Eternal solutions and heteroclinic orbits of a semilinear parabolic equation
Abstract

This dissertation describes the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Along the way, a new and elementary proof of existence and uniqueness of solutions is given. Heteroclinic orbits are shown to be characterized by a particular functional being finite. A novel asymptotic-numeric matching scheme is used to uncover delicate bifurcation behavior in the equilibria. The exact nature of this bifurcation behavior leads to a demonstration that the equilibria are degenerate critical points in the sense of Morse. Finally, the space of heteroclinic orbits is shown to have a cell complex structure, which is finite dimensional when the number of equilibria is finite.

ETERNAL SOLUTIONS AND HETEROCLINIC ORBITS OF A SEMILINEAR PARABOLIC EQUATION

A Dissertation

Presented to the Faculty of the Graduate School

of Cornell University

in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

by

Michael Robinson

May 2008

Biographical Sketch

Michael Robinson entered the field of mathematics by a rather circuitous route. In high school, he learned computer programming as a hobby. It was there that his first exposure to partial differential equations occurred, when he wrote a solver for the invicid Navier-Stokes equations in the plane under the direction of Robert Ryder (who was then with Pratt & Whitney). Feeling that he ought to understand computer hardware at a deeper level, he enrolled at Rensselaer Polytechnic Institute and completed a Bachelor of Science in Electrical Engineering. His Master of Science degree in Mathematics at Rensselaer Polytechnic Institute was completed under the direction of Dr. Ashwani Kapila. His Master’s thesis combined the Navier-Stokes equations and Maxwell’s equations to examine the scattering of plasma waves. After completing his Master’s degree, Robinson went to work for Syracuse Research Corporation and enrolled at Cornell University one year later.

This project is dedicated to my wife, Donna, who urged me to do the obvious thing and continue my graduate studies.

Acknowledgements

I would like to thank all of the members of my thesis committee for their helpful and insightful discussions concerning this project. I would especially like to thank my advisor, Dr. John Hubbard for suggesting that I study as a “summer project.” This problem has led into a surprising variety of interesting mathematics.

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Chapter 1 Introduction

1.1 The Grand Plan

This dissertation presents some recent progress towards a rather lofty (and very difficult) goal. Specifically, there is great interest in understanding the topology of solution spaces of systems of semilinear parabolic equations,

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∂u1∂t=D1(u1,u2,...up)+f1(t,x,u1,u2,...up)∂u2∂t=D2(u1,u2,...up)+f2(t,x,u1,u2,...up)...∂up∂t=Dp(u1,u2,...up)+fp(t,x,u1,u2,...up) (1.1)

where , are densely-defined linear elliptic (diffusion) operators, and satisfy reasonable smoothness conditions. Of course, a major problem for anyone interested in (1.1) is existence and uniqueness of short-time solutions! Although the existence and uniqueness in general is daunting, many interesting and important problems have the form (1.1). (Fortunately, for some special cases, existence and uniqueness can be proven, as is done in Chapter 2.) The applications of (1.1) are numerous; for instance:

• The Navier-Stokes equations, which describe fluid flow, can be put in the form of (1.1) [20]. Understanding the topology of the solution space of the Navier-Stokes equations gives insight into the onset of turbulence. This has applications to fluid amplifiers (viscous fluid logic gates, with no moving parts) [21], in which a particular geometry and set of boundary conditions allows several semistable equilibria. The orbits which connect these equilibria (the heteroclinic orbits or heteroclines) can involve turbulent flows. As a result, understanding the topology of the space of heteroclines for such a situation might provide insight into turbulence phenomena.

• Many chemical reactions are of this form, in particular those describing combustion. The precise nature of the ignition of a flame is encoded in the topology of the solution space.

• The combination of the Navier-Stokes equations with combustion equations can model turbulent combustion phenomena. Such turbulent, reacting flows are important in modeling the inside of internal combustion engines. The ignition of a turbulent combustion depends very delicately upon the exact nature of the flow, and a topological description for such events is lacking.

