In this article, we survey recent results (with Duyckaerts and Merle) on the long-time behavior of radial solutions of the energy critical nonlinear wave equation in odd dimensions.

The theory of nonlinear dispersive equations has seen a spectacular development in the last 40 years. These equations, introduced in the nineteenth century, model phenomena of nonlinear wave propagation, coming from physics and engineering. Examples of such equations are the Korteweg–de Vries (KdV) equations, the nonlinear Schrödinger (NLS) equations, and the nonlinear wave (NLW) equations. There are also dispersive geometric flows such as wave maps, Schrödinger maps, and Yang–Mills flows.

In the 1970s and 1980s, many works studied the behavior of special solutions (“nonlinear objects”) such as traveling waves/solitons and their stability (sometimes conditionally). The late 1980s and 1990s saw a systematic study of the local well-posedness theory (and the small data global well-posedness theory) using extensively tools from harmonic analysis.

The last 25 years have seen much interest in the study of the long-time behavior of solutions for large data. Issues such as blow-up, global existence, scattering, and long-time asymptotic behavior have come to the forefront, especially in critical problems.

In this article, I will focus the discussion on the energy critical wave equation in the focusing case. [In the defocusing case, it was shown (1990–2000) that all data in the energy space yield global in time solutions that scatter and there are “no nonlinear objects.”] The hope is that the results that we will describe will be a model for what to strive for in the study of other nonlinear dispersive equations,

$(NLW)p∂t2u−Δu±|u|p−1u=0(x,t)∈RN×I,0∈Iu|t=0=u0,∂tu|t=0=u1.$

We will let $p=pc=N+2N−2$, $(NLW)=(NLW)pc$, and $u0∈H1̇={u0:∇u0∈L2},u1∈L2$ (+ = defocusing, − = focusing (N = 3, |u|p−1u = u5)).

The focusing case is very different from the defocusing one since, for instance, one can have finite time blow-up or solutions that exist for all time but do not scatter.

In NLW, small data yield unique global in time solutions that “scatter.” For large data, we have solutions in $C(I;H1̇×L2)$ with a maximal interval of existence I = (T(u), T+(u)). The energy norm is “critical”: $∀λ>0,uλ(x,t)=1λN−22u(xλ,tλ)$ is also a solution,

$‖(u0,λ,u1,λ)‖H1̇×L2=‖(u0,u1)‖H1̇×L2.$

The focusing character means that the linear and nonlinear terms have “opposite signs,” and they fight each other. The conserved energy is

$E(u0,u1)=12∫|∇u0|2+u12−N−22N∫|u0|2NN−2.$

(See, for instance, Ref. 24 for details.)

One can construct solutions that blow up in finite time, say, at T = 1, by considering the ODE. For example, when $N=3,u(x,t)=341/4(1−t)−1/2$ is a solution and using finite speed of propagation, it is then possible to construct solutions with T+ = 1 such that $limt↑1‖(u(t),∂tu(t))‖H1̇×L2=∞$. This is called type I or ODE blow-up. There are also type II blow-up solutions: T+ < and $sup0. The breakdown happens by “concentration.” This is a typical feature of focusing energy critical problems.

Examples (radial): N = 3, Krieger, Schlag, and Tataru;27N = 4, Hillairet and Raphaël;19 and N = 5, Jendrej.20

In our work, we concentrate on solutions that are bounded in the energy space,

$sup0

Examples of bounded solutions with T+ = + are “scattering solutions,” that is, with T+ = + so that there exists $(u0+,u1+)∈H1̇×L2$ with

$limt↑1‖(u(t),∂tu(t))−S(t)(u0+,u1+)‖H1̇×L2=0,$

where $S(t)(u0+,u1+)$ is the solution to the associated linear problem.

Other such examples are the nontrivial stationary solutions,

$Q≢0and △Q+|Q|4/(N−2)Q=0,Q∈H1̇.$

For example, $W(x)=(1+|x|2N(N−2))(N−2)/N$ is such a solution. (Stationary solutions do not scatter because if u scatters, then $∫|x|<1|∇u(x,t)|2dx→t→∞0$.)

