Universität Konstanz
On a new class of Partial Integro-Differential Equations
Patrick Kurth
Konstanzer Schriften in Mathematik Nr. 327, Februar 2014
ISSN 1430-3558
© Fachbereich Mathematik und Statistik Universität Konstanz
Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-265395
On a new class of Partial Integro-Differential Equations
Patrick Kurth
Department of Mathematics and Statistics University of Konstanz
78457 Konstanz, Germany
Abstract
We consider various initial-value problems for partial integro-differential equations of first order that are characterized by convolution-terms in the time-variable, where all factors depend on the solutions of the equations.
The mathematical structure of such problems is based on problems for ordinary integro-differential equations that are used to describe certain glass-transition phenomena (see e.g. [12], [13], [19]). We start consider- ing problems with kernels that are not depending on the space-variable and we will prove results concerning well-posedness and asymptotic be- haviour. Afterwards, we will extend the results on problems with kernels that depend on the space-variable.
Keywords: integro-differential equations, well-posedness, asymptotic behaviour, convolution
1 Introduction
In [12], [13] and [14], initial-value-problems for ordinary integro-differential equations were studied, that are used to describe certain glass-transition-phe- nomena. The kernels of the convolution-terms of these problems are depending on the solutions of the equations, i.e. they are given by functions k = F(Φ) resp. k =F(Φ,·). This is the main difference to integral equations as studied extensively in literature (e.g. equations of Volterra-type, see [8], [9], [11] or [17]) and to mainly considered integro-differential equations from [4], [5], [6] and [7].
In this paper we aim to treat problems for partial integro-differential equations that are of comparable structure, i.e. problems of the kind
ut(t, x) +Au(t, x) +
t
R
0
F(u)(t−s)ut(s)ds = 0, (t, x)∈(0,∞)×G, u(0, x) = u0(x), x∈G,
u(t, x) = 0, (t, x)∈[0,∞)×∂G, (1)
with a bounded domain G ⊆ Rn, a kernel-function F(u) : [0,∞) → R that depends on u, u : [0,∞)×G → R, u0 : G → R and an elliptic operator
A=Pn
i,j=1−∂iaij(·)∂j+a(·) (ai,j, a∈L∞(G)), or ut(t, x) +Au(t, x) +
t
R
0
F(u(t−s, x))ut(s)ds = 0, (t, x)∈(0,∞)×G, u(0, x) = u0(x), x∈G,
u(t, x) = 0, (t, x)∈[0,∞)×∂G, (2)
with a kernel-functionF :R→Rthat is independent onu.
Partial integro-differential equations with at least one convolution-factor that is independent from ugot much attention in mathematical literature, e.g. in [21], [22], [23] and [24]. Integro-differential equations with similar nonlinearities were studied in [19], but in contrast to the problems (1) and (2), all considered equations have been of semilinear structure.
In this work, we aim at proving results concerning well-posedness and asymp- totic behaviour for the problems (1) and (2). The restriction on Dirichlet- boundary conditions and on bounded domains is not mandatory, we will give some comments on more general cases at the end.
This work is based on the Ph.D. thesis [13].
2 Space-independent kernel-functions
In this chapter, we consider problem (1).
2.1 Preliminaries
Assume aij, a ∈ L∞ (i, j = 1, . . . , n), (aij(·))ij is symmetric and uniformly positive definite, i.e.
∃p >0 ∀ξ∈Rn ∀x∈G:
n
X
i,j=1
ξiaij(x)ξj ≥p|ξ|2. We consider the following bilinear form
B(u, v) :=
n
X
i,j=1
haij(·)∂ju, ∂ivi+ha(·)u, vi, u, v∈H01(G),
whereh·,·idenotes theL2-scalar product. B(·,·) requires to be strong coercive, i.e.
∃q >0∀u∈H01(G) :<B(u, u)≥qkuk2H1. We define
D(A) :={u∈H01(G)|∃fu∈L2(G)∀v∈H01(G) :B(u, v) =hfu, vi}
and by this
A:D(A)→L2(G)
u7→fu. (3)
Ais a self-adjoint operator with positive spectrumσ(A)⊆[q,∞). Due to this, one has by using [3, Theorem 1.2.1 (p. 256)]
D(A3)⊆D(A) is dense with respect to the graph-normk · kD(A). (4)
u∈C1([0,∞), L2(G))∩C0([0,∞), D(A)) is called a solution of (1), iff ut+Au+F(u)∗ut
L2(G)
= 0, u(0)D(A)= u0, whereu0∈D(A),F ∈C0(L2(G),R) and
(F(u)∗ut)(t) =
t
Z
0
F(u(t−s))ut(s)ds, t∈[0,∞).
