The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model

0.21 MB·English

by  H. A. Erbay

The Cauchy Problem for a One Dimensional Nonlinear Elastic
Peridynamic Model
H. A. Erbay1, A. Erkip2, G. M. Muslu3∗
1 Faculty of Artsand Sciences, OzyeginUniversity,Cekmekoy 34794, Istanbul, Turkey
2 Faculty of Engineering and Natural Sciences, Sabanci University,Tuzla 34956, Istanbul, Turkey
2 3 Department of Mathematics, Istanbul Technical University,Maslak 34469, Istanbul, Turkey
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0
2

n
a
J Abstract

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2 This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic
model describing the dynamic response of an infinitely long elastic bar. The issues of

] local well-posedness and smoothness of the solutions are discussed. The existence of a
P
global solution is proved first in the sublinear case and then for nonlinearities of degree
A
at most three. The conditions for finite-time blow-up of solutions are established.
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Keywords: Nonlocal Cauchy problem, Nonlinear peridynamic equation, Global
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existence, Blow-up.
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2010 MSC: 35Q74,74B20, 74H20, 74H35
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1. Introduction
8
6
In this study, we consider the one-dimensional nonlinear nonlocal partial differential
5
equation, arising in the peridynamic modelling of an elastic bar,
.
0
1
0 utt = α(y−x)w(u(y,t)−u(x,t))dy, x∈R, t>0 (1.1)
ZR
1
:
v with initial data
i u(x,0)=ϕ(x), u (x,0)=ψ(x). (1.2)
X t
r In (1.1)-(1.2) the subscripts denote partial differentiation, u = u(x,t) is a real-valued
a function, the kernel function α is an integrable function on R and w is a twice differen-
tiablenonlinearfunctionwithw(0)=0. Wefirstestablishthelocalwell-posednessofthe
Cauchy problem (1.1)-(1.2), considering four different cases of initial data: (i) continu-
ous and bounded functions, (ii) bounded Lp functions (1 ≤ p ≤ ∞), (iii) differentiable
and bounded functions and (iv) Lp functions whose distributional derivatives arealso in
∗Correspondingauthor. Tel: [phone]Fax: [phone]
Email addresses: [email protected] (H.A.Erbay1),[email protected] (A.
Erkip2),[email protected] (G.M.Muslu3)
Preprint submitted to Journal of DifferentialEquations January 31, 2012
Lp(R). We then extend the results to the case of L2 Sobolev spaces of arbitrary (non-
integer) order for the particular form w(η)=η3. We prove global existence of solutions
for two types of nonlinearities: when w(η) is sublinear and when w(η) = |η|ν−1η for
ν ≤ 3. Lastly, for the general case we provide the conditions under which the solutions
of the Cauchy problem blow-up in finite time.
Equation (1.1) is a model proposed to describe the dynamical response of an infinite
homogeneous elastic bar within the context of the peridynamic formulation of elasticity
theory. The peridynamic theory of solids, mainly proposed by Silling [1], is an alterna-
tive formulation for elastic materials and has attracted attention of a growing number
of researchers. The most important feature of the peridynamic theory is that the force
acting on a material particle, due to interaction with other particles, is written as a
functional of the displacement field. This means the peridynamic theory is a nonlocal
continuumtheoryandregardingnonlocalityitbearsastrongresemblancetomoretradi-
tional theories of nonlocalelasticity, which are principally based on integral constitutive
relations [2, 3, 4]. As in other nonlocal theories of elasticity, the main motivation is
to propose a generalized elasticity theory that involves the effect of long-range internal
forces of molecular dynamic, neglected in the conventional theory of elasticity. Another
feature of the peridynamic theory is that the peridynamic equation of motion does not
involve spatial derivatives of the displacement field. The absence of spatial derivatives
of the displacement field in the equation of motion makes possible to use the peridy-
namic equations even at points of displacement discontinuity. Furthermore, in contrast
to the conventionaltheory of elasticity, the peridynamic theory predicts dispersive wave
propagation as a property of the medium even if the geometry does not define a length
scale.
