J Dyn Diff Equat (2012) 24:341–367DOI 10.1007/s10884-012-9241-x

Stability and Freezing of Nonlinear Waves in First OrderHyperbolic PDEs

Jens Rottmann-Matthes

Received: 4 July 2011 / Revised: 22 September 2011 / Published online: 28 February 2012© Springer Science+Business Media, LLC 2012

Abstract It is a well-known problem to derive nonlinear stability of a traveling wavefrom the spectral stability of a linearization. In this paper we prove such a result for alarge class of hyperbolic systems. To cope with the unknown asymptotic phase, the prob-lem is reformulated as a partial differential algebraic equation for which asymptotic stabilitybecomes usual Lyapunov stability. The stability proof is then based on linear estimates from(Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlin-ear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J ApplDyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited forthe long time simulation and numerical approximation of the asymptotic behavior. The theoryis illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxleyequations.

Keywords Hyperbolic partial differential equations · Traveling waves · Partial differentialalgebraic equations · Nonlinear stability · Asymptotic behavior · Freezing method

Mathematics Subject Classification (2000) 35B40 · 35P15 · 35L45 · 65P40

1 Introduction

At the latest since the work of Sattinger [23] it is well-known that for many reaction-diffusionsystems the nonlinear stability of traveling waves can be derived from their spectral stability.This is usually proved using analytic semigroup theory, see for example [10] and [26, Ch.5]. In the context of analytic semigroup theory one can apply the spectral mapping theo-rem, which yields linear stability from spectral properties of the generator. Here we considerfirst order hyperbolic problems which do not generate analytic but only strongly continuoussemigroups. In general, the spectral mapping theorem does not hold for arbitrary strongly

J. Rottmann-Matthes (B)Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germanye-mail: jrottman@math.uni-bielefeld.de

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continuous semigroups, see for example [9, Ch. IV, Sect. 3]. Therefore we will use the linearresults from [20], which are a kind of Gearhardt-Prüss-Greiner theorem for partial differentialalgebraic equations, see also [9, Ch. V, Thm. 1.11].

We analyze the Cauchy problem for hyperbolic PDEs of the form

vt = Bvx + f (v) in R × R+, v(0) = v0, v(x, t) ∈ Rm, (1.1)

that possess a traveling wave solution V (x, t) = v(x − λt), where v is the profile and λ thespeed of the wave. Our assumptions on B allow (1.1) to be a non-strictly hyperbolic systemcoupled to a system of ODEs, see below.

Equations of this form naturally arise in the context of reaction diffusion equations whenthe flux q is not given by the classical Fick law q = −vx , but by the modified Fickian law,i.e.

vt + qx = r(v), T qt + q = −vx .

This flux is the well-known Cattaneo–Maxwell flux and has been derived by Cattaneo in [7]in the context of heat transfer, to prevent the unphysical infinite speed of propagation whichappears when Fourier’s law is applied. A review article about modified Fourier’s law in thephysics literature is [12], see also [16, Ch. 2] and the references therein for generalizationsto thermodynamics.

The importance of a finite speed of propagation has been emphasized in the context ofchemical reaction systems, by King et al. [13]. They observe that for certain supply rates,the usual Fickian law leads to anomalous behavior which is caused by the infinite speed ofpropagation. Therefore, they propose in [14] to use the Cattaneo–Maxwell flux and obtain asystem of the form (1.1). In Sect. 8 we consider several examples where such a constitutiverelation is chosen.

In Sects. 3–6 we prove asymptotic stability with asymptotic phase of the traveling waveunder purely spectral assumptions. In particular, we do not assume any growth bound onthe nonlinearity, but use only local properties. This is very important for applications fromreaction diffusion systems, since the reaction term is usually smooth but unbounded.

Our stability result is closely related to results of Kreiss et al. [15]. Compared to theirresult, we relax the assumption to include also non-strictly hyperbolic problems, which is asevere restriction. Furthermore, we treat the problem of the unknown asymptotic phase bya reformulation as a partial differential algebraic equation (PDAE). This seems to be a verynatural approach which unifies the treatment of the profile and the asymptotic phase. Also,by our approach a rigorous justification of the use of the Laplace transform (see [20]) ispossible. This is not given in [15].

From a computational point of view, the asymptotic stability with asymptotic phase isof little use because the speed is unknown. Therefore, in Sect. 7 we also show asymptoticstability for the PDAE that is used numerically for the freezing method [4,22]. This gener-alizes results of [24] to hyperbolic systems. In Sect. 8 we consider (hyperbolic versions) ofthe Nagumo equation and the Hodgkin–Huxley equations [11]. These systems satisfy ourassumptions and we confirm the theoretically predicted results in numerical experiments.

2 Assumptions and Main Results

Throughout the whole paper we assume

(H0) the nonlinearity f is an element of C3(Rm,Rm) and there exists a non-constanttraveling wave solution of (1.1) with profile v and speed λ that satisfies

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v ∈ C1b(R,R

m), vx ∈ H2(R,Rm), f (v) ∈ L2(R,Rm).

As usual, C1b(R) denotes the space of bounded and continuously differentiable functions

with bounded derivatives, Hk , k ≥ 1, are the usual Sobolev spaces. Note that (H0) impliesf (v) ∈ H2.

We require that equation (1.1) is hyperbolic and, without loss of generality, we assume thatB ∈ R

m,m is a real diagonal matrix with diagonal entries b11 ≥ . . . ≥ bmm . In a co-movingframe the traveling wave becomes a stationary solution of

vt = (λI + B)vx + f (v). (2.1)

To fix notation, we denote the linearization of (2.1) about v by

vt = (λI + B)vx + fv(v)v =: (λI + B)vx + C(x)v =: Pv. (2.2)

We impose the following conditions on the linear differential operator P:

(H1) The matrix λI + B ∈ Rm,m is an invertible, real diagonal matrix with r positive and

m − r negative eigenvalues.(H2) The matrix valued function C belongs to C1

b(R,Rm,m) and the limits

limx→∞ C(x) =: C± and lim

x→∞ Cx (x) = 0 exist.

(H3) There is δ > 0 so that s ∈ σ(iω(λI + B)+ C±) for some ω ∈ R implies Re s ≤ −δ,where σ denotes the spectrum.

Note that the existence of the limits C± is a consequence of (H0). In particular, the smooth-ness of the steady state v implies that the matrix valued functions Cx and Cxx are uniformlybounded.

Because of the translational invariance of (1.1), the closed linear operator P on L2 alwayshas an eigenvalue 0 with corresponding eigenfunction vx . We assume that no other eigen-value of P lies to the right of −δ in the complex plane. More precisely, let ρ(P) denotethe resolvent set of P , let σpt (P) denote the set of isolated eigenvalues of finite algebraicmultiplicity, and let σess(P) := C \ (ρ(P) ∪ σpt (P)) denote the essential spectrum of P .The assumption then reads:

(H4) Zero is an algebraically simple eigenvalue of P and σpt (P) ∩ {Re s > −δ} = {0}.Under these assumptions holds our first main result, asymptotic stability with asymptoticphase. Before we state the theorem, we specify our notion of a solution.

Definition 2.1 A function v is called a (classical) solution of (1.1) in [0, T ] iff

v ∈ C1([0, T ]; v + L2) ∩ C0([0, T ]; v + H1),

v(0) = v0, and vt = Bvx + f (v) holds as an equality in L2(R,Rm) for all t ∈ [0, T ].Furthermore, the function v is called a solution on R+ if it is a solution on [0, T ] for allT > 0.

Theorem 2.2 (Stability of steady states) Let Assumptions (H0)-(H4) hold. Then for every0 < η < δ there is ρ = ρ(η) > 0 such that for all v0 ∈ v + H2 with ‖v0 − v‖H2 < ρ theCauchy problem (1.1) has a unique global solution.

Moreover, there are C = C(η) > 0 and ϕ∞ = ϕ∞(v0) ∈ R, with

|ϕ∞| ≤ C‖v0 − v‖H2 , (2.3)

‖v(·, t)− v(· − λt − ϕ∞)‖H1 ≤ C‖v0 − v‖H2 e−ηt ∀t ≥ 0. (2.4)

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Theorem 2.2 is important from a theoretical point of view, but in applications, one is ratherinterested in determining the asymptotic profile itself and its speed. Moreover, long-timesimulations of such systems typically face the problem that the solution eventually leavesthe computational domain. One possibility to circumvent this is, to separate the evolution ofthe solution into the evolution of a (time-dependent) profile and a (time-dependent) phasevariable. This is the principal idea of the freezing method [4,22]. The method works forgeneral symmetries, but here we consider only traveling waves, where the symmetry simplyis the spatial shift. Basically, one makes the ansatz

v(x, t) = u(x +(t), t) (2.5)

for the solution v. The new variable u is interpreted as the profile and (t) ∈ R as itsposition. If u and are sufficiently smooth with respect to time, differentiating (2.5) andsetting λ := leads to

ut = Bux + f (u)+ λux .