W has several important characterizations: up to sign and scaling,

$Wλ(x)=1λ(N−2)/2W(x/λ)$

is the only radial non-zero stationary solution. Up to translation and scaling, it is the only non-negative stationary solution. W is called the “ground state.” It minimizes the energy among all the non-zero stationary solutions (see Refs. 32, 18, and 29). There are also variable sign, non-radial stationary solutions (in great abundance).

Further examples of non-scattering bounded solutions with T+ = are the traveling wave solutions. They are obtained as Lorentz transforms of stationary solutions Q.

Let $l→∈RN$, $|l→|<1$. Then,

$Ql→(x,t)=Ql→(x−tl→,0)$
$=Q−t1−|l→|2+1|l→|11−|l→|2−1l→⋅xl→+x.$

Remark 1.

In Ref. 9, Duyckaerts, Kenig, and Merle showed that these are all the traveling wave solutions. (In the radial case, we must have $l→=0$, Q = ±Wλ.)

In Ref. 24, Kenig and Merle initiated the study of the long-time behavior of large solutions of NLW establishing the “ground-state” conjecture.

Let u be a solution of NLW with E(u0, u1) < E(W, 0):

• If $‖∇u0‖L2<‖∇W‖L2$, then T± = ±, and u scatters in both time directions.

• If $‖∇u0‖L2>‖∇W‖L2$, T+ < , T > −.

• $‖∇u0‖L2=‖∇W‖L2$ is impossible.

The threshold case E(u0, u1) = E(W, 0) was completely described by Duyckaerts and Merle.14

What happens for E(u0, u1) > E(W, 0)? When Kenig and Merle (Refs. 23–25) introduced their “concentration-completeness/rigidity theorem” method, used, for instance, in the proof of the result just mentioned, the ultimate goal was to describe the asymptotic behavior of bounded solutions as tT+, in the spirit of the “solution resolution” conjecture. Since the 1970s, there has been a widely held belief that “coherent structures” and “free radiation” describe the long-time asymptotic behavior of solutions to nonlinear dispersive equations. This came to be known as the “solution resolution” conjecture. Roughly speaking, this says that asymptotically in time, the evolution decouples as a sum of (modulated) traveling waves and a free radiation term (a dispersive term solving the associated linear equation). This is a remarkable, beautiful statement, which claims an “asymptotic simplification” in the complex, long-time dynamics of general solutions. The origin of this statement goes back to a “puzzling paradox” in a numerical simulation of Fermi, Pasta, and Ulam at Los Alamos, Ref. 17 (see also Ref. 34). In the mid 1960s, in order to explain the paradox, Kruskal28 brought the Korteweg–de Vries (KdV) equation into the picture. Kruskal and Zabusky35 were then led to an influential numerical simulation on KdV. They discovered two remarkable things:

(a) For simple data concentrated on one mode, the evolution for large time “equaled” a sum of traveling waves.

(b) If the data are a sum of two traveling waves, in the evolution they eventually “collide,” but after the collision, they re-appear unchanged, except for translation (“elastic collision”).

(a) was performed in connection with explaining the Fermi–Pasta–Ulam paradox. It gave rise to the “solution resolution conjecture.”

(b) was a consequence of “integrability,” an important new concept in nonlinear science that emerged from Ref. 36.

Integrable nonlinear equations can be reduced to a collection of linear problems. There are important examples, but it is a non-generic phenomenon. “Soliton resolution” has been proved in a few integrable cases (see Refs. 15, 31, and 2). The proofs are challenging, and issues are still unresolved. There are also many results in perturbative regimes (near traveling waves) and also in non-integrable settings.

The mechanism for relaxation to a “coherent structure” observed numerically and experimentally is the radiation of excess energy to spatial infinity. This appears in such diverse settings as the dynamics of gas bubbles in a compressible fluid and the formation of black holes in gravitational collapse. Proving this in generality, in non-integrable settings, is a major goal in the study of nonlinear dispersive equations.