In the peridynamic theory, by assuming a uniform cross-section and the absence of
body forces, the governing equation of an infinitely long elastic bar is given by
ρ u = f(u(y,t)−u(x,t),y−x)dy, (1.3)
0 tt
ZR
wheretheaxisofthebarcoincideswiththecoordinateaxis,amaterialpointontheaxisof
thebarhascoordinatexintheundeformedstate,uandf maybeinterpretedasaverages
oftheaxialdisplacementandtheaxialforcelocatedatanyxattimet,takenoveracross
sectionofthe bar,andρ isdensityofthe barmaterial[5,6]. The spaceintegralin(1.3)
0
implies that the displacement ata generic point is influenced by the displacements of all
particles of the bar (As commonly known, in the conventional theory of elasticity, the
equationgoverningthe dynamic response of aninfinitely long bar is a hyperbolic partial
differential equation that does not involve such a space integral originating from the
nonlocalcharacterofthe peridynamictheory). Equation(1.3)isobtainedbyintegrating
the equation of motion for the axial displacement over the cross-section and dividing
through by the area of the cross-section. The bar is supposed to be composed of a
homogeneous objective microelastic material [1, 5, 6] and its constitutive behavior is
described by the function f. Newton’s thirdlaw imposes the followingrestrictiononthe
form of f:
f(η,ζ)=−f(−η,−ζ) (1.4)
for all relative displacements η =u(y,t)−u(x,t) and relative positions ζ =y−x. For a
2
linear peridynamic material the constitutive relation is given by
f(η,ζ)=α(ζ)η
where α is called the micromodulus function [1, 5, 6]. It follows from (1.4) that α must
be anevenfunction. In[5,6]the dynamicresponseofalinearperidynamicbarhasbeen
investigated and some striking observations that are not found in the classical theory of
elastic bars have been made. Some results on the well-posedness of the Cauchy problem
for the linear peridynamic model have been established in [7, 8, 9, 10]. In spite of its
age, there is quite extensive literature on the linear peridynamic theory.
It is natural to think that more interesting behavior may be observed when the
attentionisconfinedtothe nonlinearperidynamic materials. Fromthis pointofview,to
thebestofourknowledge,thepresentstudyappearstobethefirststudyonmathematical
analysis of nonlinear peridynamic equations. Techniques similar to those in [11, 12,
13] enable us to answer some basic questions, like local well-posedness and lifespan of
solutions, as the groundwork of further analysis of the nonlinear peridynamic problem.
In this study we consider the case in which the constitutive behavior is described by
a class of nonlinear peridynamic models in the separable form:
f(η,ζ)=α(ζ)w(η) (1.5)
where α and w are two functions satisfying the restriction imposed by (1.4). This sepa-
rableform,whileallowingustoexploitthepropertiesofconvolution-basedtechniques,is
not a serious restrictionandit just makes the proofs easierto follow. Our results canbe
carriedovertothecaseofgeneralf(η,ζ). WeillustratethisinTheorem2.8;byimposing
certain differentiability and integrability conditions on f, we prove local well-posedness
for the general nonlinear peridynamic problem. Throughout this study we assume that
α is an integrable even function while w is a differentiable odd function so that (1.4) is
satisfied.
Substitution of the separable form of (1.5) into (1.3) and non-dimensionalization of
the resulting equation (or simply taking the mass density to be 1) gives the governing
equationoftheprobleminitsfinalform(1.1)(Henceforthweusenon-dimensionalquan-
tities but for convenience use the same symbols). The aim of this study is three-fold: to
establish the local well-posedness of the Cauchy problem, to investigate the existence of
a global solution, and to present the conditions for finite-time blow-up of solutions.
The paper is organized as follows. In Section 2, the existence and uniqueness of
the local solution for the nonlinear Cauchy problem is proved by using the contraction
mappingprinciple. ForinitialdatainfractionalSobolevspacesthegeneralcaseseemsto
involvetechnicaldifficultiesandinSection3weconsidertheparticularcasew(η)=η3 in
theL2 Sobolevspacesetting. Wenotethatthecubiccasecanbeeasilygeneralizedtoan
arbitrary polynomial of η. In Section 4, we consider the issue of global existence versus
finitetimeblow-upofsolutions. Wefirstshowthatblow-upmustnecessarilyoccurinthe
L∞-norm. We then prove two results on global existence and finally establish blow-up
criteria.
