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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

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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)(ts)ut(s)ds = 0, (t, x)∈(0,∞)×G, u(0, x) = u0(x), xG,

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

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A=Pn

i,j=1−∂iaij(·)∂j+a(·) (ai,j, aL(G)), or ut(t, x) +Au(t, x) +

t

R

0

F(u(ts, x))ut(s)ds = 0, (t, x)∈(0,∞)×G, u(0, x) = u0(x), xG,

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, aL (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)ξjp|ξ|2. We consider the following bilinear form

B(u, v) :=

n

X

i,j=1

haij(·)∂ju, ∂ivi+ha(·)u, vi, u, vH01(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)|∃fuL2(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)

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uC1([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, whereu0D(A),FC0(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 uC0([0,∞), D(A))∩C1([0,∞), L2(G)) : ut(t) +Au(t) +

t

Z

0

m(ts)ut(s)ds= 0, t∈[0,∞), u(0) =u0,

(5)

wherem: [0,∞)→R.

Theorem 1. If mC1([0,∞),R) and m(0) > −q then (5) has a unique solution uC1([0,∞), D(A))∩C2([0,∞), L2(G))for any givenu0D(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≤tN. 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.

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One proves analoguously the following

Corollary 2. If mC1([0,∞),R) and m(0) > −q then (5) has a unique solution uC1([0,∞), D(A2))∩C2([0,∞), D(A))for any given u0D(A3).

Lemma 3. Assume mC1([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(c1λ1).

Furthermore, let lim

t→∞(t) = 0 andu0D(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)

λ1c1

kλ1(c1λ1)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

e1−c1)(s−r)dseλ1rkz(r)kdr.

Application of Gronwall’s inequality finishes the proof.

Corollary 4. Letu0D(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)

λ1c1

kλ1(c1λ1)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, letu0D(A3). Furthermore, letFC0(L2(G),R) locally Lip- schitz continuous and supposeFuC1([0,∞),R) for alluC1([0,∞), L2(G)) and F(u0)>−q.

Letλ1:=q+F(u0),c1> λ1 andv2, β >0.

(i) Letk >0 such thatk < λ1(c1λ1), (ii) α1, α2>0 such that (α1+α2)k−λc1(c1−λ1)

1−λ1 ≤ −c1, (iii) v1>0 such thatv1ku0kαD(A)12

λ1−c1 k−λ1(c1−λ1)

α1

k.

(iv) Furthermore, let|F(u)| ≤v2kukβ, |F(u0)| ≤ ck

1,

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(v)

dtdF(u(t))

v1ku(t)kα1kut(t)kα2 for alluC1([0,∞), L2(G)), (vi) ∃L1>0∀u1, u2C1([0,∞), L2(G)) that satisfy

kui(t)k ≤ ku0kD(A) λ1c1

kλ1(c1λ1) 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 :=

uC0([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:=Fv. 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 u0D(A3), FC0(L2(G),R) is locally Lipschitz- continuous and supposeFuC1([0,∞),R)for alluC1([0,∞), L2(G))and F(u0)>−q. Furthermore, letλ1=q+F(u0),c1> λ1,v2, β >0 and

(i) k >0 such thatk < λ1(c1λ1).

(ii) Letα1, α2>0 such that1+α2)k−λc1(c1−λ1)

1−λ1 ≤ −c1 and (iii) v1>0 such that v1ku0kαD(A)12

λ1−c1

k−λ1(c1−λ1)

α1

k.

Furthermore, let

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(iv) |F(u)| ≤v2kukβ,|F(u0)| ≤ ck

1, (v)

dtdF(u(t))

v1ku(t)kα1kut(t)kα2 for all uC1([0,∞), L2(G))and (vi) ∃L1>0∀u1, u2C1([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 solutionuC0([0,∞), D(A))∩C1([0,∞), L2(G)) that fulfils

ku(t)k ≤ ku0kD(A)

λ1c1

kλ1(c1λ1)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 restrictionu0D(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 u0D(A),FC0(L2(G),R)as given in Theorem 5 and letu01, u02D(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 :ku1u2kNCku01u02kD(A). Proof. Letp, N >0. Direct computations lead to

e−ptku1(t)−u2(t)k ≤ku01u02kD(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 ≤ku01u02kD(A)

+ 1

p(p+λ1)v1Mα12 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

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e−ptkAu1(t)−Au2(t)k ≤ku01u02kD(A)

+ 1

p(p+λ1)v1Mα12 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

ku1u2kN,pCku01u02kD(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 u0D(A), FC0(L2(G),R) is locally Lipschitz- continuous and suppose FuC1([0,∞),R) for all uC1([0,∞), L2(G)) andF(u0)>−q. Furthermore, letλ1=q+F(u0),c1> λ1,v2, β >0 and

(i) k >0 such thatk < λ1(c1λ1).