The new variable λ introduces an additional degree of freedom into the system, and one needsan additional equation to obtain a well-posed problem again. One possibility is to requirethat u(·, t) always lies in the same hyperplane in L2. This can be achieved for example by aso called fixed phase condition [5]

0 = (u − u),

where is a linear functional in L2 and u is some reference function. The ansatz (2.5) thenleads to the system of PDAEs

ut = Bux + f (u)+ λux , u(0) = u0, (2.6a)

0 = (u − u), (2.6b)

for u and λ, which can be implemented on a computer.The functional and reference function u cannot be arbitrary and we require

(P1) is linear,

|(u)| ≤ C‖u‖L2 , ∀u ∈ L2(R,Rm), (continuity), (2.7)

(vx ) �= 0, (non-degeneracy). (2.8)

(P2) The reference function u satisfies v − u ∈ H1 and (v − u) = 0.

By (H0) and (P2), the tupel (v, λ) is a stationary solution of (2.6). The following theoremshows that the stationary solution (v, λ) is exponentially stable in the sense of Lyapunov andtherefore can be approximated by a direct long-time simulation.

Theorem 2.3 (Lyapunov stability for the freezing method) Let (H0)-(H4) and (P1), (P2)hold. Then for all 0 < η < δ there is ρ0 > 0, so that for all consistent initial data u(0) =u0 ∈ v + H2 with ‖u0 − v‖H2 < ρ0 there is a unique solution (v, λ) of the PDAE (2.6) in[0,∞). The solution satisfies the smoothness

u ∈ C1([0,∞); v + L2) ∩ C([0,∞); v + H1), λ ∈ C([0,∞); R).

Furthermore, there is a constant C = C(η), independent of the initial data, so that

‖u(t)− v‖H1 + |λ(t)− λ| ≤ C‖u0 − v‖H2 e−ηt ∀t ≥ 0. (2.9)

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Some remarks:

(a) The notion of a PDAE solution from Definition 3.6 easily adapts here.(b) When system (2.6) is solved on a computer, one may choose u = u0, so that the algebraic

condition (2.6b) is immediately satisfied.(c) A reasonable choice for the phase condition is

0 = (u − u) =∫

R

uTx (u − u) dx,

see [5]. In this case, belongs to the dual of L2 if u is an element of v + H1, whichis a reasonable assumption from a computational point of view. (See Sect. 8.1 for anexample.)

(d) Because of the non-degeneracy condition (2.8), the inverse function theorem implies: Ifu is close enough to some profile v, there is a shift ξ ∈ R, such thatψ(u −v(·−ξ)) = 0.In this sense (P2)can be understood as fixing the properly shifted profile v, to whichthe u–variable of the solution of (2.6) converges.

(e) Often one is only interested in the speed λ = of the wave, which appears in thePDAE. If also the position is needed, it can be obtained from λ by integration.

A major difficulty in the proof of Theorem 2.2 is the treatment of the unknown asymptoticphase. For this we reformulate the problem as a partial differential algebraic equation inSect. 3.2, which we prove to be (at least locally) equivalent to the original PDE problem. Bya careful analysis, exponential stability is shown for this nonlinear PDAE in Sects. 4–5. InSect. 6 we prove that this implies Theorem 2.2.

In Sect. 7 we give the proof of Theorem 2.3. Here the main problem is related to the termλux , which belongs to the principal symbol and cannot be treated as a small perturbation asin the parabolic case considered by Thümmler [25].

3 PDAE Reformulation

Without loss of generality we assume λ = 0. If λ �= 0, the equation is considered in aco-moving frame, i.e. one considers

vt = (B + λ)vx + f (v), v(0) = v0, (1.1′)

so that, v is a solution of (1.1′) if and only if v(x, t) = v(x − λt, t) is a solution of (1.1). Forthis equation the traveling wave’s profile is a stationary solution.

3.1 Existence and Uniqueness for the PDE

First we state a local existence and uniqueness result for the semilinear PDE problem (1.1).Writing the solution as a perturbation w of the steady profile, v = v + w, problem (1.1) isequivalent to the following equation for w

wt = Bwx + C(x)w + q(x, w), w(0) = w0 := v0 − v, (3.1)

where C(x) = fv(v(x)) and q(·, w) = f (v + w) − f (v) − fv(v)w. By Taylor’s formula,q(·, w) = ∫ 1

0 (1 − s)D2 f (v + sw)ds[w,w] holds as an equality in H1 for v ∈ H1. Hereand in the following we use the notation A[u, v] for a bilinear mapping A applied to u, vand Au2 for A[u, u]. The following existence and uniqueness result follows by the methodof characteristics. Similar results are given in [6, Ch. 3] and we omit the proof.

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Theorem 3.1 (Existence and uniqueness) Let the assumptions be as above. For every w0 ∈H1(Rm) there is T = T (‖w0‖H1) > 0, depending only on the H1-norm ofw0, such that theCauchy problem (3.1) possesses a unique solution w ∈ C1([0, T ]; L2) ∩ C0([0, T ]; H1).

Because T in Theorem 3.1 only depends on the H1-norm of the initial data and (3.1) doesnot explicitly depend on time, a simple contradiction argument proves the following globalcontinuation result, which we formulate for the original equation:

Theorem 3.2 (Global continuation) For every v0 ∈ v+ H1(R) there is a unique global solu-tion v∗ ∈ C1([0, T ∗); v + L2) ∩ C0([0, T ∗); v + H1) of (1.1), so that if v ∈ C1([0, T ]; v +L2)∩C0([0, T ]; v+ H1) is a solution of (1.1) it follows T < T ∗ and v∗|[0,T ] = v. Moreover,

either T ∗ = +∞ or T ∗ < ∞ and limt↗T ∗ ‖v∗(t)− v‖H1 = +∞.

A complete proof of this theorem can be found in [19, Ch. 3].

3.2 PDAE Reformulation Via Nonlinear Coordinates

Since λ = 0, v is a steady state. For the stability analysis it is convenient to rewrite theequation using nonlinear coordinates v and ϕ in the form:

v(x, t) = v(x − ϕ(t))+ v(x, t). (3.2)

This separates the evolution of the solutions shape and its location. A similar idea is used in[10, Ch. 5.1], to analyze the asymptotic stability of equilibria in parabolic evolution equa-tions. A crucial assumption in [10] is that of a sectorial principal part, which is not satisfiedhere.

The splitting (3.2) is not unique, and we restrict v to a hyperplane by imposing

ψ(v(·, t)

) = 0, ∀t ≥ 0, (3.3)

where ψ is a suitable linear functional. This is similar to the phase condition (2.6b) of thefreezing method. We assume that ψ is a continuous linear functional on H−1(R,Rm) =(H1(R,Rm))′, i.e. there is Cψ > 0 so that

|ψ(v)| ≤ Cψ‖v‖H−1 ∀v ∈ H−1(R,Rm), (3.4)

and, furthermore, ψ satisfies the non-degeneracy condition

ψ(vx ) �= 0. (3.5)

The boundedness (3.4) is important for Theorem 4.1, see also [20, Sects. 4, 5].

Lemma 3.3 Letψ be given as above. Then there are open neighborhoods U and V of 0 ∈ R

so that the function

G : U → V, G(ϕ) = ψ(v(· − ϕ)− v

)(3.6)

is a C2–diffeomorphism.

Proof The mapping ϕ �→ v(·− ϕ)−v belongs to C2(R, L2(R,Rm)) because of (H0). There-fore, G ∈ C2 by the chain rule. The assertion then follows with the inverse function theoremsince d

dϕG(ϕ)|ϕ=0 = −ψ(vx ) �= 0 by (3.5). ��Note that the proof also applies with the functional from the freezing method.

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Let := (G|U )−1 ∈ C2(V,U ) denote the inverse of G from Lemma 3.3. For v ∈ v+ L2

with ψ(v − v) ∈ V define v and ϕ by

�(v) :=(ϕ

v

):=

(

(ψ(v − v)

)v − v

(· − (ψ(v − v)

))). (3.7)

Conversely, for arbitrary ϕ ∈ R and v ∈ L2(R,Rm), define

�

(ϕ

v

):= v + v(· − ϕ). (3.8)

We consider the restrictions of the mappings � and � to the sets

D� := {v ∈ v + L2(R,Rm) : ψ(v − v) ∈ V

}, and

D� := {(ϕ, v) ∈ R × L2(R,Rm) : ϕ ∈ U, v ∈ N (ψ)}, respectively. (3.9)

It easily follows from Lemma 3.3 and the smoothness assumption (H0) on the profile:

Lemma 3.4 The restrictions �|D� of � to D� and �|D� of � to D� are inverse to eachother.

Since these transformations will be used to write solutions of (1.1) in the form (3.1), itis important to know how smoothness properties of a time dependent function v relate tosmoothness properties of ϕ and v, given by

(ϕ(t), v(t)

) = �(v(t)

). For this we define for

T > 0 the sets of functions

MT� := {

v ∈ C1([0, T ]; v + L2) ∩ C([0, T ]; v + H1) : v(t) ∈ D� ∀t ∈ [0, T ]} and

MT� := {(ϕ, v) with (3.10) holds and (ϕ(t), v(t)) ∈ D� ∀t ∈ [0, T ]} .