Throughout this paper, C denotes a generic constant. We use kuk and kuk
∞ p
to denote the norms in L∞(R) and Lp(R) spaces, respectively. The notation hu,vi
denotestheinnerproductinL2(R). Furthermore,C (R)denotesthespaceofcontinuous
b
bounded functions on R, and C1(R) is the space of differentiable functions in C (R)
b b
3
whose first-order derivatives also belong to C (R). In the spaces C (R) and C1(R) we
b b b
have the norms kuk and kuk = kuk +ku′k , respectively, where the symbol
∞ 1,b ∞ ∞
′ denotes the differentiation. The Sobolev space W1,p(R) is the space of Lp functions
whose distributional derivatives are also in Lp(R) with norm kuk = kuk +ku′k .
W1,p p p
Similarly,forintegerk ≥1,Ck(R)denotesthespaceoffunctionswhosederivativesupto
b
order k are continuous and bounded; Wk,p(R) denotes the space of Lp functions whose
derivatives up to order k are in Lp(R).
2. Local Well Posedness
Below we will give several versions of local well-posedness of the nonlinear Cauchy
problemgivenby(1.1)-(1.2). ThisisachievedinTheorems2.2-2.5forfourdifferentcases
ofinitial dataspaces,namely C (R), Lp(R)∩L∞(R), C1(R) andW1,p(R) (1≤p≤∞).
b b
The proofs will follow the same scheme given below.
If (1.1) is integrated twice with respect to t, the solution of the Cauchy problem
satisfies the integral equation u=Su where
t
(Su)(x,t)=ϕ(x)+tψ(x)+ (t−τ)(Ku)(x,τ)dτ, (2.1)
Z
0
with
(Ku)(x,t)= α(y−x)w(u(y,t)−u(x,t))dy. (2.2)
ZR
Let X be the Banach space with norm k.k , where the initial data lie. We then
X
define the Banach space X(T) = C([0,T],X), endowed with the norm kuk =
X(T)
max ku(t)k , and the closed R-ball Y(T) = {u ∈ X(T) : kuk ≤ R}. We
t∈[0,T] X X(T)
will show that for suitably chosenR and sufficiently small T,the map S is a contraction
onY(T). ThiswillbeachievedbyestimatingfirstKuandthenSuinappropriatenorms.
In each of the four cases, for u,v∈Y(T) we will get estimates of the form
kSuk ≤kϕk +TJ (R,T) (2.3)
X(T) X 1
t
(t−τ)((Ku)(τ)−(Kv)(τ))dτ ≤TJ (R,T)ku−vk (2.4)
(cid:13)Z (cid:13) 2 X(T)
(cid:13) 0 (cid:13)X(T)
(cid:13) (cid:13)
and hence (cid:13) (cid:13)
kSu−Svk ≤TJ (R,T)ku−vk (2.5)
X(T) 2 X(T)
with certain functions J and J nondecreasing in R and T. Taking R ≥ 2kϕk and
1 2 X
then choosing T small enough to satisfy TJ (R,T) ≤ R/2 will give S : Y(T) → Y(T);
1
the further choice TJ (R,T)≤ 1/2 will show that S is a contraction. This implies that
2
thereisauniqueu∈Y(T)satisfyingtheintegralequationu=Su. But,asKuisclearly
continuous in t, we can differentiate (2.1) to get
t
u (x,t)=ψ(x)+ (Ku)(x,τ)dτ
t
Z
0
4
and consequently u (x,t)=(Ku)(x,t). This shows that u∈C2([0,T],X) solves (1.1)-
tt
(1.2). Finally, if u and u satisfy (1.1)-(1.2) with initial data ϕ ,ψ for i = 1,2 we
1 2 i i
get
t
u −u =ϕ −ϕ +t(ψ −ψ )+ (t−τ)((Ku )(τ)−(Ku )(τ))dτ.
