Hyperboloidal evolution and global dynamics for the cubic wave equation

Hyperboloidal evolution and global dynamics for the focusing cubic wave equation

Abstract.

The focusing cubic wave equation in three spatial dimensions has the explicit solution . We study the stability of the blowup described by this solution as without symmetry restrictions on the data. Via the conformal invariance of the equation we obtain a companion result for the stability of slow decay in the framework of a hyperboloidal initial value formulation. More precisely, we identify a codimension-1 Lipschitz manifold of initial data leading to solutions which converge to Lorentz boosts of as . These global solutions thus exhibit a slow nondispersive decay, in contrast to small data evolutions.

Key words and phrases:
nonlinear wave equations, cubic wave equation, hyperboloidal initial value problem, global solutions, asymptotics, blowup, stability
2010 Mathematics Subject Classification:
35L05 (primary), 35L71, 58J45 (secondary)

1. Introduction

We consider the cubic wave equation

 □v(t,x)+v(t,x)3=0,x∈R3, (1.1)

with wave operator and conserved energy

 12∫R3(|∂tv(t,x)|2+|∇xv(t,x)|2)dx−14∫R3|v(t,x)|4dx.

It is well known that solutions with small -norm exist globally and scatter to zero [22], whereas for large initial data nonlinear effects dominate and generically lead to finite-time blowup [1, 4, 10, 15, 17]. The explicit solution of (1.1), on the other hand, exists globally for but is nondispersive. Since it has infinite energy, its role in the dynamics is a priori unclear. Nevertheless, solutions with such an exceptionally slow decay have been observed in numerical simulations [2] and are conjectured to sit on the threshold between dispersion and finite-time blowup. The main purpose of this paper is to develop a rigorous understanding of this phenomenon.

To this end, we follow [2, 6] and utilize the conformal invariance of the cubic wave equation (1.1) to reduce the question of stability of as to the stability analysis of the blowup described by as . In fact, the conformal transformation leads to a natural hyperboloidal initial value formulation in which the stability of as can be studied. In what follows, we carry out this transformation, and then state and review our Theorem 1.2 for the blowup stability. We then return to the decay picture. After introducing the hyperboloidal initial value problem of interest, we state two decay results for global solution of (1.1). Our main Theorem 1.9 verifies the codimension-1 stability of slow decay.

The blowup result

The cubic wave equation (1.1) in is conformally invariant. This invariance is expressed in terms of the (time-reversed) Kelvin transform , which reads

 (T,X) ↦κT(X)=κ(T,X)=(−TT2−|X|2,XT2−|X|2). (1.2)

Note that the inverse transform from to is of the same form, i.e.,

 κ−1(t,x)=κt(x)=(−tt2−|x|2,xt2−|x|2), (1.3)

and we moreover have the identity

 t2−|x|2=(T2−|X|2)−1.

The coordinate transformation maps the future light cone to the past light cone . A straightforward computation now shows the following invariance property.

Lemma 1.1.

The function

solves the cubic wave equation

 □u(T,X)+u(T,X)3=0 (1.4)

on the past light cone if and only if solves the same equation (1.1) on the future light cone .

As a consequence of this conformal invariance of (1.1), the stability analysis of as translates into the question of stability of the blowup solution

 u0(T,X)=v0∘κT(X)T2−|X|2=√2−T

of (1.4) in the backward light cone . In this blowup picture we establish the following codimension-1 stability result for the blowup described by and its Lorentz boosts with rapidity (see Section 2.2 for the exact definition). We denote by the open ball of radius in and .

Theorem 1.2 (Blowup stability).