Lemma 3.5 For v ∈ MT� holds (ϕ(t), v(t)) = �(v(t)) ∈ D� for all 0 ≤ t ≤ T and

ϕ ∈ C1([0, T ]; R) and v ∈ C1([0, T ]; L2) ∩ C0([0, T ]; H1). (3.10)

Conversely, if ϕ and v satisfy the smoothness (3.10), then

v := �(ϕ, v) = v + v(· − ϕ) ∈ C1([0, T ]; v + L2) ∩ C0([0, T ]; v + H1). (3.11)

Moreover, the restrictions �|MT�

and �|MT�

are inverse to each other.

Proof Let v ∈ MT� . The smoothness ϕ =

(ψ(v − v)

) ∈ C1([0, T ]; R) follows fromv− v ∈ C1([0, T ]; L2) and Lemma 3.3. For the smoothness of v = (

v− v)+ (v− v(·− ϕ))

note that v ∈ MT� implies v − v ∈ C1([0, T ]; L2) ∩ C0([0, T ]; H1) and, by assumption,

the mapping ϕ �→ v(·) − v(· − ϕ) is differentiable for all ϕ ∈ R with derivative at ϕgiven by vx (· − ϕ) ∈ H1. Since the shift is continuous in L2, this mapping is an elementof C1([0, T ]; L2). Moreover, R � ϕ �→ vx − vx (· − ϕ) ∈ L2 is continuous, so that alsot �→ v − v(· − ϕ(t)) is an element of C1([0, T ]; L2) ∩ C0([0, T ]; H1). This proves (3.10).The property (ϕ(t), v(t)) ∈ D� immediately follows from Lemma 3.4.

The proof of the converse statement follows by applying the same arguments to

v = v + v(· − ϕ) = v + (v(· − ϕ)− v

) + v.

The last statement follows from the above properties and Lemma 3.4. ��

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Let v be the solution of (1.1) given by Theorem 3.2 and assume v ∈ MT� . Then for

(ϕ, v) = �(v) and λ := ϕ′ ∈ C([0, T ]) holds

vt = Bvx + f (v + v(· − ϕ))− f (v(· − ϕ))+ λvx (· − ϕ) (3.12)

as an equality in L2(R,Rm) for all t ∈ [0, T ]. Because of (H0) and the smoothness of v,Taylor’s formula and the fundamental theorem of calculus show

f(v + v(· − ϕ)

) − f(v(· − ϕ)

)

= fv(v)v −1∫

0

fvv(v(· − sϕ)

)vx (· − sϕ) ds ϕv

+1∫

0

(1 − s) fvv(v(· − ϕ)+ sv

)ds [v, v], (3.13)

as an equality in H1 for all t ∈ [0, T ]. Inserting (3.13) into (3.12) yields

vt = P v + λvx + F1(ϕ, v)+ F2(ϕ, v)+ R(ϕ, λ), (3.14)

as an equality in L2 for all t ∈ [0, T ]. The nonlinear terms are given by

F1(ϕ, v) = −1∫

0

fvv(v(· − sϕ)

)[vx (· − sϕ), ϕv] ds,

F2(ϕ, v) =1∫

0

(1 − s) fvv(v(· − ϕ)+ sv

)ds [v, v], (3.15)

R(ϕ, λ) = −1∫

0

vxx (· − sϕ) ds ϕλ,

and belong to C([0, T ]; H1), see Lemmas 5.2–5.4. Thus, locally � transforms solutions of(1.1) into solutions of the following PDAE for the unknowns v, ϕ, λ:

vt = P v + λvx + F1(ϕ, v)+ F2(ϕ, v)+ R(ϕ, λ),

ϕt = λ, (3.16a)

0 = ψ(v),

subject to the consistent initial data

v(0) = v0 ∈ H1, ϕ(0) = ϕ0 ∈ R, λ(0) = λ0, (3.16b)

given by (ϕ0, v0)T = �(v0). The term consistent reflects that the initial data are not arbitrary,

but are restricted by the hidden constraints. Namely, for sufficiently small v0, ϕ0, the hiddenconstraints can be solved for λ0 in (3.16). Therefore, whenever we write consistent initialdata, the hidden constraints are respected implicitly.

Definition 3.6 We call (v, ϕ, λ) a (classical) solution of (3.16) in [0, T ], if

v ∈ C1([0, T ]; L2) ∩ C0([0, T ]; H1), ϕ ∈ C1([0, T ]; R), λ ∈ C([0, T ]; R),

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(3.16b) is satisfied and for all t ∈ [0, T ] the first equation in (3.16a) holds as an equality inL2 and the other two equations hold pointwise for all t ∈ [0, T ].

We call the triple a solution in [0,∞) if it is a solution in [0, T ] for all T > 0.

The results of this section are summarized in Proposition 3.7. It shows a one-to-one corre-spondence of solutions to the original Cauchy problem (1.1) and the PDAE (3.16). Hence, itsuffices to analyze the asymptotic behavior of solutions to the PDAE.

Proposition 3.7 (Equivalence of solutions) Let the setting be as above.If v ∈ C1([0, T ]; v + L2) ∩ C([0, T ]; v + H1) solves (1.1) with ψ(v(t)− v) ∈ V for all

t ∈ [0, T ], then (v, ϕ, λ), given by (ϕ, v) = �(v) and λ = ϕt , solves (3.16) subject to theconsistent initial data ϕ(0) = (ψ(v0 − v)), v(0) = v0 − v(· − ϕ(0)), and λ(0) given bythe hidden constraint. Moreover, ϕ(t) ∈ U for all 0 ≤ t ≤ T .

Conversely, if (v, ϕ, λ) is a solution of (3.16) in [0, T ], it follows that v, given by�(ϕ, v) =v+ v(· − ϕ), is an element of C1([0, T ]; v+ L2)∩ C([0, T ]; v+ H1) and solves the Cauchyproblem (1.1) with v(0) = v0 + v(· − ϕ0).If ϕ(t) ∈ U for all 0 ≤ t ≤ T , the two transformations are inverse to each other.

Proof The smoothness and the last statement follow from Lemma 3.5.That solutions of the Cauchy problem (1.1) lead to solutions of the PDAE (3.16) is shown

above. For the other implication, the arguments from above can be reversed because of thesmoothness assumptions and Lemma 3.5. ��

4 Linear Stability for Hyperbolic PDAEs

Let the setting be as above, in particular, impose (H0)-(H4), λ = 0, and ψ satisfies (3.4) and(3.5). Note that F1, F2, and R from (3.16a) are at least quadratic functions of their arguments.Replacing these higher order terms by time dependent inhomogeneities, the PDAE becomeslinear

vt = P v + vx λ+ F, in L2,

ϕt = λ, in R,

0 = ψ(v), in R.

(4.1)

We analyze this linear system in [20], where we assume the inhomogeneity F belongs toC(J ; H1(R)), J = [0, T ] or J = [0,∞). The equation for ϕ decouples from the otherequations in (4.1) and can be solved in an additional step. Therefore, consider

vt = P v + vx λ+ F, in L2,

0 = ψ(v), in R,(4.2a)

for v and λ, which we assume to be subject to consistent initial conditions

v(0) = v0 ∈ H1(R,Rm), λ(0) = λ0. (4.2b)

The hidden constraint implies λ0 = −ψ(P v0 + F(0)

)/ψ(vx ).

Adapting the solution-concept from Definition 3.6 to the current case, a pair (v, λ) iscalled a (classical) solution of (4.2) in [0, T ] if

v ∈ C1([0, T ]; L2(R,Rm)) ∩ C([0, T ]; H1(R,Rm)

)and λ ∈ C([0, T ]; R

),

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so that (4.2b) is satisfied, the first equation in (4.2a) holds in L2(R,Rm), and the secondequation holds pointwise for all t ∈ [0, T ]. The tuple is called a solution on [0,∞) if it is asolution on [0, T ] for every T > 0.

The estimate in the following theorem leads to a priori estimates in Sect. 5 which are thekey in the proof of nonlinear stability.

Theorem 4.1 (Linear PDAE stability [20, Thm 5.3]) In the above setting, there exists forevery consistent initial data v0 ∈ H2(R), λ0 ∈ R, a unique solution (v, λ) of the linearPDAE (4.2) on J .

Moreover, if η0 < δ, with δ from (H3) and (H4), then there is Cl = Cl(η0) > 0, indepen-dent of F and v0, so that for all η ≤ η0 the solution satisfies for all t ∈ J

‖v(t)‖2H1 + e−2ηt

t∫

0

e2ητ (‖v(τ)‖2H1 + |λ(τ)|2) dτ

≤ Cle−2ηt

[‖v0‖2

H2 +t∫

0

e2ητ‖F(τ )‖2H1 dτ

]. (4.3)

The main tool in the proof of Theorem 4.1, given in [20], is the vector-valued Laplacetransform (cf. [1]). In the proof, the case of arbitrary initial data is reduced to homogeneousinitial data by a homogenization trick, which is a well-known technique for the Laplacetransform. This homogenization introduces the H2-norm of the initial data in the estimatesand also in our main Theorem 2.2.