2.2 Linear problem
We consider the following related linear problem u∈C0([0,∞), D(A))∩C1([0,∞), L2(G)) : ut(t) +Au(t) +
t
Z
0
m(t−s)ut(s)ds= 0, t∈[0,∞), u(0) =u0,
(5)
wherem: [0,∞)→R.
Theorem 1. If m ∈ C1([0,∞),R) and m(0) > −q then (5) has a unique solution u∈C1([0,∞), D(A))∩C2([0,∞), L2(G))for any givenu0∈D(A2).
Proof. Differentiation of (5) with respect totand variation of constants formula lead to
zt+ ˜Az+
t
Z
0
m0(t−s)z(s)ds= 0, z(0) =z0:=−Au0, (6)
where z = ut and ˜A =A+m(0). Problem (6) is equivalent to the following fixed-point problem
z=T z, T z(t) :=e−tA˜z0−
t
Z
0
e−(t−s) ˜A
s
Z
0
m0(s−r)z(r)drds. (7)
We define for anyN >0
XN :=C0([0, N], D(A))∩C1([0, N], L2(G)), with the norm1
kzkN := max (
sup
t∈[0,N]
kz(t)kD(A), sup
t∈[0,N]
d dtz(t)
)
By using Banach fixed-point theorem, one proves the existence of a unique fixed- pointzN ∈XN for anyN >0. We definez(t) :=zN(t) for 0≤t≤N. Due to the uniqueness of anyzN,z: [0,∞)→D(A) is well-defined and fulfilsz∈XN
for allN >0. u:=u0+Rt
0z(s)dsis the requested solution of (5).
1k · kdenotes theL2(G)-norm.
One proves analoguously the following
Corollary 2. If m ∈ C1([0,∞),R) and m(0) > −q then (5) has a unique solution u∈C1([0,∞), D(A2))∩C2([0,∞), D(A))for any given u0∈D(A3).
Lemma 3. Assume m∈ C1([0,∞),R)with m(0)> −q and|m0(t)| ≤ ke−c1t for all t∈[0,∞), wherec1> λ1 withλ1=q+m(0)andk >0such that
k < λ1(c1−λ1).
Furthermore, let lim
t→∞(t) = 0 andu0∈D(A2). Then the solutionuof (5) fulfils for all t∈[0,∞)
kut(t)k ≤ ku0kD(A)e
k−λ1 (c1−λ1 ) c1−λ1 t
, ku(t)k ≤ ku0kD(A)
λ1−c1
k−λ1(c1−λ1)e
k−λ1 (c1−λ1 ) c1−λ1 t
.
Proof. We consider the fixed-point equation from the proof of Lemma 1
z(t) =e−tA˜z0−
t
Z
0
e−(t−s) ˜A
s
Z
0
m0(s−r)z(r)drds.
We obtain
eλ1tkz(t)k ≤ ku0kD(A)+k
t
Z
0 t
Z
r
e(λ1−c1)(s−r)dseλ1rkz(r)kdr.
Application of Gronwall’s inequality finishes the proof.
Corollary 4. Letu0∈D(A3). The same asumptions from Lemma 3 lead to kut(t)kD(A)≤ kAu0kD(A)e
k−λ1 (c1−λ1 ) c1−λ1 t
and ku(t)kD(A)≤ kAu0kD(A)
λ1−c1
k−λ1(c1−λ1)e
k−λ1 (c1−λ1 ) c1−λ1 t
.
2.3 Non-linear problem
We now aim to produce a self-mapping using the linear problem (5), so that its fixed-point solves the nonlinear problem (1).
At first, letu0∈D(A3). Furthermore, letF ∈C0(L2(G),R) locally Lip- schitz continuous and supposeF◦u∈C1([0,∞),R) for allu∈C1([0,∞), L2(G)) and F(u0)>−q.