There exists a codimension-1 Lipschitz manifold of initial data in , with , such that the Cauchy problem

 □u(T,X)+u(T,X)3 =0, u(−1,.) =u0(−1,.)+F, ∂Tu(−1,.) =∂Tu0(−1,.)+G,

with has a unique solution (in the Duhamel sense) on the truncated lightcone . For a unique rapidity and the corresponding Lorentz-boosted , denoted by , we have

 |T|12∥u(T,.)−ua(T,.)∥˙H1(B|T|) ≲|T|12−, |T|−12∥u(T,.)−ua(T,.)∥L2(B|T|) ≲|T|12−, |T|12∥∂Tu(T,.)−∂Tua(T,.)∥L2(B|T|) ≲|T|12−,

for all .

Remark 1.3.

Observe that

 ∥u0(T,.)∥L2(B|T|) ≃|T|12, ∥∂Tu0(T,.)∥L2(B|T|) ≃|T|−12,

and thus, the normalization factors on the left hand sides are natural. In particular, the solution converges to the Lorentz-boosted as the blowup time is approached.

Remark 1.4.

The instability of the blowup comes from the fact that general perturbations of will change the blowup time. Consequently, since the manifold has codimension one, it follows that the blowup profile is stable up to time translation (and Lorentz boosts). In this sense, the instability is not “real”. In fact, one could include a Lyapunov–Perron-type argument as in [5] to get rid of the codimension-1 condition and vary the blowup time instead. However, for the decay picture it turns out that the result in its present form is more useful.

Remark 1.5.

Theorem 1.2 is closely related to the seminal work by Merle and Zaag [17, 18] which established the universality of the blowup speed for the conformal wave equation. In the subconformal case they also proved the stability of the blowup profile [19, 20]. A similar stability result for superconformal equations was obtained in [5], but in a stronger topology.

The decay result

Geometrically, the conformal invariance naturally leads to a hyperboloidal initial value problem for equation (1.1). For we consider the spacelike slices defined by

 ΣT:={κT(X)∣∣X∈B|T|}={(t,x)∈(0,∞)×R3∣∣t2−|x|2t=1(−T)}.

These slices provide a foliation of the future light cone . The hyperboloidal slices in the decay picture are the pre-images of constant time slices in the blowup picture, see Figure 1. We emphasize that the slices are asymptotic to different light cones and hence “foliate” future null infinity. As a consequence, energy can escape to infinity and this provides the crucial stabilizing mechanism.

In the hyperboloidal initial value formulation the data are prescribed on the spacelike hyperboloid , see Figure 2. These initial data consist of a function in and a derivative normal to in . The function spaces and , as well as the derivative , are naturally transferred from the blowup picture and defined as follows.

Definition 1.6 (Function spaces on hyperboloids).

On each hyperboloidal slice we define the norms

 ∥v∥2L2(ΣT) :=∫B|T|∣∣ ∣∣v∘κT(X)T2−|X|2∣∣ ∣∣2dX, ∥v∥2˙H1(ΣT) :=∫B|T|∣∣ ∣∣∇Xv∘κT(X)T2−|X|2∣∣ ∣∣2dX,

and

 ∥v∥2H1(ΣT):=∥v∥2˙H1(ΣT)+|T|−2∥v∥2L2(ΣT).
Definition 1.7 (Normal derivative).

The differential operator is defined by the relation

 ∇nv∘κT(X)T2−|X|2=∂T(v∘κT)(X)T2−|X|2,

and explicitly given by

 ∇nv(t,x)=(t2+|x|2)∂tv(t,x)+2txj∂jv(t,x)+2tv(t,x).
Remark 1.8.

The principal term of is the pullback of the vector field along the inverse Kelvin transform , and well-known as the Morawetz multiplier . The zeroth order term appears due to the weight in the transformation.

The result in the decay picture reads as follows. For the domain appearing in the Strichartz norm we refer to Figure 3.

Theorem 1.9 (Stability of slow decay).