5 Nonlinear Stability of the PDAE

Without mentioning it again, we impose in this section beside (H0)-(H4) that ψ is given asin Sect. 3 and satisfies (3.4) and (3.5). We prove the nonlinear PDAE-stability result:

Theorem 5.1 (Asymptotic stability of the PDAE) For every 0 < η < δ, δ from (H3) and(H4), there are ρ0, θ0 > 0 so that for all consistent initial data v0 ∈ H2, ϕ0, λ0 ∈ R with‖v0‖H2 ≤ ρ0 and |ϕ0| ≤ θ0, there is a unique (classical) solution (v, ϕ, λ) of (3.16) on[0,∞). The solution satisfies ϕ(t) ∈ U for all t ≥ 0, U from Lemma 3.3. Moreover, there isϕ∞ ∈ R, so that with Cl = Cl(η) from Theorem 4.1 holds for all t ≥ 0

|ϕ∞| ≤ |ϕ0| +√

Cl(η)

η‖v0‖H2 , (5.1a)

|ϕ(t)− ϕ∞|2 ≤ Cl(η)

η‖v0‖2

H2 e−2ηt , (5.1b)

‖v(t)‖2H1 ≤ Cl(η)‖v0‖2

H2 e−2ηt , (5.1c)t∫

0

e2ητ |λ(τ)|2 dτ ≤ 2Cl(η)‖v0‖2H2 . (5.1d)

Note that Theorem 5.1 is a strictly local result for small v, ϕ, λ. We emphasize this in theproof by rescaling the variables and analyzing the problem in the rescaled form. The proof isgiven in the next subsection. For completeness, we collect some properties of the nonlinearterms in Sect. 5.2.

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5.1 Proof of Theorem 5.1

Assume 0 < η < δ and let Cl = Cl(η) be the constant from Theorem 4.1. Choose ε1 > 0,with ε1 ≤ (2Cl)

−1/2 and B3ε1(0) ⊂ U , U ⊂ R given in Lemma 3.3. Also let and V begiven by Lemma 3.3, i.e. : V → U , = (G|U )−1 is a C2–diffeomorphism.

Step 0: Rescaling. Assume that (v, ϕ, λ) is a (classical) solution of (3.16) in [0, T ]. Forarbitrary 0 < ε ≤ ε1 define the rescaled variables (v, ϕ, λ) by

v = εv, ϕ = εϕ, λ = ελ.

For the sake of readability we use v, ϕ, λ for the new variables, which should not be confusedwith the variables from Sect. 1. In these variables (3.16) reads

vt = Pv + λvx + εFε1 (ϕ, v)+ εFε2 (ϕ, v)+ εRε(ϕ, λ),

ϕt = λ,

0 = ψ(v),

(5.2a)

subject to the initial data

εv(0) = εv0 := v0, εϕ(0) = εϕ0 := ϕ0, ελ(0) = ελ0 := λ0. (5.2b)

The functions Fεj and Rε in (5.2a) are defined in terms of R and Fj from (3.15) as

ε2 Fεj (ϕ, v) := Fj (εϕ, εv), j = 1, 2, ε2 Rε(ϕ, λ) := R(εϕ, ελ). (5.3)

Step 1: A priori estimates. Let 0 < ε < ε1. Assume that (v, ϕ, λ) is a solution of (5.2)on [0, T ] for some T > 0. Also assume |ϕ(t)| ≤ 2 and ‖v(t)‖H1 ≤ 2 for all t ∈ [0, T ].Then, the nonlinearities in (5.2a) satisfy for all such t and ε

‖Fε1 (ϕ, v)‖2H1 + ‖Fε2 (ϕ, v)‖2

H1 ≤ Cn‖v‖2H1 , (5.4)

‖Rε(ϕ, λ)‖2H1 ≤ Cn |λ|2, (5.5)

with a uniform constant Cn > 0 by Lemmas 5.2, 5.3, and 5.4. Moreover, these lemmas alsoshow that

F : t �→ F(t) = εFε1 (ϕ(t), v(t))+ εFε2 (ϕ(t), v(t))+ εRε(ϕ(t), λ(t)) (5.6)

is an element of C([0, T ]; H1). Thus, considering the nonlinear term F(t) as an inhomoge-neity in the linear PDAE (4.2), Theorem 4.1 applies and shows for all 0 ≤ t ≤ T

‖v(t)‖2H1 + e−2ηt

t∫

0

e2ητ (‖v(τ)‖2H1 + |λ(τ)|2) dτ

≤ Cle−2ηt

[‖v0‖2

H2 + 3ε2Cn

t∫

0

e2ητ(‖v(τ)‖2

H1 + |λ(τ)|2)

dτ],

where the estimates (5.4) and (5.5) were used.If 0 < ε ≤ ε0 ≤ min(ε1, (6ClCn)

−1/2), this yields for all 0 ≤ t ≤ T the bound

‖v(t)‖2H1 + 1

2e−2ηt

t∫

0

e2ητ (‖v(τ)‖2H1 + |λ(τ)|2) dτ ≤ Cle

−2ηt‖v0‖2H2 . (5.7)

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352 J Dyn Diff Equat (2012) 24:341–367

Moreover, by Cauchy-Schwarz inequality and (5.7), ϕ(t) satisfies for all 0 ≤ t ≤ T

|ϕ(t)| ≤ |ϕ0| +t∫

0

|ϕt (τ )| dτ = |ϕ0| +t∫

0

e−ητ (eητ |λ(τ)|) dτ

≤ |ϕ0|+( t∫

0

e−2ητ dτ) 1

2( t∫

0

e2ητ |λ(τ)|2 dτ) 1

2< |ϕ0|+

(Cl

η

) 12 ‖v0‖H2 . (5.8)

Step 2: Local Existence and Uniqueness. Let 0 < ε ≤ ε0, ε0 from Step 1, and assumeϕ0 = εϕ0 with some |ϕ0| ≤ 1. By Step 0 and Proposition 3.7, a triple (v, ϕ, λ) is a solutionof (5.2) in [0, T ] with εϕ(t) ∈ U for all 0 ≤ t ≤ T if and only if

ϕ = 1

ε (ψ(w − v)), v = 1

ε(w − v(· − εϕ)), λ = ϕ′, ∀0 ≤ t ≤ T,

wherew ∈ C1([0, T ]; v+L2)∩C([0, T ]; v+H1) satisfiesψ(w(t)−v) ∈ V for all 0 ≤ t ≤ Tand solves (1.1) with initial data w(0) = εv0 + v(· − εϕ0). Therefore, unique solvability of(1.1), proved in Theorem 3.2, implies unique solvability of the PDAE (5.2).

To show existence, let w∗ be the unique global solution of (1.1) with w∗(0) = εv0 +v(· − εϕ0), obtained by Theorem 3.2 and let [0, T ∗) denote its interval of existence. Assume|ϕ0| < ε so that |ϕ0| ≤ 1. Consistency of the initial data and linearity of ψ implies v0 ∈N (ψ) and, therefore, ψ(w∗(0) − v) ∈ V . By continuity, there is 0 < T1 < T ∗ so thatψ(w∗(t) − v) ∈ V for all 0 ≤ t ≤ T1. With Proposition 3.7 follows local existence of a(classical) solution of the PDAE (5.2).

Step 3: Continuation and Global Existence. Choose ω0 = min( 1Cl,ηCl) and assume

v0 = εv0 with ‖v0‖2H2 ≤ ω0. Let w∗ and T ∗ be given as in Step 2. Define

T0 := sup{

T ∈ [0, T ∗) : | (ψ(w∗(t)− v))| < 2ε and

‖w∗(t)− v(· − (ψ(w∗(t)− v))

)‖H1 < 2ε ∀0 ≤ t ≤ T}

(5.9)

and show that for T0, thus defined, the assumption T0 < ∞ implies

h(T0) = 2ε and h(t) < 2ε for all 0 ≤ t < T0, (5.10)

where h is defined as

h : t �→ max(| (ψ(w∗(t)− v))|, ‖w∗(t)− v

(· − (ψ(w∗(t)− v)))‖H1

). (5.11)

First assume T0 < T ∗ = ∞. From w∗ ∈ C([0, T ]; v + H1) for all T > 0 and Lemma 3.3,follows the continuity of h as long as ψ

(w∗(t) − v

) ∈ V . Furthermore, h(t) < 3ε impliesψ

(w∗(t)− v

) ∈ V because of the definition of 0 < ε ≤ ε1. Therefore, by continuity (5.10)holds.

Now assume T ∗ < ∞. From Theorem 3.2 follows limt↗T ∗ ‖w∗(t) − v‖H1 = +∞. Ifthere is t ∈ (0, T ∗) with ψ(w∗(t)− v) �∈ V , there must exist T1 < T ∗ with

| (ψ(w∗(T1)− v))| = 2ε and | (ψ(w∗(t)− v))| < 2ε ∀0 ≤ t < T1.