Letλ1:=q+F(u0),c1> λ1 andv2, β >0.
(i) Letk >0 such thatk < λ1(c1−λ1), (ii) α1, α2>0 such that (α1+α2)k−λc1(c1−λ1)
1−λ1 ≤ −c1, (iii) v1>0 such thatv1ku0kαD(A)1+α2
λ1−c1 k−λ1(c1−λ1)
α1
≤k.
(iv) Furthermore, let|F(u)| ≤v2kukβ, |F(u0)| ≤ ck
1,
(v)
dtdF(u(t))
≤v1ku(t)kα1kut(t)kα2 for allu∈C1([0,∞), L2(G)), (vi) ∃L1>0∀u1, u2∈C1([0,∞), L2(G)) that satisfy
kui(t)k ≤ ku0kD(A) λ1−c1
k−λ1(c1−λ1) andkuit(t)k ≤ ku0kD(A) for allt∈[0,∞),i= 1,2 :
d
dtF(u1(t))− d
dtF(u2(t))
≤(ku1(t)−u2(t)k+ku1t(t)−u2t(t)k). We consider
X :=
u∈C0([0,∞), D(A))∩C1([0,∞), L2(G)) :u, utare bounded
together with the normkukX := max (
sup
t∈[0,∞)
ku(t)kD(A), sup
t∈[0,∞)
kut(t)k )
and the following subset
C :=
u∈X
u(0)D(A)= u0:
ku(t)k ≤ ku0kD(A) λ1−c1 k−λ1(c1−λ1)e
k−λ1 (c1−λ1 ) c1−λ1 t
, kut(t)k ≤ ku0kD(A)e
k−λ1 (c1−λ1 ) c1−λ1 t
, ku(t)kD(A)≤ kAu0kD(A) λ1−c1
k−λ1(c1−λ1)e
k−λ1 (c1−λ1 ) c1−λ1 t
.
C ⊆X is bounded, closed, convex and due to (iv) not-empty. We define
T :C →C, v7→Tv:=uv,
whereuv is the solution of (5) with kernel-functionm:=F◦v. One has
|m0(t)|
(ii),(iii),(v)
≤ ke−c1t and due to (iv)
|m(t)|t→∞−→ 0.
With respect to (i), Lemma 3 and Corollary 4, one has proved thatT is well- defined.
Using Gronwall’s inequality and the Arzelà–Ascoli theorem, we easily prove the continuity and compactness of T. By applying Schauder fixed-point theorem, we obtain the following
Theorem 5. Assume u0 ∈ D(A3), F ∈ C0(L2(G),R) is locally Lipschitz- continuous and supposeF◦u∈C1([0,∞),R)for allu∈C1([0,∞), L2(G))and F(u0)>−q. Furthermore, letλ1=q+F(u0),c1> λ1,v2, β >0 and
(i) k >0 such thatk < λ1(c1−λ1).
(ii) Letα1, α2>0 such that (α1+α2)k−λc1(c1−λ1)
1−λ1 ≤ −c1 and (iii) v1>0 such that v1ku0kαD(A)1+α2
λ1−c1
k−λ1(c1−λ1)
α1
≤k.
Furthermore, let
(iv) |F(u)| ≤v2kukβ,|F(u0)| ≤ ck
1, (v)
dtdF(u(t))
≤v1ku(t)kα1kut(t)kα2 for all u∈C1([0,∞), L2(G))and (vi) ∃L1>0∀u1, u2∈C1([0,∞), L2(G))that satisfy
kui(t)k ≤ ku0kD(A)k−λλ1−c1
1(c1−λ1) andkuit(t)k ≤ ku0kD(A)(i= 1,2):
d
dtF(u1(t))− d
dtF(u2(t))
≤L1(ku1(t)−u2(t)k+ku1t(t)−u2t(t)k).
Then problem (1) has a unique solutionu∈C0([0,∞), D(A))∩C1([0,∞), L2(G)) that fulfils
ku(t)k ≤ ku0kD(A)
λ1−c1
k−λ1(c1−λ1)e
k−λ1 (c1−λ1 ) c1−λ1 t
, kut(t)k ≤ ku0kD(A)e
k−λ1 (c1−λ1 ) c1−λ1 t
.
Comment on the proof. Uniqueness follows from a direct computation that uses Gronwall’s inequality.