Let us turn now to the “asymptotic simplification” results for bounded in energy norm solutions of NLW. For simplicity, we restrict to the case T+ = . The corresponding results hold for T+ < .

Theorem 1
(Duyckaerts, Kenig, and Merle8). Letube a radial solution of NLW,N = 3, which exists for allt ≥ 0. Then, there exists$J∈N∪{0}$, a radial solutionvLof the linear wave equation,1 ≤ jJ,ij ∈ {±1},λj(t) > 0, with
$0<λJ(t)≪λJ−1(t)≪⋯≪λ1(t)≪t,$
$u(t)=∑j=1Jijλj(t)1/2W(xλj(t))+vL(t)+o(1)$
$∂tu(t)=∂tvL(t)+o(1),$
whereλj−1(t) ≪ λj(t) means that$limt→∞λj(t)λj−1(t)=∞$.

The key ingredient in the proof is as follows: Let u be a radial solution of NLW, N = 3, defined for all t, u ≢ 0, u ≢ ±Wλ. Then, there exist R > 0, η > 0 such that ∀t ≥ 0 or ∀t ≤ 0,

$∫|x|>|t|+R|∇x,tu|2dx≥η.$
(1)

(1) quantifies “the radiation of excess energy to spatial infinity.” The proof of (1) relies on “energy channels.” For N = 3, r0 > 0, let $Pr0={(ar−1,0):a∈R,r>r0}$ be considered as a one dimensional subspace of $H1̇×L2(r>r0)$. Let $Πr0$ be the orthogonal projection to the orthogonal complement of $Pr0$. Then (DKM5), for v, a radial solution of the linear wave equation, N = 3, ∀t ≥ 0, or ∀t ≤ 0, we have

$∫|x|>|t|+r0|∇x,tv|2≥12‖Πr0(v0,v1)‖H1̇×L2(r>r0)2.$
(2)

A key point is that $W(x)=1(1+r2/3)1/2≈1r$, which is what appears in the exceptional subspace $Pr0$. It is easy to see that (2) gives (1) for solutions of the linear wave equation. The passage to nonlinear solutions uses (2) and “elliptic arguments” of iterative type.

(2) has a non-radial version, corresponding to r0 = 0: Let v be a solution of the linear wave equation in $RN×R$, N being odd. Then (DKM6), ∀t ≥ 0 or ∀t ≤ 0,

$∫|x|>|t||∇x,tv|2dx≥12∫|∇v0|2+v12.$
(3)

At this point, a natural question is what corresponding results hold for NLW, N > 3, radial and N ≥ 3, non-radial. Relevant facts are as follows:

Fact1: (Côte, Kenig, and Schlag4) (2) and (3) fail for all even N and radial solutions of the linear wave equation.

Fact2: (Kenig, Lawrie, Liu, and Schlag26) For N being odd and N > 3, the following analog of (2) holds: Let r0 > 0, $Pr0=1rN−2k1,0,0,1rN−2k2$, $1≤k1≤N+24$, $1≤k2≤N4$. Then, if v is a radial solution of linear waves in $RN$, N being odd, then ∀t ≥ 0 or ∀t ≤ 0

$∫|x|>|t|+r0|∇x,tv|2dx≥12∫|Πr0(v0,v1)|2.$
(4)

Fact 3: The analog of (2) and (4) fails in all dimensions, in the non-radial case, with infinite dimensional counterexamples.

The fact that the subspace $Pr0$ appearing in (4) is of dimension bigger than 1, when N is odd, N ≥ 5, was responsible for the failure of the method of Proof of Theorem 1, yielding the higher dimensional analog of Theorem 1 since we can only use the one parameter scaling invariance to deal with the exceptional set. This has been a major obstacle until very recently.