• Related to the Navier-Stokes and chemical reaction equations are nonlinear wave equations. Many nonlinear wave equations are of the form (1.1), and traveling waves appear as heteroclines connecting two equilibrium states. Traveling waves are often stable, in that perturbations of them tend to “wash out” over time. However, there are interesting situations where traveling waves are suddenly suppressed as a parameter is changed slightly. This is a consequence of an abrupt change in the topology of the solution space.

• In population biology, (1.1) describes a number of competing or cooperating species. One can ask about the kinds of bifurcations in stable populations when a new species is introduced, or when harvesting patterns are changed.

• One of the most pressing issues in population biology is that of non-native invasive species, which disrupt the ecology of many parts of our planet. In agriculture, they cause significant crop losses and threaten our protected areas. One of the important problems concerning invasive species is how to displace them with minimal ecological impact. Understanding the topology of the space of solutions for (1.1) would help find optimal control algorithms for eliminating (or limiting) the spread of invasive species. There is a vast literature on this subject, going back to Fisher [13], Kolmogorov, Petrovski, and Piskunov [27].

1.2 Specialization to a scalar gradient equation

Since the general setting of (1.1) is much too difficult to allow any kind of progress at this time, we must instead consider more specialized situations. To this end, we restrict attention to the case of

• a scalar equation (p=1 in (1.1)),

• in one spatial dimension, where

• the diffusion operator is the Laplacian, , and

• the reaction term is a polynomial.

In other words, consider

 ∂u(t,x)∂t=Δu(t,x)+N∑i=0ai(x)ui(t,x)=Δu+P(u), (1.2)

where and , and the are bounded and smooth. In this dissertation, we consider eternal solutions, those satisfying (1.2) that lie in for . In particular, observe that is densely defined when is not an integer.

This kind of equation provides a simple model for a number of physical phenomena. First, choosing the right side to be results in an equation which can represent a model of the population of a single species with diffusion and a spatially-varying carrying capacity, . As a second application, this equation is a very simple model of combustion. If is a positive constant, then the equation supports traveling waves. Such traveling waves can model the propagation of a flame through a fuel source.

1.3 Discussion of the literature

Equations of the form (1.2) have been of interest to researchers for quite some time. Existence and uniqueness of solutions on short time intervals (on strips ) can been shown using semigroup methods and are entirely standard [44]. However, there are obstructions to the existence of eternal solutions. Aside from the typical loss of regularity due to solving the backwards heat equation, there is also a blow-up phenomenon which can spoil existence in the forward-time solution to (1.2). Blow-up phenonmena in the forward time Cauchy problem (where one does not consider ) have been studied by a number of authors [18] [10] [42] [26] [6] [46] [47]. More recently, Zhang et al. ([45] [38] [43]) studied global existence for the forward Cauchy problem for

 ∂u∂t=Δu+up−V(x)u

for positive . Du and Ma studied a related problem in [9] under more restricted conditions on the coefficients but they obtained stronger existence results. In fact, they found that all of the solutions which were defined for all tended to equilibrium solutions.

Eternal solutions to (1.2) are rather rare. Most works which describe blow-up make the assumption that the solution is positive. Unfortunately, blow-up is much more difficult to characterize in the general situation, and understanding exactly what kind of initial conditions are responsible for blow-up in the Cauchy problem for (1.2) is an important part of the question.

The boundary value problem that results from taking for some bounded (instead of ) has also been discussed extensively in the literature [20] [24] [7]. For the boundary value problem, all bounded forward Cauchy problem solutions tend to limits as , and these limits are equilibrium solutions.

Almost all of the literature (including this dissertation) describing eternal solutions to (1.2) is restricted to discussing heteroclines. For unbounded domains and certain symmetries among the coefficients , one can find traveling waves. Since the propagation of waves in nonlinear models is of great interest in applications, there is much written on the subject. The general idea is that one makes a change of variables which reduces (1.2) to an ordinary differential equation. This ordinary differential equation describes the profile of a traveling wave. Powerful topologically-motivated techniques, such as the Leray-Schauder degree, can be used to prove existence of wave solutions to (1.2). Asymptotic methods can be used to determine the wave speed , which is often of interest in applications. See [40] for a very thorough introduction to the subject of traveling waves in (1.2).