The restrictionu0∈D(A3) was necessary to obtain a self-mapping onC by using Corollary 4. We now aim to generalize the conditions of Theorem 5.
Lemma 6. Assume u0∈D(A),F ∈C0(L2(G),R)as given in Theorem 5 and letu01, u02∈D(A)andu1, u2∈X related solutions of (1) that satisfy
kui(t)k ≤M1 andkuit(t)k ≤M2 for all t∈[0,∞)whereM1, M2>0. Furthermore, suppose
∃L1=L1(M1, M2, F)>0∀t∈[0,∞) :
d
dtF(u1(t))− d
dtF(u2(t))
≤L1 ku1(t)−u2(t)kD(A)+ku1t(t)−u2t(t)k . Then one has
∀N >0 ∃C=C(N, M1, M2, F)>0 :ku1−u2kN ≤Cku01−u02kD(A). Proof. Letp, N >0. Direct computations lead to
e−ptku1(t)−u2(t)k ≤ku01−u02kD(A)
+ 1
p(p+q)v2Mβ sup
t∈[0,N]
e−ptku1t(t)−u2t(t)k
+ 1
p(p+q)LM sup
t∈[0,N]
e−ptku1(t)−u2(t)k, e−ptku1t(t)−u2t(t)k ≤ku01−u02kD(A)
+ 1
p(p+λ1)v1Mα1+α2 sup
t∈[0,N]
e−ptku1t(t)−u2t(t)k
+ 1
p(p+λ1)L1M sup
t∈[0,N]
e−ptku1(t)−u2(t)kD(A)
+ 1
p(p+λ1)L1M sup
t∈[0,N]
e−ptku1t(t)−u2t(t)k and
e−ptkAu1(t)−Au2(t)k ≤ku01−u02kD(A)
+ 1
p(p+λ1)v1Mα1+α2 sup
t∈[0,N]
e−ptku1t(t)−u2t(t)k
+ 1
p(p+λ1)L1M sup
t∈[0,N]
e−ptku1(t)−u2(t)kD(A)
+ 1
p(p+λ1)L1M sup
t∈[0,N]
e−ptku1t(t)−u2t(t)k +1
pLM sup
t∈[0,N]
e−ptku1(t)−u2(t)k.
Chosing p >0 large enough, one obtains2
ku1−u2kN,p≤Cku01−u02kD(A)
with a constantC >0. Due to the equivalence of the normsk · kN andk · kN,p, the proof is finished.
By the help of this lemma, on can prove the following
Corollary 7. Assume u0 ∈ D(A), F ∈ C0(L2(G),R) is locally Lipschitz- continuous and suppose F ◦u ∈ C1([0,∞),R) for all u ∈ C1([0,∞), L2(G)) andF(u0)>−q. Furthermore, letλ1=q+F(u0),c1> λ1,v2, β >0 and
(i) k >0 such thatk < λ1(c1−λ1).
(ii) Letα1, α2>0 such that (α1+α2)k−λc1(c1−λ1)
1−λ1 <−c1 and (iii) v1>0 such that v1ku0kαD(A)1+α2 λ
1−c1
k−λ1(c1−λ1)
α1
< k.
Furthermore, let
(iv) |F(u)| ≤v2kukβ,|F(u0)| ≤ ck
1, (v)
dtdF(u(t))
≤v1ku(t)kα1kut(t)kα2 for all u∈C1([0,∞), L2(G))and (vi) ∃ε >0 ∃L1>0∀u1, u2∈C1([0,∞), L2(G))that satisfy
kui(t)k ≤ ku0kD(A)k−λλ1−c1
1(c1−λ1)+εandkuit(t)k ≤ ku0kD(A)+ε(i= 1,2):
d
dtF(u1(t))− d
dtF(u2(t))
≤L1(ku1(t)−u2(t)k+ku1t(t)−u2t(t)k).
Then problem (1) has a unique solutionu∈C0([0,∞), D(A))∩C1([0,∞), L2(G)) that fulfils
ku(t)k ≤ ku0kD(A)
λ1−c1
k−λ1(c1−λ1)e
k−λ1 (c1−λ1 ) c1−λ1 t
andkut(t)k ≤ ku0kD(A)e
k−λ1 (c1−λ1 ) c1−λ1 t
.