In addition, fact 3 and the fact that the non-zero solutions of the elliptic equation constitute a huge set, for which no classification is available, have been responsible for the fact that no analog of Theorem 1 has been obtained yet in the nonradial case. Despite this, some weaker results have been obtained using monotonicity after time averaging, in analogy with geometric flow problems. The results obtained in this way, instead of holding for all times, hold for “well-chosen” sequences of times.

Theorem 2.
Letube a radial solution of NLW,N > 3, such that
$sup0
Then, the decomposition analogous to Theorem 1 holds, along a “well-chosen”tn. (ForNodd this was proved by Rodriguez,29forN = 4 by Côte–Kenig–Lawrie–Schlag3and for generalNeven by Jia–Kenig.21 )

Theorem 3
(Jia;21 DKM;10 Duyckaerts, Jia, Kenig, and Merle7). Letube a (non-radial) solution of NLW in$RN,3≤N≤6$such that
$sup0
Then, there exist$J∈N∪{0}$, a linear solutionvL, a “well-chosen” sequence of times {tn},tn,∀j, 1 ≤ jJ,λj,n > 0,$xj,n∈RN$, |xj,n/tn| ≤ β,β ∈ [0, 1),$limn→∞xj,nλj,n=lj⃗$, and traveling waves$Ql⃗jj$so that
$u(tn)−vL(tn)=∑j=1J(λj,n)−N−22Qlj⃗j+o(1),$
$∂tu(tn)−∂tvL(tn)=∑j=1J(λj,n)−N2∂tQlj⃗j+o(1),$
with
$λj,nλj′,n+λj′,nλj,n+|xj,n−xj′,n|λj,n→n∞,j≠j′.$

Remark 2.

A method of proof using monotonicity after time averaging cannot give more than a decomposition for well-chosen sequences of time. This difficulty is illustrated by the harmonic map heat flow. Here, results analogous to ours are known, but the decomposition for all times need not hold in full generality. This was shown by Topping.33 The analogous result for wave maps is due to Engelstein and Mendelson.16

We now turn to recent progress in the radial case of NLW, N > 3.

In a series of three papers posted at the end of 2019, DKM (Refs. 11–13) have proved the full decomposition.

Theorem 4.
LetN > 3 and odd,ube a radial solution of NLW, and
$sup0
Then, there exist$J∈N∪{0}$, for 1 ≤ jJ,ij = {±1},λj(t), 0 ≪ λJ(t) ≪ ⋯ ≪ λ1(t) ≪ t, and a radial linear solutionvLsuch that
$u(t)=∑j=1JijWλj(t)+vL(t)+o(1),$
$∂tu(t)=∂tvL(t)+o(1).$

To prove Theorem 4, the starting point is Theorem 2, allowing us to study the dynamics close to a sum of scaled static solutions plus a linear dispersive term. We then study the “collisions” of two or more scaled static solutions (where the scales are far apart). We show that they are “inelastic” (in stark contrast with the integrable case of Kruskal and Zabusky35). Specifically, we say that the radial solution u of NLW is a “pure multisolution” (asymptotically as t → ±) when there exists J ≥ 2, 0 < λJ(t) ≪ ⋯ ≪ λ2(t) ≪ λ1(t) ≪ t, ij = {±}, 1 ≤ jJ, such that

$u(t),∂tu(t)=∑j=1JijWλj(t),0+o(1)ast→±∞.$

If u is a “pure multisolution” as t → ±, we say that the collision between “the solitons” is elastic. When N = 3, (1) rules out elastic collisions.

In this series of papers, we prove a slightly weaker version of (1), namely that a radial solution of NLW that stays close to a sum of decoupled static solutions, for a sufficiently long time, must satisfy that ∀t ≥ 0 or ∀t ≤ 0,

$∫|x|≥|t|−M|∇x,tu|2>0for some largeM.$

Hence, elastic collisions are excluded.

The M is needed to “eliminate” the extra dimensions in $Pr0$ in (4). This “inelasticity” depends heavily on the “energy channel” property for the linearized wave equation $∂t2−Δ−N+2N−2W4/N−2$, which we proved in the first paper.