1 2 1 2 1 2 1 2
Z
0
Then the estimate (2.4) shows that
ku −u k ≤kϕ −ϕ k +tkψ −ψ k +TJ (R,T)ku −u k .
1 2 X(T) 1 2 X 1 2 X 2 1 2 X(T)
When TJ (R,T)≤1/2,
2
ku −u k ≤2kϕ −ϕ k +2tkψ −ψ k
1 2 X(T) 1 2 X 1 2 X
for t ∈ [0,T]. This shows that, locally, solutions of (1.1)-(1.2) depend continuously on
initial data; thus the problem (1.1)-(1.2) is locally well posed.
The Mean Value Theorem for nonlinear estimates and the following lemma for con-
volution estimates will be our main tools:
Lemma 2.1. Let 1 ≤ p ≤ ∞ and f ∈ L1(R), g ∈ Lp(R). The convolution (f ∗g)(x) =
f(y−x)g(y)dy is well defined and f ∗g ∈Lp(R) with
R
R
kf ∗gk ≤kfk kgk .
p 1 p
In the estimates below, we will often encounter the nondecreasing function M(R)
defined for R>0 as
M(R)= max |w′(η)|. (2.6)
|η|≤2R
We now state and prove (i.e. show that the estimates (2.3) and (2.5) hold) the four
theorems of local well posedness.
Theorem 2.2. Assume that α ∈ L1(R) and w ∈ C1(R) with w(0) = 0. Then there is
some T > 0 such that the Cauchy problem (1.1)-(1.2) is well posed with solution in
C2([0,T],C (R)) for initial data ϕ,ψ ∈C (R).
b b
Proof. Take X = C (R). For u ∈ Y(T), clearly Ku is continuous in x and t and hence
b
Su∈C2([0,T],X). Since w(0)=0 and
|u(y,t)−u(x,t)|≤2ku(t)k ,
∞
the Mean Value Theorem implies
|w(u(y)−u(x))|≤ sup |w′(η)| |u(y)−u(x)|=M(kuk )(|u(y)|+|u(x)|),
∞
|η|≤2kuk∞
where we have suppressed the t variable for convenience. Then
|(Ku)(x,t)| ≤ M(ku(t)k ) |α(y−x)|(|u(y,t)|+|u(x,t)|)dy
∞
ZR
= M(ku(t)k )[(|α|∗|u|)(x,t)+kαk |u(x,t)|], (2.7)
∞ 1
5
and
k(Ku)(t)k ≤2M(ku(t)k )kαk ku(t)k (2.8)
∞ ∞ 1 ∞
where we have used Lemma 2.1. Then
t
|(Su)(x,t)|≤|ϕ(x)|+t|ψ(x)|+ (t−τ)|(Ku)(x,τ)|dτ,
Z
0
and
t
k(Su)(t)k ≤kϕk +tkψk +2kαk (t−τ)M(ku(τ)k )ku(τ)k dτ. (2.9)
∞ ∞ ∞ 1Z ∞ ∞
0
As u∈Y(T), this gives M(ku(τ)k )≤M(R) and hence
∞
t
kSuk ≤ kϕk +Tkψk +2M(R)kαk kuk sup (t−τ)dτ
X(T) ∞ ∞ 1 X(T) Z
t∈[0,T] 0
≤ kϕk +Tkψk +M(R)Rkαk T2. (2.10)
∞ ∞ 1
Thisproves(2.3)withJ (R,T)=kψk +M(R)Rkαk T. Nowletu,v ∈Y(T). Westart
1 ∞ 1
by estimating Ku−Kv. Again suppressing t,
|w(u(y)−u(x))−w(v(y)−v(x))|≤M(R)(|u(y)−v(y)|+|u(x)−v(x)|),
and
|(Ku)(x,t)−(Kv)(x,t)| ≤ M(R)(|α|∗|u−v|)(x,t)
+M(R)kαk |u(x,t)−v(x,t)|. (2.11)
1
Similar to (2.10) we get
t
k(Su)(t)−(Sv)(t)k ≤2M(R)kαk (t−τ)ku(τ)−v(τ)k dτ (2.12)
∞ 1Z ∞
0
and
kSu−Svk ≤M(R)kαk T2ku−vk (2.13)
X(T) 1 X(T)
which proves (2.5) with J (R,T) = M(R)kαk T. According to the scheme described
2 1
above, this completes the proof.