2We define the weighted normkukN,p:= max
sup
t∈[0,N]
e−ptku(t)kD(A), sup
t∈[0,N]
e−ptkut(t)k
(N, p >0).
Proof. It follows from (4)
∃(u0n)n∈N⊆D(A3) :ku0−u0nkD(A)n→∞−→ 0.
Due to the continuous dependence ofq+F(u) onu∈L2(G), there existsn0∈N such that problem (1) with kernel-functionF and initial-valueu0nhas a unique solutionun ∈X for alln≥n0, that satisfies
kun(t)k ≤ ku0nkD(A)
λn−c1
k−λn(c1−λn)e
k−λn(c1−λn) c1−λn t
and kunt(t)k ≤ ku0nkD(A)e
k−λn(c1−λn) c1−λn t
for allt∈[0,∞),
(8)
where λn := q+F(u0n) (n ≥2). We define M1 :=ku0kD(A) λ1−c1
k−λ1(c1−λ1)+ε and M2 :=ku0kD(A)+ε. Then there exists n1 ≥n0 such that kun(t)k ≤ M1
andkunt(t)k ≤M2for alln≥n1and t∈[0,∞). It follows from Lemma 6
∃u∈X ∀N >0 :ku−unkN
n→∞−→ 0.
Limit of (1) and (8) as n→ ∞finally proves, that uis the requested solution of (1) with kernel-functionF and initial-valueu0.
2.4 Remark and example
Remark 8. It is easy to extend the methods from the previous chapter on inhomogenious problems of the kind
ut(t) +Au(t) +
t
Z
0
F(u)(t−s)ut(s)ds=f(t), u(0) =u0
with f ∈C1([0,∞), D(A2)). To obatin a self-mapping as above, we will need the following conditions on F
kf0(t)kD(A)≤M e−dt, lim
t→∞kf(t)kD(A)= 0,
withM ≥0,d >0andd(c1−λ1)> kin case ofd < λ1.The smallness-conditions will additionally depend on M andd.
Example 9. We consider so-called radial-symmetric functions. Assume u0 ∈ D(A)andf ∈C1([0,∞),R)and supposef(ku0k)>−q, withf0locally Lipschitz- continuous. Furthermore, let |f(x)| ≤ v2|x|β, |f(ku0k)| ≤ ck
1 and |f0(x)| ≤ v1|x|α1. Then problem (1) with kernel-function F, defined byF(u) :=f(kuk), has a unique solution u∈X that decays exponentially.
3 Space-dependent kernel-functions
In this chapter, we consider problem (2).
3.1 Preliminaries
We start proving some auxiliaries that will be used later. Assume G ⊆Rn is open and bounded and has aCk-boundary∂Gwherek∈Nis chosen sufficiently large enough for the following results.
V 1. Let k ∈ N, aij ∈ C2k−1(G) (i, j = 1, . . . , n), a ∈ C2k−2(G) and u∈D(Ak)∩C2k(G). Then one can easily prove by induction
Aku= X
β∈Nn0,|β|≤2k
Pβ(∂γ1aij, ∂γ2a;i, j= 1, . . . , n,|γ1| ≤2k−1,|γ2| ≤2k−2)·∂βu,
wherePβ are polynomials with degrees degPβ≤k.
V 2. Ellipctic regularity: Letk ∈N, aij, a∈C2k−1( ¯G) (i, j = 1, . . . , n). One can prove by using V 1 and [10, Theorem 5 (p. 340)]
∃C1, C2>0∀u∈D(Ak)∩C∞(G) :C1kukH2k(G)≤ kukD(Ak)≤C2kukH2k(G).
V 3. Assume k ∈ N satisfies 4k > n. Furthermore, let F ∈ C2k(R,R) with
|F(i)(x)| ≤v1|x|α fori= 0, . . . ,2kand givenv1, α >0.
We consider the following improved formula from [18, proof of Lemma 4.7, p.