1.4 A Morse-theoretic approach

A somewhat less traditional approach to studying (1.2) exists. This method attempts to directly compute topological invariants for the space of heteroclinic orbits of (1.2). It makes use of the fact that equation (1.2) defines the flow of the gradient of a certain action functional,

 A(f)=∫Rn12∥∇f∥2+N∑i=0ai(x)i+1fi+1(x)dx. (1.3)

It is then evident that along a solution to (1.2), is a monotonic function in . As an immediate consequence, nonconstant -periodic solutions to (1.2) do not exist. This kind of behavior suggests that a Morse-theoretic framework might be helpful.

Morse theory is concerned with the computation of homotopy or homology groups of a Riemannian or Hilbert manifold by “exploring” it with a suitable scalar function . The function is selected to satisfy the Morse-Smale (-Floer) conditions, namely

• a nondegeneracy condition: if is a critical point (), then the Hessian at is nonsingular,

• the Morse index, which is the number of negative eigenvalues of the Hessian is finite for each critical point,

• stable and unstable manifolds for the gradient flow of are transverse (the Smale condition), and

• if is noncompact, there is a compact isolating neighborhood for each pair of critical points under the gradient flow [17].

The function can be thought of as a special kind of “height function” on . One then examines the topology of sets , which form a cover for . It is straightforward to show that the homotopy class of remains constant on when there are no critical points in . The homotopy class of changes abruptly, however, when contains a critical point. Morse theory describes how this homotopy class changes by the attachment of handles to . A very readable introduction to Morse theory is [30].

There is a dual formulation of Morse theory, which uses Witten’s complex to compute homology instead of homotopy. This approach is better suited to understanding differential equations, as it focuses not on level sets, but rather on the flow of

 dxdt=∇f(x(t)). (1.4)

Using this flow, one constructs a chain complex in which the are free modules generated by the critical points of Morse index . The boundary maps are then constructed by the formula

 ∂k(q)=∑p∈Ck−1n(q,p)p,

where is the number of heteroclinic orbits of (1.4) which connect to , counted with sign. The surprising thing is that this chain complex computes the homology of ! A thorough, modern treatment of Morse theory can be found in [3].

1.5 Floer homology

A similar theory can be made to work even if the Morse index of all critical points is infinite. Instead of relying on the critical points to supply an index directly, one constructs a “relative” index based on the structure of connecting manifolds. This theory was first assembled by Floer for the purpose of understanding the homology of the space of orbits for an exact symplectomorphism [15]. More recently, Ghrist et al. [19] have done work on a similar theory for a certain evolution of braids. What is crucial to Floer theories is that the manifold of heteroclinic orbits which connect a given pair of equilibria (a connecting manifold) is finite dimensional, and compact modulo time translation. Suppose that the connecting manifold for a given pair of equilibria has dimension . One shows that the following two relations hold for this dimension:

 μ(x,x)=0,μ(x,z)=μ(x,y)+μ(y,z), (1.5)

where are distinct critical points. This relation allows one to assign indices to the critical points such that . Evidently, is only defined uniquely up to an additive constant. However, one can use the index in place of the Morse index to construct a Witten complex in the usual way. However, the ambiguity in the definition of means that the degrees of the resulting complex are only defined up to this additive constant. In this way, one obtains a kind of ambiguous-degree homology theory, which is called “Floer homology.” Alternatively, one may take the dual approach, and use the finite dimensional connecting manifolds to assemble a cell complex structure for the space of heteroclines. The attaching maps of this cell complex are evidently related to the boundary maps of the Witten complex.

1.6 How to construct a Floer theory for parabolic equations

For the case of semilinear parabolic equations, the following must be established in order to construct a Floer theory:

1. One must compute the Morse index of each critical point, or more properly, show that the Morse index is not well-defined due to degeneracy.

2. One must show that connecting manifolds are finite dimensional, and that they form a cell-complex structure for the space of heteroclinic orbits .