Then h is continuous on [0, T1], T0 ≤ T1, and (5.10) follows.If instead ψ(w∗(t)− v) ∈ V for all 0 ≤ t < T ∗, then h is continuous on [0, T ∗). More-

over, because ‖v − v(· − (ψ(w∗(t) − v))

)‖H1 is uniformly bounded for 0 ≤ t < T ∗, itfollows

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J Dyn Diff Equat (2012) 24:341–367 353

limt↗T ∗ ‖w∗(t)− v

(· − (ψ(w∗(t)− v)))‖H1 = ∞,

which implies T0 < T ∗ and again proves (5.10).Now let T0 be given by (5.9) and assume T0 < ∞. This implies T0 < T ∗ by the above

calculation. Define (v, ϕ, λ) by

ϕ = 1

ε

(ψ(w∗ − v)

), v = 1

ε

(w∗ − v(· − εϕ)

), λ = ϕ′ ∀0 ≤ t ≤ T0,

which is a solution of (5.2) on [0, T0] by Step 2. By definition of T0 hold

|ϕ(t)| ≤ 2 and ‖v(t)‖H1 ≤ 2 ∀t ∈ [0, T0],so that the a priori estimates (5.7) and (5.8) apply in [0, T0]. The choice ofω0 and‖v0‖2

H2 ≤ ω0

then shows for all 0 ≤ t ≤ T0 the bounds

‖v(t)‖2H1 ≤ Cle

−2ηt‖v0‖2H2 ≤ 1 and |ϕ(t)| < |ϕ0| +

(Cl

η

)1/2 √ω0 ≤ 2. (5.12)

This contradicts (5.10), and T0 = T ∗ = ∞ follows.Step 4: Rate of convergence. Steps 2 and 3 show that for initial data with ‖v0‖2

H2 ≤εω0 =: ρ2

0 and |ϕ0| ≤ ε = θ0, the solution of the PDAE (5.2) exists for all positive timesand the (scaled) variables satisfy |ϕ(t)| < 2 and ‖v(t)‖H1 < 2 for all t ≥ 0. Therefore, thea priori estimates (5.7) and (5.8) apply to every compact interval [0, T ], T < ∞, and yieldfor all t ≥ 0 the estimates (5.1c) and (5.1d):

‖v(t)‖2H1 = ε2‖v(t)‖2

H1 ≤ ε2Cle−2ηt‖v0‖2

H2 = Cle−2ηt ‖v0‖2

H2 ,

t∫

0

e2ητ |λ(τ)|2 dτ = ε2

t∫

0

e2ητ |λ(τ)|2 dτ ≤ ε22Cl‖v0‖2H2 = 2Cl ‖v0‖2

H2 .

Finally, define ϕ∞ := εϕ∞, where ϕ∞ = ϕ0 + ∫ ∞0 λ(τ) dτ . The integral is absolutely con-

vergent by (5.8) which also implies estimate (5.1a). Estimate (5.1b) follows from (5.7) andan application of the Cauchy-Schwarz inequality

|ϕ(t)− ϕ∞|2 ≤ e−2ηt

2η

∞∫

t

e2ητ |λ(τ)|2 dτ ≤ Cl

η‖v0‖2

H2 e−2ηt .

��5.2 Smoothness and Estimates of Nonlinear Terms

For the analysis of the nonlinear terms in (5.2a), we assume

v ∈ C([0, T ]; H1(R,Rm)), ϕ ∈ C([0, T ]; R), λ ∈ C([0, T ]; R),

in addition to (H0). This is in accordance with the smoothness of the PDAE solution obtainedin Step 2 via Theorem 3.1 and Proposition 3.7. We now present estimates for the nonlinearterms. In the proofs we restrict to smooth functions and only give the ideas. For the technicaldetails we refer to [21] and [19].

Lemma 5.2 For ε > 0 the function R � t �→ Fε1 (ϕ(t), v(t)) is an element of C([0, T ]; H1)

and there is a constant C = C( f, v), independent of ε, so that

‖Fε1 (ϕ(t), v(t))‖2∗ ≤ C |ϕ(t)|2‖v(t)‖2∗, ∗ ∈ {L2(R,Rm), H1(R,Rm)}. (5.13)

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354 J Dyn Diff Equat (2012) 24:341–367

Proof Assume v ∈ C∞0 (R,R

m) and ϕ ∈ R. From Hölder’s inequality and the shift invarianceof the L∞–norm follows the ε-independent estimate

‖Fε1 (ϕ, v)‖2 =∫

R

∣∣∣1∫

0

fvv(v(x − sεϕ))vx (x − sεϕ) dsϕv(x)∣∣∣2

dx ≤ C |ϕ|2‖v‖2.

Similarly, one obtains by using v ∈ C2b , which holds by (H0),

‖Fε1 (ϕ, v)x‖2 =∫

R

∣∣∣1∫

0

fvvv(v(x − sεϕ))vx (x − sεϕ)2 ds ϕv(x)

+1∫

0

fvv(v(x − sεϕ))vxx (x − sεϕ) ds ϕv(x)

+1∫

0

fvv(v(x − sεϕ))vx (x − sεϕ) ds ϕvx (x)∣∣∣2

dx

≤ C |ϕ|2‖v‖2H1 .

This proves (5.13) for smooth functions. For general v ∈ H1(R,Rm) one uses a suitableapproximating sequence vn ∈ C∞

0 , n ∈ N with vn → v in H1.The technical details can be found in [19, App. D], where also a proof of the continuity

of t �→ Fε1 (ϕ(t), v(t)) is given. ��For the rescaled nonlinearity Fε2 we have the following

Lemma 5.3 Assume 0 < ε ≤ ε0 and v is as above with ‖v‖L∞([0,T ];H1) ≤ K . Then thefunction R � t �→ Fε2

(ϕ(t), v(t)

) ∈ H1 is an element of C([0, T ]; H1). Furthermore, thereis a constant C = C( f, v, ε0, K ), independent of ε, such that

‖Fε2 (ϕ(t), v(t))‖2∗ ≤ C‖v(t)‖2H1‖v(t)‖2∗, ∀t ∈ [0, T ], ∗ ∈ {L2, H1}. (5.14)

Proof We again only prove (5.14) for ϕ ∈ R and v ∈ C∞0 (R,R

m) and refer to [21] and [19,App. D] for the details and for a proof of the continuity.

Assume ‖v‖H1 ≤ K1, where K1 > 0 is some constant. By Hölder’s inequality,

‖Fε2 (ϕ, v)‖2 =∫

R

∣∣∣1∫

0

(1 − s) fvv(v(x − εϕ)+ sεv(x)) ds [v(x), v(x)]∣∣∣2

dx

≤ sup|w|≤‖v‖∞+ε‖v‖∞

| fvv(w)|2‖v‖2∞‖v‖2.

Similarly, estimating Fε2 (ϕ, v)x with Hölder’s inequality, yields

‖Fε2 (ϕ, v)x‖2 ≤ 3 sup0≤s≤1

‖ fvvv(v(· − εϕ)+ sεv(·))vx (· − εϕ)‖2∞‖v‖2∞‖v‖2

+3 sup0≤s≤1

‖ fvvv(v(· − εϕ)+ sεv(·))s‖2∞|ε|2‖v‖4∞‖vx‖2

+6 sup0≤s≤1

‖ fvv(v(· − εϕ)+ sεv(·))‖2∞‖v‖2∞‖vx‖2

≤ const ‖v‖2∞‖v‖2H1 .

Estimate (5.14) then follows from Sobolev embedding. ��

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J Dyn Diff Equat (2012) 24:341–367 355

Lemma 5.4 Let the assumptions be as above. Then for every ε > 0 the function t �→Rε(ϕ(t), λ(t)) is an element of C([0, T ]; H1) and satisfies the estimate

‖Rε(ϕ(t), λ(t))‖2H1 ≤ C |ϕ(t)|2|λ(t)|2, (5.15)

with C independent of ϕ, λ and ε.

Proof Monotonicity of the Bochner integral and vx ∈ H2 immediately imply (5.15) andcontinuity follows since the shift is a continuous operation in H1. ��

6 Proof of Theorem 2.2

Choose some ψ that satisfies (3.4) and (3.5), and let U , V , G, be given as in Sect. 3.First consider the case λ = 0. Choose 0 < η < δ. Then Theorem 5.1 applies and let

ρ0 = ρ0(η), θ0 = θ0(η) > 0 be given by that result.The mapping v �→ ◦ψ(v− v) is continuously differentiable in a neighborhood of v in

v + H1(R,Rm). Therefore, there are ρ1 > 0 and Clip > 0 so that∣∣ (ψ(v − v))

∣∣ ≤ Clip‖v − v‖H1 ∀v ∈ v + H1(R,Rm) with ‖v − v‖H1 ≤ ρ1. (6.1)

Let ρ = min(ρ1,θ0

Clip,ρ02 ,

ρ02Clip‖vx ‖H2

), so that for all v0 ∈ v + H2(R,Rm) satisfying

‖v0 − v‖H2 < ρ, the initial data ϕ(0) = (ψ(v0 − v)

)and v(0) = v0 − v

(· − ϕ(0))

of the PDAE-reformulation (3.16) can be estimated by

|ϕ(0)| = | (ψ(v0 − v))| ≤ Clip‖v0 − v‖H1 ≤ θ0, (6.2)

‖v(0)‖H2 ≤ ‖v0 − v‖H2 + ‖v − v(· − ϕ(0))‖H2 ≤ (1 + Clip‖vx‖H2)‖v0 − v‖H2 ≤ ρ0.