47]3
∂β(F(u)) =
|β|
X
µ=1
F(µ)(u) X
γ∈N|β|0 ,|γ|=µ,
|β|
P
j=1 jγj=|β|
(µ+1)|β|−1
X
p=1
Cµ,γ,p
|β|
Y
i=1 γi
Y
l=1
∂αil,p,γu, (9)
with β ∈Nn0 such that |β| ≤ 2k, Cµ,γ,p ≥0 and αil,p,γ ∈ Nn0 with αil,p,γ ≤ β and|αil,p,γ|=i. A similar calculation as [18, proof of Lemma 4.7] leads to
k∂β(F(u))k ≤
|β|
X
µ=1
X
γ∈N|β|0 ,|γ|=µ,
|β|
P
j=1 jγj=|β|
(µ+1)|β|−1
X
p=1
Cµ,γ,pkF(µ)(u)k∞
|β|
Y
i=1
kukγi
Wi, 2|β|
i
.
One obtains from Gagliardo-Nirenberg inequality ([18, Theorem 4.4]) k∂β(F(u))k
≤
|β|
X
µ=1
X
γ∈N|β|0 ,|γ|=µ,
|β|
P
j=1 jγj=|β|
(µ+1)|β|−1
X
p=1
Cµ,γ,pv1kukα∞
|β|
Y
i=1
C(i,|β|)kuk
iγi
|β|
H|β|kukγi−
iγi
|β|
∞
and from Sobolev embedding theorem and V 2 kF(u)kH2k ≤C5v1
( kukαD(Ak), kukD(Ak)≤1 kukα+2kD(Ak), kukD(Ak)>1
) ,
3For a proof, see [13, Lemma 1.13].
where C5 = P
|β|≤2k
C(C0, C1,|β|)2
!12
with the Sobolev embedding constant C0.
V 4. Assumek andF as in V 3 and u∈D(Ak)∩C∞(G).
Due to Sobolev embedding theorem and V 2, one has u ∈ C0( ¯G) and from [18, Lemma 4.7] F(u) ∈ H2k(G). We consider the (unique) trace-operator S : H1(G) → L2(∂G) from [10, Theorem 1 (p. 272)], that maps continuous functions on their restrictions on the boundary. Due to u ∈ H01(G), one has Su= 0 and it follows u|∂G = 0. One obtainsF(u)|∂G = 0 from F(0) = 0 and due to [10, Theorem 2 (p. 273)]F(u)∈H01(g), resp. F(u)∈D(A)∩C2k( ¯G). In case ofF(u)∈D(Am)∩C2k( ¯G) for a fixedm∈ {1, . . . , k−1}, application of V 1 and (9) lead toS(Am(F(u))) = 0, i.e. F(u)∈D(Am+1). It follows iteratively F(u)∈D(Ak) and from V 3
kF(u)kD(Ak)≤C4v1
( kukαD(Ak), kukD(Ak)≤1 kukα+2kD(Ak), kukD(Ak)>1
) ,
whereC4=C2C5.
V 5. Assume k∈ Nsuch that 4k > nand F ∈ C2k(R,R) withF(2k) locally Lipschitz-continuous and F(i)(0) = 0 (i = 0, . . . ,2(k−1)). Furthermore, let u1, u2∈D(Ak)∩C∞(G) andM >0 such thatkuikD(A)≤M (i= 1,2). Using (9), one obtains
∂β(F(u1)−F(u2)) =
|β|
X
µ=1
X
γ∈N|β|0 ,|γ|=µ,
|β|
P
i=1 iγi=|β|
(µ+1)|β|−1
X
p=1
F(µ)(u1)−F(µ)(u2) Cµ,γ,p
|β|
Y
i=1 γi
Y
l=1
∂αil,p,γu1 +
|β|
X
µ=1
X
γ∈N|β|0 ,|γ|=µ,
|β|
P
i=1 iγi=|β|
(µ+1)|β|−1
X
p=1
F(µ)(u2)Cµ,γ,p
·
|β|
X
i=1
γi
P
l=1
∂αil,p,γu1−∂αil,p,γu2
l−1 Q
m=1
∂αim,p,γu2 γi
Q
m=l+1
∂αim,p,γu1
·
i−1
Q
m=1 γm
Q
l=1
∂αml,p,γu2
|β|
Q
m=i+1 γm
Q
l=1
∂αml,p,γu1
.