3. The connecting manifolds must obey the additivity relation (1.5).

4. The space of heteroclinic orbits must be compact moduluo time translation.

5. One must construct the boundary maps in the Witten complex, and verify that they actually form a chain complex that computes the homology of .

This dissertation contains proofs of items 1, 2, and most of 3 for a special case of (1.2). (In the case of a bounded spatial domain, all but item 5 are standard [32], [24].) Rather than working with equation (1.2) in full generality, the later chapters use the following special case:

 ∂u(t,x)∂t=∂2u(t,x)∂x2−u2(t,x)+ϕ(x), (1.6)

where tends to zero as . The resulting questions and techniques have obvious generalizations to (1.2), though there are many technical obstacles. Higher degree polynomial nonlinearities may of course have more than two roots, which creates the possibility for more complicated equilibrium structure than we analyze here.

This simpler model still provides insight into applications, as it is still a model of the population of a single species, with a spatially-varying carrying capacity, . Indeed, one easily finds that under certain conditions the behavior of solutions to (1.6) is reminiscent of the growth and (admittedly tenuous) control of invasive species [4]. It is the control of invasive species that is of most interest, and it is also what the structure of the boundary maps reveals. In the example given in Chapter 8, there is one more stable equilibrium, and several other less stable ones. The more stable equilibrium can be thought of as the situation where an invasive species dominates. The task, then, is to try to perturb the system so that it no longer is attracted to that equilibrium. An optimal control approach is to perturb the system so that it barely crosses the boundary of the stable manifold of the the undesired equilibrium, and thereby the invasive species is eventually brought under control with minimal disturbance to the rest of the environment.

1.7 Outline and prerequisite results

While Chapters 6 and 7 contain the results of most interest to constructing a Floer theory for (1.6), the other chapters provide a number of results that are prerequisites.

1.7.1 Prerequisites that concern the higher degree case

Chapters 2 and 3 are devoted to the general equation (1.2). The first of these provides a new proof of short-time existence and uniqueness for (1.2), and as a side-effect provides a numerical method for approximating its solutions. While existence and uniqueness for (1.2) is standard [44], the usual proofs are not suited to computation.

The first novel result for (1.2) is obtained in Chapter 3, where a decay condition on the allows us to classify heteroclinic orbits of (1.2). In particular, if an eternal solution exists and converges uniformly to equilibria as if and only if has finite energy (supremum of the difference in the action functional (1.3) over all time). Without the decay condition on the , finite energy classifies those eternal solutions that connect finite action equilibria.

The result in Chapter 3 is actually quite important for connecting the Morse-theoretic results with the analysis. Morse theory, and in particular Witten’s complex, requires the flow to have a gradient structure. As a result, the space on which the flow acts must have an inner product structure, so a natural solution space would be . However, the proofs of the cell-complex structure (Chapter 7) do not work with timeslices in . In particular, one needs this space to have a Banach algebra structure (in which the Laplacian is densely defined), so the Hölder space for is more natural. What follows from the results of Chapter 3 is that the space of heteroclines lies in the intersection , so in fact there is no difficulty (Corollary 30).

1.7.2 Prerequisites that concern the quadratic case

In the remaining chapters (Chapters 4 through 8), only the special case (1.6) is considered. This is quite sufficient to obtain interesting results about the structure of .

It is important to understand the collection of equilibrium solutions for (1.6), which are global solutions to

 0=Δu−u2+ϕ (1.7)

on all of . Like (1.2), there are obstructions to global existence in (1.7) [39]. Indeed, there are fairly few global solutions to (1.7). We examine solutions to this problem under asymptotic decay conditions for in Chapter 4. The solution reveals delicate bifurcation behavior in the number of equilbria as is varied. Further, the asymptotic behavior is such that all global solutions to (1.7) have finite action (see (1.3)).

Since they are rare, it is reassuring to construct an example of heteroclinic orbits, which is done in Chapter 5. This example makes specific use of the structure of the equilibrium solutions, in particular, their asymptotic decay is crucial.

