A Necessary and Sufficient Condition for Global Existence for Nonlinear Semigroups

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

2 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 (if g : [0, ) X X so that g(t, x) = T (t)x, t 0, x X then g is continuous.)

3 Conventional Generator for T : X : subset of a Banach space Y T : a semigroup on X B = {(x, y) X Y : 1 y = lim t 0+ t (T (t)x x) } B called conventional generator of T.

4 Lie Generator for T on X: CB(X) : B-space of all bounded continuous functions X R. A = {(f, g) CB(X) 2 : g(x) = lim t 0+ 1 (f (T (t)x) f (x)), x X}. t

5 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 (lim t s T (t)x exists = s < m(x))

6 A Word from Sophus Lie:...all problems related to the one-parameter group may be solved by means of the infinitesimal transformation of the group.

7 ...alle auf die eingliedrige Gruppe bezüglichen Probleme durch Benutzung der infinitesimalen Transformation derselben allein gelöst werden können.

8 Generators and Semigroups: Hille-Yosida Theorem characterizes strongly continuous linear semigroups in terms of their conventional generators. Dorroh-N Theorem characterizes nonlinear jointly continuous semigroups on X in terms of their Lie generators.

9 Definitions CB(X): Bounded, continuous real functions on X. β convergence on X: Uniform convergence on compacta. SG(X): Global semigroups on X. LG(X): Linear derivations, β dense domain, nonexpansive resolvents with equicontinuity property.

10 Theorem (Dorroh-N): If A LG(X), ) there is a unique T SG(X) with Lie generator A. Moreover f (T (t)x) = lim n ((I t n A) n f )(x), x X, t 0, f CB(X). Conversely, if T SG(X) and A its Lie generator, then A LG(X).

11 Example of a Local Semigroup: X = [0, ). T (t)x = x 1 tx, t large as possible, x 0. Generated by solutions z to z(0) = x, z (t) = z(t) 2, t large as possible.

12 When is a Semigroup Local, Global? Theorem: Suppose T is either a local or global jointly continuous semigroup and A is the Lie generator of T. Then, A has a positive eigenvalue, with eigenfunction in CB(X), if and only if T is local.

13 Moreover If f (x) = exp( m(x)), x X, then f CB(X). f is an eigenvector of A if and only if T is local (m(x) < for some x X.)

14 Example for Theorem: For X = [0, ), B(x) = x 2, x X, corresponding semigroup T is given by T (t)x = x 1 tx, x 0, t [0, 1 x ). Eigenvector of Lie generator A is f : f (x) = exp( 1 ), x > 0, f (0) = 0. x 1 is an eigenvalue of A.

15 Characterization of Eigenfunctions of A: Theorem: Suppose that T is a local semigroup on X, A is the Lie generator of T and f (x) = exp( m(x)), x X. (1) If g CB(X), Ag = g and x X, then there is c R so that g(t (t)x) = cf (T (t)x), t [0, m(x)).

16 A Numerical Attack: Compute spectra of discrete versions of Lie generators. Use discrete versions of Sobolev spaces X and of CB(X). Single ODE examples. Pairs of ODEs

17 Some Test Cases in One Dimension: u = u 2 u = u(u 1)(u 2) u = u u = u 2 u = exp(u/10) u = u 3 u

18 8 n=1000, b(u) = u Figure: u = u 2

19 100 n=2000, b(u) = u*(u 1)*(u 2) Figure: u = u(u 1)(u 2)

20 1.2 n=2000, b(u) = u Figure: u = u

21 1 x 10 8 n=10000, b(u) = u Figure: u = u 2

22 45 n=20000, b(u) = exp(u/10) Figure: u = exp(u/10)

23 120 n=2000, b(u) = u 3 u Figure: u = u 3 u

24 A Decoupled System of Two ODEs u(0) = x 0, u = u 2, v(0) = y 0, v = v 2. Observe m(x, y) = min( 1 x, 1 ) x, y 0. y

25 J. W. Neuberger A 0 Necessary and Sufficient Condition for Global Existence for Non

26 A Second Decoupled Pair u(0) = x 0, u = u 2, v(0) = y 0, v = v 2. Observe m(x, y) = 1, x, y 0. x

27 J. W. Neuberger A 0 Necessary and Sufficient Condition for Global Existence for Non

28 A Coupled System u(0) = x 0, u = v 2, v(0) = y 0, v = u 2. An expression for the corresponding stoping time function is not available, but it can be shown that the underlying semigroup is local.

29 J. W. Neuberger A 0 Necessary and Sufficient Condition for Global Existence for Non

30 John M. Neuberger, Jim Swift and Nandor Sieben are collaborating on numerics in more than one dimension.

31 References: J. R. Dorroh, J. W. Neuberger, A Theory of Strongly Continuous Semigroups in Terms of Lie Generators, J. Functional Analysis, 136 (1996), J. W. Neuberger, Lie Generators for Local Semigroups, Contemporary Mathematics, 513 (2010),

32 J. W. Neuberger, How to distinguish local semigroups from global semigroups, Discrete amd Continuous Dynamical Systems-A, 33, (2012), Preprint at arxiv.org/abs/ J. W. Neuberger, A Sequence of Problems on Semigroups, Springer Problem Book Series, (2012).

33 John M. Neuberger, J. W. Neuberger, James W. Swift, A Linear Condition Determining Local or Global Existence for Nonlinear Problems, Central European J. Mathematics, 11 (2013),