One has for allµ∈Nn0 with|µ| ≤2k
F(µ)(u1)−F(µ)(u2) ∞≤Lµ
C0
C1
ku1−u2kD(Ak),
with Lipschitz-constants Lµ>0 ofF(µ)on the intervalh
−CC0
1M,CC0
1Mi . Addi-
tionally, one has by using Hölder’s inequality
F(µ)(u2)
∂αil,p,γu1−∂αil,p,γu2
l−1 Q
m=1
∂αim,p,γu2 γi
Q
m=l+1
∂αim,p,γu1 i−1
Q
m=1 γm
Q
l=1
∂αml,p,γu2
·
|β|
Q
m=i+1 γm
Q
l=1
∂αml,p,γu1
≤
F(µ)(u2) ∞
∂
αil,p,γ
u1−∂αil,p,γu2
2|β|
i
l−1
Q
m=1
∂αim,p,γu2
2|β|
i(l−1)
·
γi
Q
m=l+1
∂αim,p,γu1
2|β|
i(γi−l) i−1
Q
m=1
γm
Q
l=1
∂αml,p,γu2
2|β|
mγm
|β|
Q
m=i+1
γm
Q
l=1
∂αml,p,γu1
2|β|
mγm
Application of Gagliardo-Nirenberg inequality, Sobolev embedding theorem and V 2 lead to
∂αil,p,γu1−∂αil,p,γu2
2|β|
i
≤C(i,|β|)C1−
i
|β|
0
C1
ku1−u2kD(Ak)
and
l−1
Y
m=1
∂αim,pu2
2|β|
i(l−1)
,
γi
Y
m=l+1
∂αim,pu1
2|β|
i(γi−l)
,
γm
Y
l=1
∂αml,pu2
2|β|
mγm
,
γm
Y
l=1
∂αml,pu1
2|β|
mγm
,
|β|
Y
i=1 γi
Y
l=1
∂αil,pu1
≤ C(|β|, C0, C1, M).
Altogether, one has proved
∃K=K(C0, C1, F, k, M)>0∀u1, u2∈D(Ak)∩C∞(G),kuikD(Ak)≤M (i= 1,2) :
∂β(F(u1)−F(u2))
≤Kku1−u2kD(Ak). It follows from V 2 and V 4
∃K=K(C0, C1, F, k, M)>0∀u1, u2∈D(Ak)∩C∞(G),kuikD(Ak)≤M (i= 1,2) : kF(u1)−F(u2)kD(Ak)≤Kku1−u2kD(Ak).
V 6. Assumek∈Nwith 4k > n,u, v, w∈D(Ak)∩C∞(G) andF∈C(2k)(R,R) such that F(i)(0) = 0 (i = 1, . . . ,2(k−1)). Due to V 2 and V 3, one has F(u)∈H2k(G)∩C∞(G). H2k(G) is a Banach algebra (see [1, Theorem 4.39]), i.e. F(u)v, F(u)vw ∈ H2k(G)∩C∞(G). One can prove with similar methods as used in V 4: F(u)v, F(u)vw∈D(Ak)∩C∞(G).
V 7. We obtain from [3, Theorem 1.2.1 (p. 256)] for allk∈N
\
j∈N0
D(Ak+j)⊆D(Ak) is dense with respect tok · kD(Ak).
Due to V 2 and Sobolev embedding theorem, one has the density of D(Ak)∩ C∞(G) inD(Ak).
Application of V 7 on V 2–V 6 leads to the following
Lemma 10. Assume k ∈ N with 4k > n, F ∈ C(2k)(R,R) with |F(i)(x)| ≤ v1|x|α (i= 0, . . . ,2k, v1, α >0) andF(2k)is locally Lipschitz-continuous. Fur- thermore, letG⊆Rn be open and bounded with a C2k-boundary.
(i)
∃C1, C2>0∀u∈D(Ak) : u∈H2k(G),
C1kukH2k(G)≤ kukD(Ak)≤C2kukH2k(G). (ii)
∃C3>0∀u, v, w∈D(Ak) : F(u)v, F(u)vw∈D(Ak),
kF(u)vkD(Ak)≤C3kF(u)kD(Ak)kvkD(Ak). (iii)
∃C4>0∀u∈D(Ak) : F(u)∈D(Ak),
kF(u)kD(Ak)≤C4v1
kukα
D(Ak), kukD(Ak)≤1, kukα+2kD(Ak), kukD(Ak)>1.