In order to construct a Morse theory for (1.2), understanding the dimension of the stable, center, and unstable manifolds of equilibria is important. In Chapter 6 it is shown that the the center/stable manifold’s dimension is typically infinite, and later in Chapter 7 it is shown that the unstable manifold has finite dimension. Each equilibrium solution is in fact unstable, even if its linearization is stable. This implies that each equilibrium is a degenerate critical point. This neatly derails any hope of using a standard Morse theory, or even using any of its extensions to infinite dimensional dynamical systems [31]. (In Chapter 9, Conjecture 104 suggests that restriction of the flow to may correct the degeneracy.)

The most important result of this work is obtained in Chapter 7, where the space of heteroclinic orbits is shown to have a cell-complex structure (with finite dimensional cells). The dimension of each cell is determined, under a standing assumption of transversality (Conjecture 95). From the formula for the dimension of the cells, it is clear that an additivity rule like (1.5) will hold. This result is further explained by an example in Chapter 8. Finally, in Chapter 9, several important future directions are outlined.

Chapter 2 Short-time existence and uniqueness

2.1 Introduction

(This chapter has already been published as [35].)

Existence and uniqueness of solutions for (2.1) under reasonable initial conditions have been known for some time. For instance, [20] and [44] contain straightforward proofs using semigroup methods. The purpose of this chapter is to show how a more elementary proof can be obtained from a sequence of explicitly computed discrete-time approximations.

Due to their theoretical and computational stability, implicit iteration schemes are often prefered over their easier-to-implement explicit analogues. However, in the case of semilinear equations, one can form a hybrid implicit-explicit (IMEX) method which offers computational and theoretical benefits. The use of IMEX methods for approximating semilinear parabolic equations is well-established [2]. Many of the recent works on these methods employ discretizations in both space and time. These fully discrete approximations can be computed directly by a computer. However, one can obtain a stronger condition for convergence of the approximation if only the time dimension is discretized [8]. We show how an even stronger condition for convergence is met by the Cauchy problem for

 ∂u(x,t)∂t=Δu(x,t)+∞∑i=0ai(x)ui(x,t), (2.1)

where , and how convergence of this method provides an elementary proof of existence and uniqueness of solutions.

The Cauchy problem for (2.1) arises in a variety of settings. Notably, some reaction-diffusion equations are of this form [12]. Another application is the special case

 ∂u(x,t)∂t=Δu(x,t)−u2(x,t)+a0(x),

where is a nonzero function of . This situation corresponds to a spatially-dependent logistic equation with a diffusion term, which can be thought of as a toy model of population growth with migration.

Following [8], the approximation to be used is

 un+1=(I−hΔ)−1(un+h∞∑i=0aiuin), (2.2)

which is obtained by inverting the linear portion of a discrete version of (2.1). For brevity, we shall call (2.2) the implicit-explicit method. (In the summary paper [2], this is called an SBDF method, to distinguish it from other implicit-explicit methods.) One can compute the operator explicitly using Fourier transform methods, and obtain a proof of the numerical stability of the iteration as a whole.

2.2 A version of the fundamental inequality

In order to simplify the algebraic expressions, we make the following definitions.

Definition 1.

Let

 F(u(x,t))=Δu(x,t)+∞∑i=0ai(x)ui(x,t), (2.3)

and

 G(u(x,t))=∞∑i=0ai(x)ui(x,t). (2.4)
Definition 2.

Define the analytic functions

 g1(z)=∞∑i=0∥ai∥1zi, (2.5)

and

 g∞(z)=∞∑i=0∥ai∥∞zi. (2.6)

Since we do not discretize the spatial dimension, we can employ some of the theory of ordinary differential equations. We therefore first prove a variant of the fundamental (Gronwall) inequality for (2.1) as is done in [23]. The fundamental inequality gives a sufficient condition for approximate solutions to converge. A slightly weaker version of Lemma 3 was obtained in Theorem 3.1 of [8], where the existence of solutions was required.

Lemma 3.