Theorem 2.3. Let 1≤p≤∞. Assume that α∈L1(R) and w ∈C1(R) with w(0)=0.
Then there is some T >0 such that the Cauchy problem (1.1)-(1.2) is well posed with
solution in C2([0,T],Lp(R)∩L∞(R)) for initial data ϕ,ψ ∈Lp(R)∩L∞(R).
Proof. LetX =Lp(R)∩L∞(R)withnormkuk =kuk +kuk . Aswealreadyhavethe
X p ∞
L∞ estimates givenin (2.9) and(2.12), we nowlook for the correspondingLp estimates.
Lemma 2.1 implies k(|α|∗|u|)(t)k ≤kαk |ku(t)k so
p 1 p
k(Ku)(t)k ≤2M(ku(t)k )kαk ku(t)k (2.14)
p ∞ 1 p
6
and Minkowski’s inequality for integrals will yield
t
k(Su)(t)k ≤kϕk +tkψk +2kαk (t−τ)M(ku(τ)k )ku(τ)k dτ. (2.15)
p p p 1Z ∞ p
0
Adding this to the L∞ estimate (2.9), we get
kSuk ≤kϕk +Tkψk +M(R)Rkαk T2.
X(T) X X 1
Similarly we have
t
k(Su)(t)−(Sv)(t)k ≤2M(R)kαk (t−τ)ku(τ)−v(τ)k dτ. (2.16)
p 1Z p
0
Adding this to (2.12) gives
kSu−Svk ≤M(R)kαk T2ku−vk
X(T) 1 X(T)
and concludes the proofs of (2.3) and (2.5).
Theorem 2.4. Assume that α ∈ L1(R) and w ∈ C2(R) with w(0) = 0. Then there is
some T > 0 such that the Cauchy problem (1.1)-(1.2) is well posed with solution in
C2([0,T],C1(R)) for initial data ϕ,ψ ∈C1(R).
b b
Proof. We now take X = C1(R) for which the norm is kuk = kuk +ku′k . Since
b 1,b ∞ ∞
we have the sup norm estimates (2.9) and (2.12) all we need is estimates for their x
derivatives. Throughout this proof we will suppress t (or τ) to keep the expressions
shorter, whenever it is clear from the context. Differentiating (2.2) gives
∂ ∂
(Ku)(x) = α(y−x)w(u(y)−u(x))dy
∂x ∂xZR
∂
= α(z)w(u(x+z)−u(x))dz
∂xZR
= α(z)w′(u(x+z)−u(x))(u (x+z)−u (x))dz
x x
ZR
= α(y−x)w′(u(y)−u(x))(u (y)−u (x))dy.
x x
ZR
Recall that |w′(u(y)−u(x))|≤M(ku(t)k ) due to (2.6). Then
∞
|(Ku) (x)| ≤ M(kuk ) |α(y−x)|(|u (y)|+|u (x)|)dy
x ∞ x x
ZR
≤ M(kuk )[(|α|∗|u |)(x)+kαk |u (x)|]. (2.17)
∞ x 1 x
Since
t
|(Su) (x,t)|≤|ϕ′(x)|+t|ψ′(x)|+ (t−τ) |(Ku) (x,τ)|dτ, (2.18)
x x
Z
0
we have
t
k(Su) (t)k ≤kϕ′k +tkψ′k +2kαk (t−τ)M(ku(τ)k )ku (τ)k dτ.
x ∞ ∞ ∞ 1Z ∞ x ∞
0
7
But M(ku(τ)k )≤M(R) so adding up with the estimate (2.9) proves (2.3)
∞
kSuk = max (k(Su)(t)k +k(Su) (t)k )
X(T) ∞ x ∞
t∈[0,T]
≤ kϕk +Tkψk +M(R)Rkαk T2.