(iv)
∀M >0∃K >0∀u1, u2∈D(Ak)withkuikD(Ak)≤M (i= 1,2) : kF(u1)−F(u2)kD(Ak)≤Kku1−u2kD(Ak).
Remark 11. Most of the previous results will not need aC2k-boundary of G.
This regularity comes from [10, Theorem 5 (p. 340)] to prove V 2.
3.2 Linear problem
Assume k ∈ N such that 4k > n and G ⊆ Rn is a bounded domain with a C2k-boundary. Furthermore, let q > 0 such thatσ(A)⊆[q,∞). We consider the following linear problem
u∈C0([0,∞), D(A))∩C1([0,∞), L2(G)) : ut(t) +Au(t) +
t
Z
0
m(t−s)ut(s)ds= 0, t∈[0,∞), u(0) =u0∈D(A),
(10)
wherem: [0,∞)→L2(G).
Theorem 12. If m∈ C1([0,∞), D(Ak)) with m(0)(x)≥ −q+ε for a ε > 0 and for all x∈ G,u0 ∈ D(Ak+1) and mt(t)v ∈D(Ak) for all t ∈ [0,∞) and v ∈ D(Ak), then problem (2) has a unique solution u ∈ C1(0,∞), D(Ak))∩ C2([0,∞), D(Ak−1)).
Proof. We define fort∈[0,∞) andu∈C1([0,∞), D(Ak))
w(t) :=
t
Z
0
mt(t−s)ut(s)ds.
By the help of Lemma 10 (ii), we easily seew∈C0([0,∞), D(Ak)). By this, the theorem can be proved analogously to Theorem 1 by replacingλ1byε,k · kD(A) byk · kD(Ak)andk · kL2(G) byk · kD(Ak−1).
Lemma 13. Assumeu0∈D(Ak+1)andm∈C1([0,∞), D(Ak))as in Theorem 12. Suppose additionally kmt(t)kD(Ak)≤ωe−c1t and limt→∞km(t)kD(Ak) = 0 wherec1> εandω >0such that
C3ω < ε(c1−ε) and C0
C1ω < ε(c1−ε).
Then the solution uof (2) fulfils
kut(t)kD(Ak)≤ kAu0kD(Ak)e
C3ω−ε(c1−ε) c1−ε t
, ku(t)kD(Ak)≤ kAu0kD(Ak)
ε−c1
C3ω−ε(c1−ε)eC3
ω−ε(c1−ε) c1−ε t
and kut(t)k ≤ ku0kD(A)e
C0
C1ω−ε(c1−ε) c1−ε t
, ku(t)k ≤ ku0kD(A) ε−c1
C0
C1ω−ε(c1−ε)e
C0
C1ω−ε(c1−ε) c1−ε t
.
Proof. Differentiation of (2) with respect to t leads to the operator ˜A :=A+ m(0,·) with spectrum σ( ˜A) ⊆ [ε,∞). Using Lemma 10 (i) and (ii), one can follow the same steps as in the proof of Lemma 3 to prove the requested results.
Remark 14. To obtain a comparable result to Corollary 4, one will need higher regularity for the kernel-function. But this is not sensible for the futher theory.
3.3 Non-linear problem
Assume k ∈ N such that 4k > n, G ⊆ Rn is a bounded domain with C2k- boundary, u0 ∈D(Ak+1) andF ∈C2k+1(R,R) withF(2k+1) locally Lipschitz- continuous andF(u0(x))>−q+εfor aε >0 and for allx∈G.
i) Letc1> ε.
ii) Supposeω >0 such thatC3ω < ε(c1−ε) and CC0
1ω < ε(c1−ε).
iii) Assumeα >0 such that (α+ 1)C3ω−ε(cc 1−ε)
1−ε ≤ −c1. iv) Letv1>0 such that
v1C3C4kAu0kα+1D(Ak)
ε−c1
C3ω−ε(c1−ε) α
≤ω
and v1C3C4kAu0kα+2k+1D(Ak)
ε−c1 C3ω−ε(c1−ε)
α+2k
≤ω.
Furthermore, suppose v)
F(i)(x)
≤v1|x|α,i= 0, . . . ,2k+ 1,x∈R. vi) kF(u0)kD(Ak)≤cω
1 for allx∈G.