Suppose is a sequence of piecewise functions , such that

1. there exist so that for each and , and ,

2. for each and , the series and converge,

3. for each , and , and

4. for all

Then for each , is a Cauchy sequence in .

Proof.

Let be given. Let . Notice that the fourth condition in the hypothesis gives .

 η′(t)=2∫(u′i(t)−u′j(t))(ui(t)−uj(t))dx.

But, is equivalent to the statement that for each and ,

 F(ui(x,t))−ϵi

giving

 η′(t) ≤ 2∫(F(ui(t))−F(uj(t)))(ui(t)−uj(t))dx +2(ϵi+ϵj)∫|ui(t)−uj(t)|dx ≤ 2∫(Δui(t)+G(ui(t))−Δuj(t)−G(uj(t)))(ui(t)−uj(t))dx +2(ϵi+ϵj)∥ui(t)−uj(t)∥1 ≤ 2∫(Δ(ui(t)−uj(t)))(ui(t)−uj(t))dx +2∫(G(ui(t))−G(uj(t)))(ui(t)−uj(t))dx+2(ϵi+ϵj)∥ui(t)−uj(t)∥1 ≤ −2∫∥∇(ui(t)−uj(t))∥2dx+2∥G(ui(t))−G(uj(t))∥2∥ui(t)−uj(t)∥2 +2(ϵi+ϵj)∥ui(t)−uj(t)∥1 ≤ 2∥G(ui(t))−G(uj(t))∥2∥ui(t)−uj(t)∥2+2(ϵi+ϵj)∥ui(t)−uj(t)∥1.

Now also

 ∥G(ui(t)) − G(uj(t))∥2=∥∥ ∥∥∞∑k=0ak(uki(t)−ukj(t))∥∥ ∥∥2 ≤ ∞∑k=0∥ak∥∞∥∥uki(t)−ukj(t)∥∥2 ≤ ∞∑k=0∥ak∥∞√∫(uki(x,t)−ukj(x,t))2dx ≤ ∞∑k=0∥ak∥∞ ⎷∫(ui(x,t)−uj(x,t))2(k−1∑m=0umi(x,t)uk−m−1j(x,t))2dx ≤ ∞∑k=0∥ak∥∞∥∥ ∥∥k−1∑m=0umi(t)uk−m−1j(t)∥∥ ∥∥∞∥ui(t)−uj(t)∥2 ≤ (∞∑k=0∥ak∥∞kBk−1)∥ui(t)−uj(t)∥2 ≤ g′∞(B)∥ui(t)−uj(t)∥2,

which allows

 η′(t) ≤ 2g′∞(B)∥ui(t)−uj(t)∥22+2(ϵi+ϵj)∥ui(t)−uj(t)∥1. ≤ 2g′∞(B)η(t)+2(ϵi+ϵj)∥ui(t)−uj(t)∥1.
 η′(t)−2g′∞(B)η(t) ≤ 2(ϵi+ϵj)∥ui(t)−uj(t)∥1 ddt(η(t)e−2g′∞(B)t) ≤ 2(ϵi+ϵj)e−2g′∞(B)t∥ui(t)−uj(t)∥1,

so (recall )

 η(t) ≤ [2(ϵi+ϵj)∫t0e−2g′∞(B)s∥ui(s)−uj(s)∥1ds]e2g′∞(B)t ≤ [2(ϵi+ϵj)∫t0∥ui(s)−uj(s)∥1ds]e2g′∞(B)t. ≤ 4(ϵi+ϵj)Ate2g′∞(B)t.

Hence as , for each . Thus for each , is a Cauchy sequence in . ∎

Remark 4.

Since and is complete, Lemma 3 gives conditions for existence and uniqueness of a short-time solution to (2.1).

Lemma 5.

Suppose is the sequence of functions defined in Lemma 3, and that in . Then

 u′(t,x)=limi→∞u′i(t,x) for almost % every x, (2.7)

wherever the limit exists.

Proof.