1,b 1,b 1
Next, for |η |≤2R and |µ |≤2R for (i=1,2), we estimate
i i
|w′(η )µ −w′(η )µ | ≤ |w′(η )||µ −µ |+|w′(η )−w′(η )||µ |
1 1 2 2 1 1 2 1 2 2
≤ M(R)|µ −µ |+2Rmax|w′′(η)||η −η |
1 2 1 2
η≤2R
≤ M(R)|µ −µ |+2RN(R)|η −η |
1 2 1 2
where N(R)=max |w′′(η)|. Then
η≤2R
|(Ku−Kv) (x)| ≤ M(R) |α(y−x)|(|u (y)−v (y)|+|u (x)−v (x)|)dy
x x x x x
ZR
+2RN(R) |α(y−x)|(|u(y)−v(y)|+|u(x)−v(x)|)dy
ZR
≤ M(R)((|α|∗|u −v |)(x)+kαk |u (x)−v (x)|)
x x 1 x x
+2RN(R)((|α|∗|u−v|)(x)+kαk |u(x)−v(x)|) (2.19)
1
and
t
k(Su−Sv) (t)k ≤ 2M(R)kαk (t−τ)ku (τ)−v (τ)k dτ
x ∞ 1 x x ∞
Z
0
t
+4RN(R)kαk (t−τ)ku(τ)−v(τ)k dτ
1 ∞
Z
0
≤ (M(R)+2RN(R))kαk T2ku−vk (2.20)
1 X(T)
Finally, adding this to (2.12) we get (2.5) in the form
kSu−Svk ≤ max (k(Su−Sv)(t)k +k(Su−Sv) (t)k )
X(T) ∞ x ∞
t∈[0,T]
≤ 2(M(R)+RN(R))kαk T2ku−vk .
1 X(T)
Theorem 2.5. Let 1≤p≤∞. Assume that α∈L1(R) and w ∈C2(R) with w(0)=0.
Then there is some T >0 such that the Cauchy problem (1.1)-(1.2) is well posed with
solution in C2([0,T],W1,p(R)) for initial data ϕ,ψ ∈W1,p(R).
Proof. Let X = W1,p(R)⊂ L∞(R). Since kuk = kuk +ku′k , we need derivative
W1,p p p
estimates only in addition to the Lp estimates (2.15) and (2.16). For u,v ∈Y(T), from
(2.17)-(2.18) and Minkowski’s inequality we have
t
k(Su) (t)k ≤kϕ′k +tkψ′k +2kαk (t−τ)M(ku(τ)k )ku (τ)k dτ.
x p p p 1Z ∞ x p
0
8
We note that the term kuk∞ can be eliminated by using kuk∞ ≤ CkukW1,p due to the
Sobolev Embedding Theorem. So M(ku(τ)k ) ≤ M(CR) and adding up the above
∞
estimate with (2.15) proves (2.3);
kSuk = max (k(Su)(t)k +k(Su) (t)k )
X(T) p x p
t∈[0,T]
≤ kϕkW1,p +TkψkW1,p +M(CR)Rkαk1T2.
Again from (2.19) we get
t
k(Su−Sv) (t)k ≤ 2M(CR)kαk (t−τ)ku (τ)−v (τ)k dτ
x p 1 x x p
Z
0
t
+4RN(CR)kαk (t−τ)ku(τ)−v(τ)k dτ.
1 p
Z
0
Together with (2.16), we conclude the proof:
kSu−Svk ≤2(M(CR)+RN(CR))kαk T2ku−vk .
X(T) 1 X(T)
Remark 2.6. We remark that the investigation can also continue for smoother data in
along the samelines. That is, for initial data in Ck(R) or Wk,p(R) with integer k wecan
b
prove higher-order versions of Theorems 2.4-2.5. Also, the proofs clearly indicate that in
Theorems 2.2 and 2.3 we can replace the assumption w∈C1(R) with its weaker form: w
is locally Lipschitz. Similarly, in Theorems 2.4 and 2.5 the assumption w ∈ C2(R) can
be weakened to the condition: w′ is locally Lipschitz.