There exists a codimension-1 Lipschitz manifold of initial data in , with , such that the hyperboloidal initial value problem

 □v(t,x)+v(t,x)3 =0, v|Σ−1 =v0|Σ−1+f ∇nv|Σ−1 =∇nv0|Σ−1+g,

with and , has a unique solution (in the Duhamel sense) defined on the future domain of dependence . For a unique rapidity and the corresponding Lorentz-boosted , denoted by , we have the decay

 |T|12(∥v−va∥H1(ΣT)+∥∇nv−∇nva∥L2(ΣT))≲|T|12−

for all . Moreover, for any , the decay in Cartesian coordinates is1

 ∥v−va∥L4(t,2t)L4(B(1−δ)t)≲t−12+

as .

Remark 1.10.

As with the blowup result, the normalizing factor on the left-hand side reflects the behavior of the solution in the respective norm, i.e., we have

 ∥va∥H1(ΣT)+∥∇nva∥L2(ΣT)≃|T|−12

for all .

Remark 1.11.

Contrary to the blowup result, the instability is now “real” in the sense that generic evolutions will either disperse (i.e., decay faster, see below) or blow up in finite time. This is easily understood by noting that solutions in the blowup picture with blowup time larger than correspond to dispersive solutions in the decay picture. On the other hand, solutions in the blowup picture with blowup time less than correspond to finite-time blowup in the decay picture. Only those solutions in the blowup picture that blow up precisely at time lead to slow decay in the decay picture.

For quantitative comparison with the small data evolution we finally note the following result.

Theorem 1.12 (Small data dispersive decay).

There exists an such that the hyperboloidal initial value problem

 □v(t,x)+v(t,x)3 =0, v|Σ−1 =f, ∇nv|Σ−1 =g.

for initial data with has a unique global solution (in the Duhamel sense) in the future domain of dependence which, for any , satisfiesfootnote 1

 ∥v∥L4(t,2t)L4(B(1−δ)t)≲t−12.
Remark 1.13.

Theorem 1.12 and the fact that

 ∥va∥L4(t,2t)L4(B(1−δ)t)≃1

show that the decay of the solutions constructed in Theorem 1.9 is slower than the decay of generic small data evolutions. In particular, the solutions from Theorem 1.9 are nondispersive.

Our main Theorem 1.9 rigorously establishes the codimension-1 stability of without symmetry assumptions, which was numerically observed in [2] in the case of spherical symmetry. A direct precursor of Theorem 1.9 is [6], where the codimension-4 stability of slow decay was established. The additional three unstable directions are caused by the Lorentz symmetry. In the present paper we use modulation theory to deal with this issue. In its present form, our result crucially relies on the conformal invariance of equation (1.1) which necessitates the cubic power in three spatial dimensions. In general dimensions , the conformally invariant wave equation is

 □v(t,x)±v(t,x)|v(t,x)|4d−1=0

with the explicit solution ( is a suitable constant). Our methods can be generalized to this situation. It is an interesting open question whether similar results can be proved for equations that are not conformally invariant. We also hope that our paper is interesting from the general perspective of hyperboloidal methods which receive increasing attention (see, for example, [3, 7, 9, 12, 13, 14, 16, 23, 24, 25]).

Outline

An overview of the methods and organization of the paper follows. We go through the proof of Theorem 1.9, the adjustments for the proof of Theorem 1.12 are explained and carried out in Section 5.6. The proof of Theorem 1.2 is part of that of Theorem 1.9, see also Remark 5.11.

Via the Kelvin transform we have seen that equation (1.1) is exactly the same focusing cubic wave equation (1.4) in hyperboloidal coordinates, however, instead of treating the asymptotics of global solutions in the future light cone we are led to look at solutions in the past light cone of the origin. In the preliminary Section 2 we further transform the cubic wave equation (1.4) in hyperboloidal coordinates to similarity coordinates . Similarity coordinates are a natural choice of coordinates for the selfsimilar solution which simply transforms to the constant solution . Moreover, we introduce the Lorentz boosts of , that is, the solutions of (1.1).

