A Necessary and Sufficient Condition for Global Existence ......A Necessary and Sufﬁcient Condition for Global Existence for Nonlinear Semigroups J. W. Neuberger University of North

A Necessary and Sufficient Condition forGlobal Existence for Nonlinear Semigroups

J. W. Neuberger

University of North Texas

J. W. Neuberger A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups

Global Semigroups:

X : Polish Space.T (t) : X → X , t ∈ [0,∞).T (0)x = x , x ∈ X .T (t)T (s) = T (t + s), t , s ≥ 0.T jointly continuous (ifg : [0,∞)× X → X so thatg(t , x) = T (t)x , t ≥ 0, x ∈ X then gis continuous.)


Conventional Generator for T :

X : subset of a Banach space YT : a semigroup on XB = {(x , y) ∈ X × Y :y = limt→0+ 1t (T (t)x − x) }B called conventional generator of T .


Lie Generator for T on X :

CB(X ) : B-space of all boundedcontinuous functions X → R.A = {(f ,g) ∈ CB(X )2 :

g(x) = limt→0+

1t(f (T (t)x)−f (x)), x ∈ X}.


Local Semigroup T :

m, continuous X → (0,∞],x ∈ D(T (t)) ⇐⇒ t ∈ [0,m(x)).If t , s ≥ 0, x ∈ X , then T (t)T (s)x =T (t + s)x ⇐⇒ t + s < m(x).T jointly continuous, maximal(limt→s−T (t)x exists =⇒ s < m(x))


A Word from Sophus Lie:‘....all problems related tothe one-parameter groupmay be solved by means ofthe infinitesimaltransformation of the group.’


‘.....alle auf die eingliedrigeGruppe bezüglichen Problemedurch Benutzung derinfinitesimalen Transformationderselben allein gelöst werdenkönnen.’


Generators and Semigroups:

Hille-Yosida Theorem characterizesstrongly continuous linearsemigroups in terms of theirconventional generators.Dorroh-N Theorem characterizesnonlinear jointly continuoussemigroups on X in terms of their Liegenerators.


Definitions

CB(X ): Bounded, continuous realfunctions on X .β convergence on X : Uniformconvergence on compacta.SG(X ): Global semigroups on X .LG(X ): Linear derivations, β densedomain, nonexpansive resolventswith equicontinuity property.


Theorem (Dorroh-N):If A ∈ LG(X ), ) there is a uniqueT ∈ SG(X ) with Lie generator A.Moreover

f (T (t)x) = limn→∞

((I − tn

A)−nf )(x),

x ∈ X , t ≥ 0, f ∈ CB(X ).

Conversely, if T ∈ SG(X ) and A its Liegenerator, then A ∈ LG(X ).


Example of a Local Semigroup:

X = [0,∞).T (t)x = x1−tx , t large as possible,x ≥ 0.Generated by solutions z toz(0) = x , z ′(t) = z(t)2, t large aspossible.


When is a Semigroup Local, Global?

Theorem: SupposeT is either a local or global jointlycontinuous semigroup andA is the Lie generator of T .

Then, A has a positive eigenvalue, witheigenfunction in CB(X ), if and only if Tis local.


Moreover . . . . . .If f (x) = exp(−m(x)), x ∈ X ,

then f ∈ CB(X ).f is an eigenvector of Aif and only if T is local(m(x)

Example for Theorem:

For X = [0,∞),B(x) = x2, x ∈ X ,corresponding semigroup T is givenby T (t)x = x1−tx , x ≥ 0, t ∈ [0,

1x ).

Eigenvector of Lie generator A is f :

f (x) = exp(−1x), x > 0, f (0) = 0.

1 is an eigenvalue of A.


Characterization of Eigenfunctions of A:

Theorem: Suppose that T is a localsemigroup on X , A is the Lie generatorof T and

f (x) = exp(−m(x)), x ∈ X . (1)

If g ∈ CB(X ), Ag = g and x ∈ X , thenthere is c ∈ R so that

g(T (t)x) = cf (T (t)x), t ∈ [0,m(x)).


A Numerical Attack:

Compute spectra of discrete versionsof Lie generators.Use discrete versions of Sobolevspaces X and of CB(X ).Single ODE examples.Pairs of ODEs


Some Test Cases in One Dimension:

u′ = u2

u′ = u(u − 1)(u − 2)u′ = uu′ = −u2

u′ = exp(u/10)u′ = u3 − u


0 2 4 6 8 10 12 14 16 18 20−1

0

1

2

3

4

5

6

7

8n=1000, b(u) = u2

Figure: u′ = u2


0 2 4 6 8 10 12 14 16 18 20−20

0

20

40

60

80

100n=2000, b(u) = u*(u−1)*(u−2)

Figure: u′ = u(u − 1)(u − 2)


0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2n=2000, b(u) = u

Figure: u′ = u


0 2 4 6 8 10 12 14 16 18 20−6

−5

−4

−3

−2

−1

0

1x 10−8 n=10000, b(u) = −u

2

Figure: u′ = −u2


0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45n=20000, b(u) = exp(u/10)

Figure: u′ = exp(u/10)


0 5 10 15 20−20

0

20

40

60

80

100

120n=2000, b(u) = u3 − u

Figure: u′ = u3 − u


A Decoupled System of Two ODEs

u(0) = x ≥ 0, u′ = u2,v(0) = y ≥ 0, v ′ = v2.

Observe

m(x , y) = min(1x,1y) x , y ≥ 0.


02

46

810

0

2

4

6

8

10−0.01

0

0.01

0.02

0.03

0.04

Figure: u′ = u2, v ′ = v2


A Second Decoupled Pair

u(0) = x ≥ 0, u′ = u2,v(0) = y ≥ 0, v ′ = −v2.

Observe

m(x , y) =1x, x , y ≥ 0.


02

46

810

0

2

4

6

8

10−0.01

0

0.01

0.02

0.03

0.04

0.05

Figure: u′ = u2, v ′ = −v2


A Coupled System

u(0) = x ≥ 0, u′ = v2,v(0) = y ≥ 0, v ′ = u2.

An expression for the correspondingstoping time function is not available,but it can be shown that the underlyingsemigroup is local.


02

46

810

0

2

4

6

8

10−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Figure: u′ = v2, v ′ = u2


John M. Neuberger, Jim Swiftand Nandor Sieben are collaborating onnumerics in more than one dimension.


References:

J. R. Dorroh, J. W. Neuberger, A Theoryof Strongly Continuous Semigroups inTerms of Lie Generators, J. FunctionalAnalysis, 136 (1996), 114-126.J. W. Neuberger, Lie Generators forLocal Semigroups, ContemporaryMathematics, 513 (2010), 233-238.


J. W. Neuberger, How to distinguishlocal semigroups from globalsemigroups, Discrete amd ContinuousDynamical Systems-A, 33, (2012),5293-5303.Preprint at arXiv.org/abs/1109.2184.J. W. Neuberger, A Sequence ofProblems on Semigroups, SpringerProblem Book Series, (2012).


John M. Neuberger, J. W. Neuberger,James W. Swift, A Linear ConditionDetermining Local or Global Existencefor Nonlinear Problems, CentralEuropean J. Mathematics, 11 (2013),1361-1374.


Documents

A Necessary and Sufficient Condition for Global Existence ......A Necessary and Sufﬁcient Condition for Global Existence for Nonlinear Semigroups J. W. Neuberger University of North