(ii) Letα1, α2>0 such that1+α2)k−λc1(c1−λ1)

1−λ1 <−c1 and (iii) v1>0 such that v1ku0kαD(A)12 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 uC1([0,∞), L2(G))and (vi) ∃ε >0 ∃L1>0∀u1, u2C1([0,∞), L2(G))that satisfy

kui(t)k ≤ ku0kD(A)k−λλ1−c1

1(c1−λ1)+εandkuit(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 solutionuC0([0,∞), D(A))∩C1([0,∞), L2(G)) that fulfils

ku(t)k ≤ ku0kD(A)

λ1c1

kλ1(c1λ1)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).

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Proof. It follows from (4)

∃(u0n)n∈ND(A3) :ku0u0nkD(A)n→∞−→ 0.

Due to the continuous dependence ofq+F(u) onuL2(G), there existsn0∈N such that problem (1) with kernel-functionF and initial-valueu0nhas a unique solutionun ∈X for allnn0, that satisfies

kun(t)k ≤ ku0nkD(A)

λnc1

kλn(c1λn)e

k−λn(c1λn) c1λn t

and kunt(t)k ≤ ku0nkD(A)e

k−λn(c1−λn) c1−λn t

for allt∈[0,∞),

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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 n1n0 such that kun(t)k ≤ M1

andkunt(t)k ≤M2for allnn1and 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)(ts)ut(s)ds=f(t), u(0) =u0

with fC1([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 u0D(A)andfC1([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).

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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, aijC2k−1(G) (i, j = 1, . . . , n), aC2k−2(G) and uD(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 degPβk.

V 2. Ellipctic regularity: Letk ∈N, aij, aC2k−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 FC2k(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 j=|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,γβ 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 j=|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 j=|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)kH2kC5v1

( kukαD(Ak), kukD(Ak)≤1 kukα+2kD(Ak), kukD(Ak)>1

) ,

3For a proof, see [13, Lemma 1.13].

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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 uD(Ak)∩C(G).

Due to Sobolev embedding theorem and V 2, one has uC0( ¯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 uH01(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 FC2k(R,R) withF(2k) locally Lipschitz-continuous and F(i)(0) = 0 (i = 0, . . . ,2(k−1)). Furthermore, let u1, u2D(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 i=|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 i=|β|

(µ+1)|β|−1

X

p=1

F(µ)(u2)Cµ,γ,p

·

|β|

X

i=1

γi

P

l=1

αil,p,γu1αil,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

ku1u2kD(Ak),

with Lipschitz-constants Lµ>0 ofF(µ)on the intervalh

CC0

1M,CC0

1Mi . Addi-

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tionally, one has by using Hölder’s inequality

F(µ)(u2)

αil,p,γu1αil,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

u1αil,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,γu1αil,p,γu2

2|β|

i

C(i,|β|)C1−

i

|β|

0

C1

ku1u2kD(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, u2D(Ak)∩C(G),kuikD(Ak)M (i= 1,2) :

β(F(u1)−F(u2))

Kku1u2kD(Ak). It follows from V 2 and V 4

∃K=K(C0, C1, F, k, M)>0∀u1, u2D(Ak)∩C(G),kuikD(Ak)M (i= 1,2) : kF(u1)−F(u2)kD(Ak)Kku1u2kD(Ak).

V 6. Assumek∈Nwith 4k > n,u, v, wD(Ak)∩C(G) andFC(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

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Lemma 10. Assume k ∈ N with 4k > n, FC(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) : uH2k(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, u2D(Ak)withkuikD(Ak)M (i= 1,2) : kF(u1)−F(u2)kD(Ak)Kku1u2kD(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

uC0([0,∞), D(A))∩C1([0,∞), L2(G)) : ut(t) +Au(t) +

t

Z

0

m(ts)ut(s)ds= 0, t∈[0,∞), u(0) =u0D(A),

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wherem: [0,∞)→L2(G).

Theorem 12. If mC1([0,∞), D(Ak)) with m(0)(x)≥ −q+ε for a ε > 0 and for all xG,u0D(Ak+1) and mt(t)v ∈D(Ak) for all t ∈ [0,∞) and vD(Ak), then problem (2) has a unique solution uC1(0,∞), D(Ak))∩ C2([0,∞), D(Ak−1)).

Proof. We define fort∈[0,∞) anduC1([0,∞), D(Ak))

w(t) :=

t

Z

0

mt(t−s)ut(s)ds.

By the help of Lemma 10 (ii), we easily seewC0([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).

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Lemma 13. Assumeu0D(Ak+1)andmC1([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(c1ε) and C0

C1ω < ε(c1ε).

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)

εc1

C3ωε(c1ε)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) εc1

C0

C1ωε(c1ε)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, u0D(Ak+1) andFC2k+1(R,R) withF(2k+1) locally Lipschitz- continuous andF(u0(x))>−q+εfor aε >0 and for allxG.

i) Letc1> ε.

ii) Supposeω >0 such thatC3ω < ε(c1ε) and CC0

1ω < ε(c1ε).

iii) Assumeα >0 such that (α+ 1)C3ω−ε(cc 1−ε)

1−ε ≤ −c1. iv) Letv1>0 such that

v1C3C4kAu0kα+1D(Ak)

εc1

C3ωε(c1ε) α

ω

and v1C3C4kAu0kα+2k+1D(Ak)

εc1 C3ωε(c1ε)

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 allxG.

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