(6.3)

By Theorem 5.1 there is a unique solution (v, ϕ, λ) of (3.16) on [0,∞) and ϕ(t) ∈ U forall t ≥ 0. Therefore, by Proposition 3.7, there is a unique solution v of the Cauchy problem(1.1) on [0,∞), given by

v(t) = v(t)+ v(· − ϕ(t)) for all t ≥ 0. (6.4)

With ϕ∞ = ϕ∞ from Theorem 5.1 follows from (5.1a), (6.2), and (6.3)

|ϕ∞| ≤ Clip‖v0 − v‖H1 +√

Cl

η(1 + Clip‖vx‖H2)‖v0 − v‖H2 ≤ C‖v0 − v‖H2 ,

and the constant C only depends on Clip , Cl , η and ‖vx‖H2 . This proves (2.3).For the asymptotic behavior of v, representation (6.4) yields with (5.1b), (5.1c):

‖v(t)− v(· − ϕ∞)‖H1 ≤ ‖v(t)‖H1 + ‖v(· − ϕ(t))− v(· − ϕ∞)‖H1 ,

≤ √Cl ‖v0‖H2 e−ηt + ‖vx‖H1

√Cl

η‖v0‖H2 e−ηt =

(1 + ‖vx‖H1

η1/2

) √Cl ‖v0‖H2 e−ηt .

Inserting (6.3), this implies

‖v(t)− v(· − ϕ∞)‖H1 ≤ C‖v0 − v‖H2 e−ηt ∀0 ≤ t,

where the constant C again depends on Clip , Cl , η, and ‖vx‖H2 , but is independent of v0 sothat (2.4) holds. This finishes the proof in the case λ = 0.

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356 J Dyn Diff Equat (2012) 24:341–367

In the case λ �= 0, consider the problem in a co-moving frame, moving with speed λ.Then the result for the case λ = 0 applies to this new problem. ��

7 Stability of the Freezing Method

In this section we prove Theorem 2.3. The proof proceeds in several steps and the principalidea is the ansatz

u(x, t) = vpde(x +(t), t),

where vpde is a solution of (1.1), such that (u, ) is a solution of the PDAE

ut = Bux + f (u)+ ux , 0 = (u − u).

The hidden constraint then yields an ODE for . Because we allow for general from thedual space of L2 (see the remarks following Theorem 2.3), we need Peano’s Theorem toprove existence. Uniqueness is analyzed in an additional step by using the algebraic equation(2.6b). Asymptotic stability is a consequence of Theorem 2.2.

Proof of Theorem 2.3 Let 0 < η < δ be given.

Step 0: Without loss of generality assume λ = 0. If λ �= 0, the equation is considered ina co-moving frame and becomes

ut = (B + λI )ux + f (u)+ λux , 0 = (u − u), (2.6′)

so that (u, λ) solves (2.6′) if and only if (u, λ+ λ) solves (2.6). Let

F(u) := Bux + f (u),

Rρ := {u0 ∈ v + H2(R,Rm) : ‖u0 − v‖H2 < ρ and (u − u0) = 0}.Step 1: [Solution of the PDE] By Theorem 2.2 exist ρ,C pde > 0 so that for all u0 ∈ Rρ

exists a unique solution vpde ∈ C1([0,∞); v + L2) ∩ C([0,∞); v + H1) of (1.1) withv(0) = u0. Moreover, there is a number ϕ∞ ∈ R so that

|ϕ∞| ≤ C pde‖u0 − v‖H2 ,∥∥vpde(·, t)− v(· − ϕ∞)∥∥

H1 ≤ C pde‖u0 − v‖H2 e−ηt , ∀t ≥ 0. (7.1)

We sometimes write v[u0]pde to emphasize the dependence on the initial value u0,.

Step 2: [Ansatz for a solution of (2.6)]

Lemma 7.1 Let u0 ∈ Rρ . If ∈ C1([0, T ]; R) solves the ODE

(v

[u0]pde,x (· +(t), t)

)(t) = −(

F(v[u0]pde(· +(t), t))

), (0) = 0, (7.2)

in [0, T ], then (u, ) with u(·, t) = v[v0]pde(· +(t), t) is a solution of (2.6) in [0, T ].

Proof The smoothness of vpde and imply u ∈ C1([0, T ]; v + L2) and

d

dtu(·, t) = F(u)(t)+ (t)ux (·, t) in L2 for all t ≥ 0, (7.3)

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J Dyn Diff Equat (2012) 24:341–367 357

by the chain rule. Together with (P2) follows t �→ u − u(t) ∈ C1([0, T ]; L2) so that (P1)implies that t �→ (u − u(t)) belongs to C1([0, T ]; R). Because of (7.3)

d

dt

(u − u(t)

) = −(ut ) = −(F(u)+ ux

)

= −(F(vpde(· +(t), t))

) − (t)(vpde,x (· +(t), t)

) = 0 for all 0 ≤ t ≤ T .

Consistency of the initial data, i.e. (u − u(·, 0)

) = (u − vpde(·, 0)

) = 0, then impliesthat the algebraic condition

(u − u(t)

) = 0 holds for all 0 ≤ t ≤ T . ��Step 3: [Local existence of a solution] Step 2 reduces the local existence problem for

(2.6) to finding a solution of (7.2). The next lemma and its corollary show that this is possibleif u0 is sufficiently close to v.

Lemma 7.2 There are ρ1, ρ > 0, ρ1 ≤ ρ, so that for all u0 ∈ Rρ1 ,

∣∣(v

[u0]pde,x (· +, t)

)∣∣ ≥ |(vx )|2

> 0 ∀|| ≤ ρ, t ≥ 0, (7.4)

and the function r : Bρ(0)× [0,∞) → R, where Bρ(0) = {|| < ρ}, given by

r : (, t) �→ r(, t) = −(F(v[u0]

pde(· +, t)))

(v

[u0]pde,x (· +, t)

) , (7.5)

is an element of C(Bρ(0)× [0,∞); R).

Proof of Lemma 7.2 By Step 1, v[u0]pde ∈ C([0,∞); v + H1) for u0 ∈ Rρ , so that F(vpde) ∈

C([0,∞); L2). The continuity of the shift in L2(R,Rm) implies

(, t) �→ (F(vpde(· +, t))

)belongs to C(R × [0,∞); R),

(, t) �→ (vpde,x (· +, t)

)belongs to C(R × [0,∞); R).

For the denominator of r holds∣∣(v

[u0]pde,x (· +, t)

)∣∣ ≥ |(vx )| −[∣∣(

v[u0]pde,x (· +, t)− vx (· − ϕ∞ +)

)∣∣ +∣∣(

vx (· − ϕ∞ +)− vx

)∣∣] (7.6)

where the [ ]–term in (7.6) can be estimated using (7.1) and the continuity of , by

|(v[u0]pde,x (· +, t)− vx (· − ϕ∞ +))| + |(

vx (· − ϕ∞ +)− vx

)|≤ C

(C pde

∥∥u0 − v∥∥

H2 e−ηt + ‖vx‖H1(|ϕ∞| + ||))

≤ Cη(‖u0 − v‖H2 + ||) .

The constant Cη is independent of u0. With ρ from Step 1 define

ρ1 = min

(ρ,

|(vx )|4Cη

)and ρ = |(vx )|

4Cη.

Then for all u0 ∈ Rρ1 and all ∈ Bρ(0) = {|| ≤ ρ} the estimate (7.6) yields

∣∣(v

[u0]pde,x (· +, t)

)∣∣ ≥ |(vx )|2

> 0, for all t ≥ 0

for the denominator. Therefore, for u0 ∈ Rρ1 the function r is well-defined on Bρ(0)×[0,∞)

and, in particular, r is continuous on this set. ��

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358 J Dyn Diff Equat (2012) 24:341–367

Rewriting (7.2) as = r(, t),(0) = 0, local existence of a solution to (7.2) is a simplecorollary by using Peano’s Theorem (e.g. [27, II. Sect. 6]):

Corollary 7.3 In the setting of Lemma 7.2 there is a solution of (7.2) and

• either is a solution for all times t ≥ 0 with ∈ C1([0,∞); Bρ(0)),• or there is 0 < T ∗ < ∞ and limt↗T ∗ |(t)| = ρ.

Step 4: [Global existence of a solution] The next lemma shows that for u0 sufficientlyclose to v, the ODE (7.2) has a global solution.

Lemma 7.4 There is ρ0 > 0, ρ0 ≤ ρ1, so that for all u0 ∈ Rρ0 , every solution of (7.2),satisfying the dichotomy from Corollary 7.3, also satisfies ∈ C1

([0,∞); B ρ2(0)

).

This proves with Lemma 7.1 global solvability of the Freezing Equation (2.6):

Corollary 7.5 (Global existence for the freezing method) Let ρ0 be given as in Lemma7.4. Then for every u0 ∈ Rρ0 the PDAE (2.6) has a global solution (u, λ) which satisfies

u ∈ C1([0,∞); v + L2) ∩ C0([0,∞); v + H1) and λ ∈ C([0,∞); R).

Proof of Lemma 7.4 Lemma 3.3 applied with ψ = yields open neighborhoods U, V ⊂ R

of 0 so that G : U → V, ϕ �→ G(ϕ) = (v(·− ϕ)−v) is a C2 diffeomorphism. Let V0 ⊂ V

be a compact neighborhood of 0 and let C be the Lipschitz constant of G−1|V0 = |V0 .Choose 0 < ρ0 ≤ ρ1 with

(1 + C C)C pdeρ0 ≤ ρ

2and BCC pdeρ0(0) = {|v| ≤ CC pdeρ0} ⊂ V0.