Remark 2.7. The above theorems of local well-posedness can be easily adapted to the
general peridynamic equation (1.3). Theorem 2.8 below extends Theorem 2.2 to the gen-
eral peridynamic equation (1.3). Clearly, similar extensions are also possible in the cases
of Theorems 2.3-2.5.
Theorem 2.8. Assume that f(ζ,0) = 0 and f(ζ,η) is continuously differentiable in η
for almost all ζ. Moreover, suppose that for each R > 0, there are integrable functions
ΛR, ΛR satisfying
1 2
|f(ζ,η)|≤ΛR(ζ), |f (ζ,η)|≤ΛR(ζ)
1 η 2
for almost all ζ and for all |η|≤2R. Then there is some T > 0 such that the Cauchy
problem (1.3)-(1.2) is well posed with solution in C2([0,T],C (R)) for initial data
b
ϕ,ψ ∈C (R).
b
Proof. We proceed as in the proof of Theorem 2.2. By the Dominated Convergence
Theorem, the condition |f(ζ,η)| ≤ ΛR(ζ) implies that Ku is continuous in x so that
1
S : X(T) → X(T). Using the second inequality |f (ζ,η)| ≤ ΛR(ζ) the estimates for
η 2
k(Su)(t)k and k(Su)(t)−(Sv)(t)k follow as in (2.10) and (2.13), just replacing the
∞ ∞
term M(R)kαk by kΛRk and kΛRk respectively, completing the proof.
1 1 1 2 1
9
Remark 2.9. To finish this section let us briefly mention the issue of multidimensional
case in the general three-dimensional peridynamic theory. Although our analysis in this
section has been presented for the one-dimensional case of the peridynamic formulation,
thetechniquesusedcanbeextendedtothecaseofasystemofthreeperidynamic equations
in three space variables without any additional complication. Namely, if we replace the
scalars x, y, u, w and α in (1.1)-(1.2) by the vectors x, y, u, w(u) and the matrix α(x),
respectively, the local existence theorems given above will still be valid.
3. The Cubic Nonlinear Case in Hs(R)
We now want to consider the Cauchy problem (1.1)-(1.2) in the L2 Sobolev space
setting. We will denote the L2 Sobolev space of order s on R by Hs(R) with norm
kuk2 = (1+ξ2)s|u(ξ)|2dξ
Hs ZR
b
where u denotes the Fourier transform of u. For integer k≥0, Hk(R)=Wk,2(R).
As mentioned in Remark 2.6, the proof in the case of H1(R) can be extended to
Hk(R).bOntheotherhand,fornon-integers,Hsestimatesofthenonlineartermw(u(y)−
u(x))involvetechnicaldifficulties. Nevertheless,thecaseofpolynomialnonlinearitiescan
behandledinastraightforwardmanner. Weillustratethisinthetypicalcasew(η)=η3.
Then, the integralonthe right-handside of(1.1) can be computed explicitly in terms of
convolutions and the Cauchy problem (1.1)-(1.2) becomes
u =α∗u3−3u(α∗u2)+3u2(α∗u)−Au3 (3.1)
tt
u(x,0)=ϕ(x), u (x,0)=ψ(x), (3.2)
t
where A= α(y)dy.
R
For theRestimates below we need the following lemmas.
Lemma 3.1. Let α∈L1(R) and u∈Hs(R) for s≥0. Then α∗u∈Hs(R) and
kα∗uk ≤kαk kuk .
Hs 1 Hs
Lemma 3.2. [14] Let s≥0 and u,v ∈Hs(R)∩L∞(R). Then uv ∈Hs(R) and for some
constant C (independent of u and v)
kuvk ≤C(kuk kvk +kvk kuk ).
Hs ∞ Hs ∞ Hs
For the space Hs(R)∩L∞(R) we use the norm kuk =kuk +kuk . In general,
s,∞ Hs ∞
Lemma 3.2 implies that Hs(R)∩L∞(R) is an algebra
kuvk ≤Ckuk kvk , (3.3)
s,∞ s,∞ s,∞
and, by Lemmas 2.1 and 3.1, for α∈L1(R)
kα∗uk ≤kαk kuk . (3.4)
s,∞ 1 s,∞
We are now ready to prove the following theorem.
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