In Section 3 we rewrite the cubic wave equation as an evolution system of the form

 ∂τΨ(τ)=LΨ(τ)+N(Ψ(τ)),

where is a linear operator and is nonlinear. To account for the Lorentz symmetry we use a modulation ansatz

 Ψ(τ)=Ψa(τ)+Φ(τ)

around the Lorentz transformations of the selfsimilar solution . We allow for the (unknown) rapidity to depend on , set initially and assume (and later verify) that . This ansatz leads to an equivalent description as an evolution system for the perturbation term , i.e.,

 ∂τΦ(τ)−LΦ(τ)−L′a∞Φ(τ)=[L′a(τ)−L′a∞]Φ(τ)+Na(τ)(Φ(τ))−∂τΨa(τ), (1.5)

where denotes the linearized part of the nonlinearity and the remaining full nonlinearity.

In Section 4 we study the linearized part of the system (1.5), that is

 ∂τΦ(τ)=[L+L′a∞]Φ(τ), (1.6)

and control the asymptotics of the solutions. To this end we employ semigroup theory and spectral theory. More precisely, the operator generates a strongly continuous semigroup , and, since is bounded, there also exists a semigroup generated by . A careful analysis of the spectrum of yields decay estimates for the linearized evolution (1.6).

Finally, the nonlinear terms are controlled by standard Sobolev embedding, and the full nonlinear equation is solved by several fixed point arguments in Section 5. For this purpose we first rewrite equation (1.5) with as a weak integral equation

 Φ(τ)=Sa∞(τ)u−∫τ0Sa∞(τ−σ)[(L′a(σ)−L′a∞)Φ(σ)+Na(σ)(Φ(σ))−∂σΨa(σ)]dσ (1.7)

using Duhamel’s principle. The terms in the integrand are shown to be small and Lipschitz continuous with respect to and . The instabilities arising from the Lorentz symmetry of the cubic wave equation are suppressed by choosing the rapidity in a suitable way. In contrast, the time-translation instability is isolated by adding a correction term and first solving a modified weak equation of the form

 Φ(τ)=Ku(Φ,a)(τ)−Sa∞(τ)Cu(Φ,a),

where denotes the right hand side of (1.7), by means of contraction arguments. Solutions to this modified equation with vanishing correction term thus satisfy the original equation (1.7). The condition is shown to describe a codimension-1 manifold of initial data, which we locally represent as a graph of a Lipschitz function.

Notation

By we denote the open unit ball and by the unit sphere in . We write for the open ball with radius around the origin. The domain of an (unbounded) operator is written as . The spectrum of a linear operator is denoted by , the point spectrum by . Its resolvent is the operator , i.e., for in the resolvent set . We assume that are generic and small, however implicit constants may depend on them. Einstein’s summation convention is used throughout the manuscript. This means that if an index appears twice in a summation term (once as subscript and once as superscript), then we automatically sum over all values of that index, e.g., is written instead of . Finally, the notation indicates that there exists a constant (possibly depending on a parameter) such that . If and holds, then we simply write . The decay estimate of Theorem 1.9 means that for any there exists a constant such that .

2. Preliminaries

We have reformulated the original cubic wave equation (1.1) on the future light cone of the origin stated in Cartesian coordinates as a problem in hyperboloidal coordinates on the past light cone . Since we are interested in solutions close to the selfsimilar solution , we further employ selfsimilar coordinates , and we obtain an equivalent second order equation. In Section 3 this equation is then further transformed to an evolution problem of first order in .

2.1. The equation in similarity coordinates

Since is a selfsimilar solution, it is natural to employ the similarity coordinates

 τ:=−log(−T)=−logtt2−|x|2,ξ:=X−T=xt, (2.1)

with and , see Figure 4. The inverse transforms read

 T=−e−τ,X=e−τξ

and

 t=eτ1−|ξ|2,x=eτξ1−|ξ|2,

respectively. The cubic wave equation transforms accordingly.

Lemma 2.1.

