteτ t

knl(dteτ−s)vnl(s) ds,ϕi

H10(Ω)×H10(Ω)

dt.

We apply two times Hölder’s inequality on the previous inequality and we obtain that Z T

0

h(k∗δu)n

l(t)(k∗vnl)(t),ϕiH1

0()×H10()

dt 6

ϕ H1

0(Ω)

"Z T 0

Z t 0

knl(dteτ−s)k(t−s) ds

! dt

#12

×

"Z T 0

Z t 0

knl(dteτ−s)k(t−s)

vnl(s)2H1 0() ds

! dt

#12

(11)

+ϕ H1

0(Ω)





 Z T

0





 Z dteτ

t

knl(dteτs) ds





dt





12





 Z T

0





 Z dteτ

t

knl(dteτ−s)vnl(s)2H1 0(Ω) ds





dt





12

. From Lemma2, it follows that (fort∈(ti1,ti])

Z dteτ t

knl(dteτ−s)vnl(s)2H1

0() ds6C Z ti

0

knl(ti−s)vnl(s)2 ds=C

k∗ kδuk2

i6C, and

Z T

0

Z t 0

knl(dteτ−s)k(t−s)

vnl(s)2H1 0(Ω) ds

! dt

(?)

6 kkk

L1(0,T)

vnl

L2((0,T),H10(Ω))+ Z T

0

Z t 0

knl(dteτ−s)vnl(s)2H1 0(Ω) ds

! dt6C, where we employed Young’s inequality for convolutions at position(?). Moreover, fort∈(ti1,ti], by the Lebesgue dominated theorem, we obtain that

Z dteτ t

knl(dteτs) ds= Z tit

0







 Z 2

1

τ1nlβ

Γ(2−β)µ(β)dβ







 dξ6

Z 2 1

τ2nlβµ(β)dβ→0 asτnl →0,

sinceΓ(z)>Γ(1) =1 for allz∈(0, 1). We also have fort∈(ti1,ti]that Z t

0

knl(dteτ−s)knl(t−s) ds

= Z ti1

0

knl(ts)knl(tis) ds+ Z t

ti1

knl(ts)knl(tis) ds

6 Z ti1

0

knl(ti1−s)knl(ti−s) ds+2 Z t

ti1

knl(t−s) ds 6

Z ti1

0

(k(ti1−s)k(ti+1−s)) ds+2 Z t

ti1

k(t−s) ds

6 Z 2

1







 t2i1β

Γ(3−β)+(2τnl)2β Γ(3−β)

t2i+1β

Γ(3−β)+2(t−ti1)2β Γ(3−β)







µ(β)dβ

6C Z 2

1

τ2nlβµ(β)0 asτnl 0,

usingknl 6 k(sincekis decreasing in time),Γ(z)> 12 forz∈ (1, 2)and theα-Hölder continuity of f(x) =xαwhenα∈(0, 1). In a similar way, we get that

Z t 0

knl(t−s)k(t−s) ds6C

Z 2 1

τ2nlβµ(β)dβ→0 asτnl →0 and thus

Z t 0

knl(dteτs)k(ts) ds6C

Z 2 1

τ2nlβµ(β)dβ→0 asτnl →0.

Hence, the limit transition (ii) is valid. Now, we integrate Equation (15) again in time over η∈(0,ξ)(0,T), that is,

Z ξ

0

h(k∗δu)nl(η),ϕi

H10(Ω)×H10(Ω)dη− Z ξ

0

Z η

0

hv˜0knl(t),ϕi

H10(Ω)×H10(Ω)dtdη

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