Notice that since each and , the dominated convergence theorem allows for each

 ∫t0limi→∞u′i(τ,x)dτ = limi→∞∫t0u′i(τ,x)dτ = limi→∞(ui(t,x)−ui(0,x)) = u(t,x)−u(0,x) for almost every x.

Hence, by differentiating in ,

 u′(t,x)=limi→∞u′i(t,x) for almost % every x.

2.3 The implicit-explicit approximation

In this section, we consider the case of a 1-dimensional spatial domain, that is, . There is no obstruction to extending any of these results to higher dimensions, though it complicates the exposition unnecessarily.

As is usual, the first task is to define the function spaces to be used. Initial conditions will be drawn from a subspace of , as suggested by Lemma 3, and the first four spatial derivatives will be prescribed, for use in Lemma 10.

Definition 6.

Let

 W=L1∩C4(R),

where we interpret as being the space of bounded functions with four continuous bounded derivatives. For the remainder of this chapter, we consider the case where each of the coefficients . Then let . We consider the case where the initial condition is drawn from .

An approximate solution given by the implicit-explicit iteration will be the piecewise linear interpolation through the iterates computed by (2.2). A smoother approximation will prove to be unnecessary, as will be shown in Lemma 11.

Definition 7.

Suppose and are given. Put

 fn+1=(I−hΔ)−1(fn+hG(fn)). (2.8)

The function

 u(t)=(1−(th−n(t)))fn(t)+(th−n(t))fn(t)+1, (2.9)

where , is called the implicit-explicit iteration of size beginning at .

Calculation 8.

We explicitly compute the operator using Fourier transforms. Suppose

 (I−hΔ)u(x)=u(x)−hΔu(x)=f(x).

Taking the Fourier transform (with transformed variable ) gives

 ^u(ω)+hω2^u(ω)=^f(ω),
 ^u(ω)=^f(ω)1+hω2.

The Fourier inversion theorem yields

 u(x) = 12π∫eiωx1+hω2∫f(y)e−iωydydω = ∫f(y)(12π∫eiω(x−y)1+hω2dω)dy.

Using the method of residues, this can be simplified to give

 u(x)=((I−hΔ)−1f)(x)=12√h∫f(y)e−|y−x|/√hdy. (2.10)
Calculation 9.

Bounds on the and operator norms of are now computed. First, let . Then

 |((I−hΔ)−1f)(x)| = ∣∣∣12√h∫f(y)e−|y−x|/√hdy∣∣∣ ≤ ∥f∥∞12√h∫e−|y−x|/√hdy ≤ ∥f∥∞1√h∫∞0e−s/√hds ≤ ∥f∥∞,

so .

Now, let . So then

 ∥(I−hΔ)−1f∥1 = ∫∞−∞∣∣∣12√h∫∞−∞f(y)e−|y−x|/√hdy∣∣∣dx ≤ 12√h∫∞−∞∫∞−∞|f(y)|e−|y−x|/√hdydx ≤ 1√h∫∞−∞|f(y)|∫∞0e−|y−x|/√hdxdy ≤ ∫∞−∞|f(y)|dy=∥f∥1,

which means .

The third condition of Lemma 3 is a control on the slope error of the approximation. A bound on this error may be established for the implicit-explicit iteration as follows.

Lemma 10.

Suppose , . Put , where

 D=(I−hΔ)−1(f0+hG(f0))−f0h

Then for every ,

 ∥f′(t)−F(f(t))∥∞=O(h). (2.11)
Proof.

Recall every function in will have bounded partial derivatives up to fourth order from Definition 6.

 ∥f′(t)−F(f(t))∥∞ = ∥D−(Δ(f0+tD)+G(f0+tD))∥∞ = ∥∥ ∥∥D−(Δ(f0+tD)+∞∑i=0ai(f0+tD)i)∥∥ ∥∥∞ ≤ ∥∥ ∥∥D−Δf0−tΔD−∞∑i=0ai(i∑j=0(ij)fj0(tD)i−j)∥∥ ∥∥∞ ≤ ∥∥ ∥∥D−Δf0−tΔD−∞∑i=0aifi0∥∥ ∥∥∞+O(h) ≤ ∥∥∥(I−hΔ)−1−Ihf0−Δf0 +((I−hΔ)−1−I)G(f0)∥∥∞+O(h)