Let u0 ∈ Rρ0 and let be a solution of (7.2) given by Corollary 7.3. Define

T ∗ = sup{T0 > 0 : |(t)| < ρ ∀0 ≤ t ≤ T0}.Then, 0 =

(u − vpde(· +(t), t)

)for all 0 ≤ t < T ∗ and because of (u − v) = 0,

G(ϕ∞ −(t)) = (v(· − ϕ∞ +(t))− vpde(· +(t), t)

). (7.7)

The bound (2.7) for and (7.1) show for all 0 ≤ t < T ∗∣∣(

v(· − ϕ∞ +(t))− v[u0]pde(· +(t), t)

)∣∣ ≤ CC pde‖u0 − v‖H2 e−ηt ≤ CC pdeρ0,

so that G(ϕ∞ −(t)) ∈ V0. Application of to (7.7) therefore yields

|(t)− ϕ∞| ≤ C CC pde‖u0 − v‖H2 e−ηt for all t ∈ [0, T ∗). (7.8)

In particular, with |ϕ∞| ≤ C pde‖u0 − v‖H2 from Step 1 and the choice of ρ0,

|(t)| ≤ |ϕ∞| + C CC pde‖u0 − v‖H2 e−ηt ≤ C pde(1 + CϕC

)‖u0 − v‖H2 ≤ ρ

2,

(7.9)

so that Corollary 7.3 implies the existence of the solution for all positive times. Finally,(7.9) shows ∈ C1

([0,∞); B ρ2(0)

). ��

Step 5: [Unique solvability of the ODE (7.2)] Since we used Peano’s Theorem for theexistence, we still have to show uniqueness of the solution. The idea is that the solution mustsatisfy the algebraic constraint which is locally uniquely solvable. This is the content of thefollowing lemma:

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J Dyn Diff Equat (2012) 24:341–367 359

Lemma 7.6 Let u0 ∈ Rρ0 and let 0 ∈ C1([0,∞); B ρ2(0)) be a solution of (7.2) on

[0,∞). Define the mapping H : [0,∞)× R → R by

H : (t,) �→ H(t,)=+(v

[u0]pde,x (·+0(t), t)

)−1

(u−v[u0]

pde(· +, t)). (7.10)

Then for every t0 ∈ [0,∞) there are δt = δt (t0) > 0 and δ = δ(t0) > 0 such that for allt ∈ Bδt (t0) ∩ [0,∞) and all ∈ Bδ(0(t)),

∣∣H(t,)− H(t,0(t))∣∣ ≤ 1

2|−0(t)|. (7.11)

Note that for fixed t the function H(t, ·) from (7.10) describes one iteration step of a quasi-Newton method for the equation F() = 0, where F() =

(u − v

[u0]pde(· +, t)

).

Proof Let t0 ∈ [0,∞). Since vpde ∈ C0([0,∞); v + H1), there is δt > 0, so that

∥∥vpde(·, t0)− vpde(·, t)∥∥

H1 ≤ |(vx )|12C

∀t ∈ Bδt (t0) ∩ [0,∞). (7.12)

Furthermore, by continuity of the shift in H1(R,Rm) there exists a δ > 0 with

∥∥vpde(·, t0)− vpde(· + ξ, t0)∥∥

H1 ≤ |(vx )|12C

∀ξ ∈ Bδλ(0). (7.13)

For the ease of notation we simply write 0 for 0(t) in the sequel. The definition of Hand estimate (7.4) from Lemma 7.2 imply for all t ≥ 0 and all ∈ R

∣∣∣H(t,)− H(t,0)

∣∣∣ ≤ 2

|(vx )|∣∣∣∣

(vpde,x (· +0, t)

)(−0

)

+[

(u−vpde(·+, t)

)−(u−vpde(· +0, t)

)]∣∣∣∣. (7.14)

Because of the smoothness vpde(·, t) ∈ H1(R,Rm),

vpde(· +0, t)− vpde(· +, t) = −1∫

0

vpde,x(· +0 + s(−0), t

)ds (−0)

holds as an equality in L2. Inserting this into (7.14) and using the monotonicity of the Boch-ner-integral and (P1), the right hand side of (7.14) is bounded by

2

|(vx )|1∫

0

∣∣∣(vpde,x (· +0, t)

) −(vpde,x (· +0 + s(−0), t)

)∣∣∣ ds∣∣−0

∣∣

≤ 2C|(vx )|

sups∈[0,1]

∥∥vpde(·, t)− vpde(· + s(−0), t

)∥∥H1

∣∣−0∣∣.

The assertion then follows, since for t ∈ Bδt (t0)∩[0,∞) and ∈ Bδ(0(t)) the inequalities(7.12) and (7.13) show for all s ∈ [0, 1]∥∥vpde(·, t)− vpde

(· + s(−0(t)), t)∥∥

H1

≤ 2∥∥vpde(·, t)− vpde(·, t0)

∥∥H1 + ∥∥vpde(·, t0)− vpde(· + s(−0(t)), t0)

∥∥H1

≤ |(vx )|6C

+ |(vx )|12C

= |(vx )|4C

.

��

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360 J Dyn Diff Equat (2012) 24:341–367

Lemma 7.7 For u0 ∈ Rρ0 the solution ∈ C1([0,∞); Bρ(0)) of (7.2) is unique.

Proof Let 0 and 1 be global solutions of the ODE (7.2). By Lemma 7.4 the functionssatisfy 0,1 ∈ C1([0,∞); Bρ(0)). For every solution of (7.2) holds

(u − vpde(· +

(t), t)) = 0 for every t ≥ 0 so that H(t,(t)) = (t) for every t ≥ 0.

Let M := {t ≥ 0 : 0(t) = 1(t)}, which contains 0 and is a closed subset of [0,∞).Moreover, M is an open subset of [0,∞): Assume t0 ∈ M and let δt and δ be as in Lemma7.6. There is εt0 > 0, εt0 ≤ δt , so that |1(t)−0(t)| < δ for all t ≥ 0 with |t − t0| < εt0 .It follows 1(t) = 0(t) for all such t , since by Lemma 7.6

|1(t)−0(t)| = |H(t,1(t))− H(t,0(t))| ≤ 1

2|1(t)−0(t)|.

��

Step 6: [Unique solvability of the PDAE (2.6)] Let u ∈ C1([0, T ); v+L2)∩C([0, T ); v+H1), λ ∈ C([0, T ); R), be a solution of (2.6). Define (t) := ∫ t

0 λ(τ ) dτ and vpde(·, t) :=v(·−(t), t). Then vpde solves (1.1) in [0, T )with vpde(0) = u0. But because of uniqueness

for (1.1) vpde = v[u0]pde . Moreover, the hidden constraint shows that is a solution of the ODE

(7.2), which is unique by Lemma 7.7.Step 7: [Exponential convergence] The unique solution of (2.6) is

(u, λ) = (v

[u0]pde(· +(t), t), (t)

),

where solves (7.2). Using this representation, shift-invariance of the H1–norm shows∥∥u(·, t)−v∥∥H1 ≤ ∥∥vpde(·, t)−v(· − ϕ∞)

∥∥H1 +∥∥v(· − ϕ∞)− v(· −(t))

∥∥H1 . (7.15)

Because of (7.1), the first summand is bounded by C pde∥∥u0 − v

∥∥H2 e−ηt and the second

symmand by ‖vx‖H1 const ‖u0 − v‖H2 e−ηt , because of (7.8). Therefore,

‖u(·, t)− v‖H1 ≤ const ‖u0 − v‖H2 e−ηt . (7.16)

Since λ(t) = (t), the ODE (7.2) and estimate (7.4) from Lemma 7.2 imply

|λ(t)| = |(F(u(t))

)||(u(t))| ≤ 2

|(vx )|∣∣(

F(u(t))− F(v))∣∣

≤ 2C|(vx )|

‖F(u(t))− F(v)‖L2 ≤ C‖u(t)− v‖H1 ≤ C‖v0 − v‖H2 e−ηt ,

where F(v) = 0 and the local Lipschitz continuity of F in v + H1 were used. This finishesthe proof of Theorem 2.3. ��

Note that the global results from Step 1 are only needed for global existence of a solutionto (7.2) and, in particular, for (7.4), which is important for globally solving the ODE. Butthese estimates are not needed for local existence, which only relies on the well-definednessof the fraction (7.5). Therefore, the proof shows that the PDAE–system (2.6) is locally solv-able if we have consistent initial conditions for u0, satisfying (u0x ) �= 0. If the solution uremains bounded but (ux ) becomes singular, a possible idea is to update the phase condi-tion and continue the solution of the PDAE with the new phase condition. One can considerthe minimizing phase condition from [5] as one, which is updated in each time-step. But wedid not pursue this any further.

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J Dyn Diff Equat (2012) 24:341–367 361

8 Numerical Experiments

8.1 Hyperbolic-Nagumo Equation

Our first example is a simple hyperbolic test equation with a cubic nonlinearity:

ut + qx = u(1 − u)(u − β), q + T qt = −Dux . (8.1)

For T = 0 this is a parabolic PDE, known as Nagumo’s equation, which can be derived as asimplification of a population dynamics model (cf. [2]). For T > 0 the second equation in(8.1) is a modified Fickian law, known in the physics literature as Cattaneo–Maxwell flux[12]. It was proposed in [14] to use this flux in the context of chemical reaction problems asa way to prevent the unphysical infinite speed of propagation. These authors use it in [17]for the generalized Fisher’s equation.