The rescaled function

 ψ(τ,ξ):=e−τu(−e−τ,e−τξ)

solves the equation

 [∂2τ+3∂τ+2ξj∂ξj∂τ−(δjk−ξjξk)∂ξj∂ξk+4ξj∂ξj+2]ψ(τ,ξ)=ψ(τ,ξ)3 (2.2)

on the domain if and only if solves the cubic wave equation (1.4) on the past light cone of the origin.

Proof.

The Jacobi matrix of the transformation from to reads, for ,

This implies that

 ∂Tu(T,X)=e2τ(1+∂τ+ξj∂ξj)ψ(τ,ξ) (2.3)

and

 ∂2Tu(T,X)=e3τ[2+3∂τ+4ξj∂ξj+∂2τ+2ξj∂τ∂ξj+ξjξk∂ξj∂ξk]ψ(τ,ξ),

as well as

 ∂Xiu(T,X)=e2τ∂ξiψ(τ,ξ),∂2Xiu(T,X)=e3τ∂2ξiψ(τ,ξ).

Consequently, being a solution of is equivalent to being a solution to the equation

 e3τ[2+3∂τ+4ξj∂ξj+∂2τ+2ξj∂τ∂ξj+(ξjξk−δjk)∂ξj∂ξk]ψ(τ,ξ)=e3τψ(τ,ξ)3,

which is (2.2). ∎

The fundamental selfsimilar solution of (2.2) is the constant solution .

2.2. The 3-parameter family of the selfsimilar solutions

The cubic wave equation (1.4) is invariant under time translations and Lorentz transformations. The time translation symmetry yields the one-parameter family of solutions. Moving the blowup time of away from leads to either finite-time blowup () or dispersion (). On the other hand, the blowup surface of is invariant under spatial translations and rotations. We fix the origin and allow for hyperbolic rotations of by applying Lorentz boosts for each direction. For the rapidity the Lorentz transformation is given by

 Λj(aj):R×R3 →R×R3,j∈{1,2,3}, (TXk) ↦(Tcoshaj+XjsinhajXk+δjk(Tsinhaj+Xjcoshaj−Xj)).

Note that maps the past light cone into itself. Applied to the solution of (1.4), the Lorentz transformation generates a 3-parameter family

 ua(T,X):=u0(Λ(a)(T,X))=√2A0(a)(−T)−Aj(a)Xj,a∈R3, (2.4)

of explicit blowup solutions, given by

 A0(a) =cosha1cosha2cosha3, A1(a) =sinha1cosha2cosha3, A2(a) =sinha2cosha3, A3(a) =sinha3.

The Lorentz transformations of the fundamental selfsimilar solution of (2.2) are

 ψa(ξ):=ψa(τ,ξ)=√2A0(a)−Aj(a)ξj. (2.5)

In the original coordinates the Lorentz boosts applied to yield

 va(t,x)=1t2−|x|2ua(−tt2−|x|2,xt2−|x|2)=√2A0(a)t−Aj(a)xj. (2.6)
Remark 2.2 (H1×L2-norm of selfsimilar solution on ΣT).

The conserved energy for fixed time ,

 E(v(t,.),∂tv(t,.)):=12∥(v(t,.),∂tv(t,.))∥2˙H1×L2(R3)−14∥v(t,.)∥4L4(R3),

is infinite for the selfsimilar solution . The -norm of and its normal derivative on a hyperboloidal slice (introduced in Definition 1.7) grows like

 ∥v0∥H1(ΣT)+∥∇nv0∥L2(ΣT) =1|T|∥v0∥L2(ΣT)+∥∇nv0∥L2(ΣT) =1|T|√2|T|2vol(B|T|)+√2|T|4vol(B|T|)≃|T|−12.

The same growth rate holds for the Lorentz transformations of which is why we normalize the decay estimate of in Theorem 1.9 by the factor .

Remark 2.3 (L4tL4x-Strichartz norm of selfsimilar solution).