Now, using the fact that ,

 ∥f′(t)−F(f(t))∥∞ ≤ ∥∥(I−hΔ)−1Δf0−Δf0 +(I−hΔ)−1(hΔ)G(f0)∥∥∞+O(h) ≤ ∥(I−hΔ)−1(hΔ)(Δf0+G(f0))∥∞+O(h) ≤ h∥(I−hΔ)−1(ΔF(f0))∥∞+O(h) ≤ h∥(I−hΔ)−1∥∞∥(ΔF(f0))∥∞+O(h)=O(h)

Lemma 11.

Suppose . Let be the implicit-explicit iteration of size beginning at on . Then provided there exist such that for each and , and , then the sequence converges pointwise to a function in . The limit function is piecewise differentiable in .

Proof.

Let be the implicit-explicit iteration of size . By Lemma 10, the slope error is bounded:

 ∥u′i(t)−F(ui(t))∥∞=O(hi)=ϵi.

Notice that . Then, since , Lemma 3 applies, giving a pointwise limit function . Finally, since the slope error uniformly vanishes, Lemma 5 implies that the solution is piecewise differentiable. ∎

2.4 “A priori estimates” for the approximate solutions

Now we demonstrate that the implicit-explicit method converges for all initial conditions in . Specifically, for each , there exist such that for each and , and , given sufficiently small . We begin by recalling that from Calculation 9, the -norm of is less than one. This means that for the implicit-explicit iteration,

 ∥fn+1∥∞ ≤ ∥fn+hG(fn)∥∞ ≤ ∥fn∥∞+h∥∥ ∥∥∞∑i=0aifin∥∥ ∥∥∞ ≤ ∥fn∥∞+h∞∑i=0∥ai∥∞∥fin∥∞ ≤ ∥fn∥∞+h∞∑i=0∥ai∥∞∥fn∥i∞ ≤ ∥fn∥∞+hg∞(∥fn∥∞)

Hence the norm of each step of the implicit-explicit iteration will be controlled by the behavior of the recursion

 fn+1=fn+hg∞(fn), (2.12)

for . Since we are only concerned with short-time existence and uniqueness, we look specifically at and , for fixed and .

Remark 12.

The recursion defined by (2.12) is an Euler solver for

 dydt=g∞(y), with y(0)=f0. (2.13)

This equation is separable, and is analytic near , so there exists a unique solution for the initial value problem (2.13) for sufficiently short time. Also, whenever

 d2ydt2=g′∞(y(t))>0,

the function is concave up. As a result, the exact solution to (2.13) provides an upper bound for the recursion (2.12). More precisely, we have the following result.

Lemma 13.

Suppose in (2.13). Let be given so that is continuous on , and let . Then for each , , where satisfies (2.12) with .

Proof.

Since the right side of (2.13) is strictly positive, the maximum of is attained at on any interval where is continuous. Furthermore, since , it follows from Remark 12 that is concave up on all of . Therefore, is a convex function on . Hence Euler’s method, (2.12), will always underestimate the true value of . Another way of stating this is that

 fn≤y(nh)≤y(T).

Using Lemma 13, the growth of iterates to (2.12) may be controlled independently of the step size. This provides a uniform bound on the sequence of implicit-explicit approximations.

Lemma 14.

Suppose for . Let be the implicit-explicit iteration of size beginning at on . Then there exists a such that for each and , we have for sufficiently small .

Proof.

Suppose is the -th step of the implicit-explicit iteration of size . If we let , Lemma 13 implies that for any and any

 ∥fin∥∞≤y(T)

for sufficiently small T. Hence by (2.9) and the triangle inequality, for all and . ∎

With the bound on the suprema of the approximations, we can obtain a bound on the 1-norms.

Lemma 15.

Suppose for . Let be the implicit-explicit iteration of size beginning at on . Then there exists an such that for each and , we have for sufficiently small .

Proof.

First, notice that Lemma