Theorem 5
(Ref. 12). For N ≥ 5 andodd, let (u0, u1) be radial. Then, there existsC > 0 such that for allt ≥ 0 or for allt ≤ 0,
$∫|x|>|t||∇x,tu|2≥C‖ΠZ(u0,u1)‖H1̇×L22,$
whereusolves$∂t2u−Δu−N+2N−2W4/(N−2)u=0$with initial data (u0, u1) and if$ΛW=x⋅∇W+N2−1W$,Z = span{ΛW} × span{ΛW}. (NoteLWW) = 0, where$LW=Δ+N+2N−2W4/(N−2)$)

In the third paper, this is extended to the linearized operator around “multisolutions.” Another important ingredient in the proof is the notion of “non-radiative solution.” We say that u is non-radiative (at t = 0) if

$limt→±∞∫|x|>|t||∇x,tu|2=0.$

It is weakly non-radiative if for large R > 0,

$limt→±∞∫|x|≥|t|+R|∇x,tu|2=0.$

Note that (3) shows that non-zero solutions of the linear wave equation are not non-radiative (odd N). For N being even, this also holds and is proved in the second paper. Theorem 5 classifies the non-radiative solutions of the linearized operator (odd N), and (4) classifies the weakly non-radiative solutions of the linear wave equation (odd N).

In the second paper, we give decay estimates for the weakly non-radiative solutions of NLW for N being odd.

Theorem 6
(DKM11). Let$m=N−12$be the dimension ofPRandξkandk = 1, …, mbe generators ofPRnormalized to have normCk/Rk−1/2. There existsϵ0 > 0 such that ifuis a radial weakly non-radiative solution of NLW, with$‖(u(0),∂tu(0))‖H1̇×L2(|x|>R0)<ϵ0$and$limt→±∞∫|x|>R0+|t||∇x,tu(t)|2=0$, there exist$l∈R$,$k0∈[1,m]∩N$such that for allR > R0,
$‖(u(0),∂tu(0))−lξk0‖H1̇×L2(|x|>R)≲maxR0R(k0−1/2).N+2N−2,R0Rk0+1/2.$

Ifuis non-radiative,k0 = m, thenu = ±Wλ.

Ifk0 < m, thenl ≠ 0.

The proofs of Theorems 5 and 6 depend strongly on (4). For the Proof of Theorem 4, as in the Proof of Theorem 2, we first extract the radiation term vL, which is the unique solution of the linear wave equation such that for all $A∈R$,

$limt↑+∞∫|x|>A+t|∇x,t(u−vL)|2dx=0.$

We then work with (U(t), tU(t)) = (u(t) − vL(t), tu(t) − tvL(t)). By Theorem 2, we find a sequence {tn} → for which the decomposition of U into “multisolitons” holds and, arguing by contradiction, another sequence {tn′} for which it does not hold but on which U is still very close to a “multisoliton.” Using the property above of vL, we prove that the “nonlinear profiles” in the nonlinear profile decomposition1 of U(tn′), the “building blocks” of U(tn′), are all “non-radiative” and close to a “multisoliton” (see Ref. 13). Using the “energy channel” for the linearized equation at a “multisoliton,” working with each nonlinear profile (that are all non-radiative), we obtain a lower bound on the exterior scaling parameter of the corresponding stationary solution, in terms of l in Theorem 6, and we derive an (approximate) system of the ODEs on the scaling parameters of the stationary solutions. The fact that at {tn′}, the decomposition does not hold means that at least one nonlinear profile is not a stationary solution, and hence, l ≠ 0. After rescaling, this gives a uniform lower bound on the exterior scaling parameter of the “multisolitons.” The proof is concluded by analyzing the system of ODEs, finding a monotonicity formula for it, and showing that using it in conjunction with the lower bound on the exterior scaling parameter, we obtain an upper bound on the time of existence, which contradicts the fact that u exists for all positive times. See Ref. 13 for details.

Dedicated to the memory of Jean Bourgain.

This work was supported, in part, by NSF under Grant No. DMS-1800082.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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