The selfsimilar solution satisfies

 ∥v0∥4L4(t,2t)L4(B(1−δ)t) =∫2tt∫(1−δ)s04s4r2drds=∫2tt4s4(1−δ)3s33ds =(1−δ)334log2≃1,

and therefore also .

3. Modulation ansatz

We now rewrite (1.1) as a first-order evolution problem of the form

 ∂τΨ(τ)=LΨ(τ)+N(Ψ(τ)). (3.1)

and then insert the modulation ansatz corresponding to the -parameter family of solutions , , to study its stability in similarity coordinates introduced in Section 2.1. This leads to an evolution equation for the residual term .

3.1. Evolution problem

Let be defined by

 ψ1 =ψ, (3.2) ψ2 =∂0ψ+ξj∂jψ+ψ,

where is chosen in a way to compensate for the -derivative (cf. equation (2.3) in the proof of Lemma 2.1). Then (2.2) is equivalent to the system

 ∂0ψ1 =−ξj∂jψ1−ψ1+ψ2, (3.3) ∂0ψ2 =∂j∂jψ1−ξj∂jψ2−2ψ2+ψ31.

which admits the family of static solutions

 ψa,1(ξ) =ψa(ξ)=√2A0(a)−Aj(a)ξj, (3.4) ψa,2(ξ) =√2A0(a)(A0(a)−Aj(a)ξj)2,

being obtained from in (2.5). Thus let be the term corresponding to the solution and its Lorentz boosts . From (3.3) we read off the formal linear differential operator

 ˜Lu(ξ)=(−ξj∂ju1(ξ)−u1(ξ)+u2(ξ)∂j∂ju1(ξ)−ξj∂ju2(ξ)−2u2(ξ)) (3.5)

and nonlinear term

 N(u)(ξ)=(0u31(ξ)). (3.6)

3.2. The free evolution

For the operator we define a suitable domain in a Hilbert space and show how to obtain the linear operator from . Since the system

 ddτΨ(τ)=˜LΨ(τ)

equals the system for the residual term corresponding to the free wave equation (4-4) in [6][p. 474] we proceed analogously. Let denote the open unit ball in and the Hilbert space be the completion of with respect to the induced norm of the inner product

 ⟨u,v⟩:=∫B∂ju1(ξ)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∂jv1(ξ)dξ+∫∂Bu1(ω)¯¯¯¯¯¯¯¯¯¯¯¯v1(ω)dσ(ω)+∫Bu2(ξ)¯v2(ξ)dξ. (3.7)

By [6][Lemma 3.1] the first two terms are equivalent to the standard -norm, i.e.,

 ∥f∥H1(B)≃∥f∥˙H1(B)+∥f∥L2(∂B),

hence the space is equivalent to as a Banach space. The domain of is the subspace . is dense in since is dense in both. From the semigroup approach carried out in [6][Prop. 4.1] we obtain the following result.

Proposition 3.1.

The operator is densely defined and closable. Its closure generates a strongly continuous semigroup which satisfies

 ∥S(τ)∥≤e−τ2,τ∈[0,∞).

The spectrum of is contained in the shifted half plane . ∎

This result implies a bound on the resolvent of [8][Theorem II.1.10].

Corollary 3.2.

The resolvent of is a bounded operator for all that satisfies

 ∥RL(λ)∥≤1Rλ+12,Rλ>−12.\qed

3.3. Modulation ansatz

It is the aim of this section to write the system (2.2) in the abstract form

 ∂τΨ(τ)=LΨ(τ)+N(Ψ(τ)) (3.8)

for a function and to study the stability of the 3-parameter family derived in (3.4). This involves the modulation ansatz

 Ψ(τ)=Ψa(τ)+Φ(τ) (3.9)

for a function , where we allow for the rapidity to depend on . The Lorentz boosts are static solutions of (3.8), thus we know that satisfy