For concreteness, let T = 1, D = 1 and β = 0.25, so that (8.1) becomes(

uq

)t=

(0 −1

−1 0

)(uq

)x

+(

u(1 − u)(u − 0.25)−q

)=: B

(uq

)x

+ f (u, q). (8.2)

We call this system the hyperbolic Nagumo equation. Rest states of (8.2) are (0, 0)T and(1, 0)T and there is a traveling wave solution with profile (u, q)T and speed λ. The profile

is a heteroclinic orbit of ddx (u, q)T = −(B + λI )−1 f (u, q) =: F(u, q), connecting the two

rest states. If |λ| < 1, both are hyperbolic fixed points of the ODE and the profile approachesthe rest states exponentially fast as |x | → ∞.

In this case the profile satisfies (H0) and it is also easy to verify (H1) and (H2). To verify(H3), one diagonalizes to see that (H3) is equivalent to S ⊂ {Re s < −δ},

S ={

s ∈ C : det

(s I − iω

(−1 + λ 00 1 + λ

)− C±

)= 0, ω ∈ R

}, (8.3)

C− = 1

2

(−1.75 −0.25−0.25 −1.75

), C+ = 1

2

(−1.25 −0.75−0.75 −1.25

).

One easily calculates that, the spectral set S consists of the four curves

s+± (ω) = −1

2− 1

8+ λωi ± 1

2

√(1 − 0.25)2 − 4ω2, ω ∈ R,

s−± (ω) = −1 + 1

8+ λωi ± 1

2

√0.252 − 4ω2, ω ∈ R.

These lie entirely in {Re s ≤ −0.25}, so that (H3) follows.In Fig. 1 we show the result of a direct simulation of the freezing system (2.6) for this

example. The spatial domain is the interval [−20, 20] and we take Neumann boundary con-ditions. The initial data are

u0(x) =

⎧⎪⎨⎪⎩

1, x < −10,

0.5 − 0.05x, −10 ≤ x ≤ 10,

0, x > 10,

q0(x) = 0, ∀x ∈ R,

and the reference function is the same as the initial data and is chosen as

(u − u, q − q) =∫

R

ux (u − u)+ qx (q − qx ) dx .

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362 J Dyn Diff Equat (2012) 24:341–367

(a)

(b)

(c) (d)

Fig. 1 Evolution of the frozen Nagumo system. a Time evolution of the u variable. b Time evolution of the qvariable. c Time evolution of the algebraic variable λ, which corresponds to the waves speed. d Time evolutionof ‖(ut , qt )‖L2 as an indicator for the covergence to a steady state

Note that is an element of (L2)′ but not of the dual of H−1. One visually observes aconvergence of (u, q, λ) to the steady state which is specified in Fig. 1d, where we plot theevolution of ‖(ut , qt )‖L2 that indicates the convergence. We observe a rate of convergenceof approximately e−0.35t . This is a bit better than the theoretically predicted rate of e−0.25t .

The final states of the direct simulation are used as initial data for a numerical computa-tion of the profile and speed with the method from [3]. As asymptotic boundary conditionswe choose projection boundary conditions and the boundary value problem is solved withbvp4c from Matlab. As a result we obtain λ ≈ 0.3754.

In Fig. 2 we plot the spectral set S and a numerical approximation of the spectrum of thelinearized operator. The computation is done with the profile and speed computed with themethod described before. The system is discretized using upwinding with step–size�x = 0.1

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J Dyn Diff Equat (2012) 24:341–367 363

Fig. 2 Left: Closeup of the numerical spectrum (crosses) on [−30, 30] with step size�x = 0.1 and periodicboundary conditions together with the spectral set S from (8.3) (continuous lines). Right: The first eigenvalueswith largest real part

and periodic boundary conditions on the interval [−30, 30]. One observes a simple eigenvaluethat is close to zero and then a spectral gap.

8.2 Hyperbolic Hodgkin–Huxley System

Our second example is a hyperbolic version of the Hodgkin–Huxley model. It is formallyobtained from the original problem (see [11]) by using a modified Fickian law as above. Sowe call the following system the hyperbolic Hodgkin–Huxley system:

Vt = −qx − gK n4(V − VK )− gNam3h(V − VNa)− gl(V − Vl),

qt = − 12 Vx − q,

nt = αn (1 − n)− βnn,

mt = αm (1 − m)− βmm,

ht = αh (1 − h)− βhh.

Here the nonlinearities are

αn = 1100 (V + 10)

(exp( V +10

10 )− 1)−1

, βn = 18 exp( V

80 ),

αm = 110 (V + 25)

(exp( V +25

10 )− 1)−1

, βm = 4 exp( V18 ),

αh = 7100 exp( V

20 ), βh = (exp( V +30

10 )+ 1)−1

.

For the constants we choose the values from the original paper [11]

VNa = −115, VK = 12, Vl = −10.613, gNa = 120, gK = 36, gl = 0.3.

It is important to note that the system is hyperbolic, but not strictly hyperbolic, i.e. in thenotation of Sect. 2 the matrix (λI + B) has multiple eigenvalues. Note that this case is notcovered by the results of Kreiss et al. [15].

The system has a rest point approximately at V∞ = −0.0036, q∞ = 0, n∞ = 0.3177,m∞ = 0.0530, h∞ = 0.5960, we simply write v∞ := (V∞, q∞, n∞,m∞, h∞)T . The

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364 J Dyn Diff Equat (2012) 24:341–367

(a) (b)

Fig. 3 The time-evolution of the V -component of the solution to the hyperbolic-Hodgkin–Huxley equation.The plots show −V (x, t). a A plot of the V -variable for different times. b The same data, here V is color -coded

(a) (b)

Fig. 4 Convergence to a steady state and the profile of the traveling wave. a Convergence of the freezingmethod to a stationary solution, which is indicated by the norm ‖vt (t)‖L2 . b The profile of the traveling wave.The corresponding speed of the wave is λ ≈ 0.6676

numerical computations show that there is a traveling wave solution whose profile is a ho-moclinic connection of the rest point v∞ to itself.

In Fig. 3 we present the results of a numerical simulation of the problem with the freezingmethod (we only plot the V component). Again, one observe the convergence to a steadystate. The computation was done by the same method as for the hyperbolic-Nagumo equation,presented above. But here we updated the reference function (and also , which is derivedfrom the reference function) in each time-step. This leads to the so-called minimizing phasecondition from [5]. Note that the authors show how to transform the solutions for differentphase conditions into each other.

In Fig. 4a the value ‖vt (t)‖L2 , v = (V, q,m, n, h), is plotted against time, which illus-trates the convergence to the steady state. On the right, in Fig. 4b, we show the asymptotic

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J Dyn Diff Equat (2012) 24:341–367 365

(a) (b)

Fig. 5 The dispersion relation for the traveling wave solution of the hyperbolic Hodgkin–Huxley equation. aThe spectral set S. b A zoom into the spectral set S around 0

profile of the traveling wave which is again computed using the techniques from [3]. Thefinal values of the forward integration were used as initial data for the boundary value solver.

To justify the spectral Assumption (H3) for this problem, we resolved the dispersionrelation, i.e. we display the set

S = {s ∈ C : det

(s I − iω(λI + B)− fv(v∞)

) = 0},

in Fig. 5. It is well-known that the set S belongs to the essential spectrum of the linearizedoperator and there is no essential spectrum to the right of the rightmost curve. The set isthe union of five algebraic curves. For its numerical computation, we used the continuationtoolbox MATCONT [8] and solved for the eigenvalues of iω(λI + B)+ fv(v∞)with respectto ω.

Because the equation

G(s, ω) := det(s I − iω(λI + B)− fv(v∞)

) = 0,

is not well suited for the continuation method, we use the following real version of a complexeigenvector-eigenvalue system,

(Re(s)I − fv(v∞) ω(λI + B)− Im(s)I

−ω(λI + B)+ Im(s)I Re(s)I − fv(v∞)

) (Re(v)Im(v)

)= 0,

‖ Re(v)‖2 + ‖ Im(v)‖2 − 1 = 0,

Re(v)T Im(v) = 0.

(8.4)

The last two equations are needed to determine a unique complex eigenvector. The lastequation can be derived from the assumption

argminϕ∈[0,2π)‖ Im(eiϕv)‖ ∈ {0, π},which roughly states that we want the eigenvector to be “as real as possible”.

The idea of computing the essential spectrum by continuation is not new, see for example[18]. There the authors use a different condition to choose a unique eigenvector, namelyvT

oldv = 0, where vold is the eigenvector from the last continuation step. The advantage of

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366 J Dyn Diff Equat (2012) 24:341–367

our conditions is that it automatically normalizes the eigenvector and the toolbox MATCONTcan directly be applied.

Acknowledgements The author would like to thank Wolf-Jürgen Beyn for his helpful comments on a firstversion of this paper. This research was supported by CRC 701 “Spectral Analysis and Topological Methodsin Mathematics”.

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