The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

Abstract

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.

Full text

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arXiv:1002.1756v1 [math.AP] 9 Feb 2010
THE RADIAL DEFOCUSING ENERGY-SUPERCRITICAL
NONLINEAR WAVE EQUATION IN ALL SPACE DIMENSIONS
ROWAN KILLIP AND MONICA VISAN
Abstract. We consider the defocusing nonlinear wave equation utt− ∆u +
|u|pu = 0 with spherically-symmetric initial data in the regime
(which is energy-supercritical) and dimensions 3 ≤ d ≤ 6; we also consider
d ≥ 7, but for a smaller range of p >
blowup (or failure to scatter) must be accompanied by blowup of the critical
Sobolev norm. An equivalent formulation is that maximal-lifespan solutions
with bounded critical Sobolev norm are global and scatter.
4
d−2< p <
4
d−3
4
d−2.The principal result is that
1. Introduction
We consider the initial value problem for the defocusing nonlinear wave equation
in d ≥ 3 space dimensions:
?
u(0) = u0, ut(0) = u1,
utt− ∆u + F(u) = 0
(1.1)
where the nonlinearity F(u) = |u|pu is energy-supercritical, that is, p >
The class of solutions to (1.1) is left invariant by the scaling
4
d−2.
u(t,x) ?→ λ
2
pu(λt,λx),(1.2)
which determines the critical Sobolev space for initial data, namely, (u0,u1) ∈
˙Hsc
x
where the critical regularity is sc :=
precisely corresponds to sc> 1; since the energy
x×˙Hsc−1
d
2−2
p. Notice that p >
4
d−2
E(u) =
?
Rd
1
2|ut|2+1
2|∇u|2+
1
p+2|u|p+2dx (1.3)
scales like s = 1, this regime is known as energy-supercritical.
Let us start by making the notion of a solution more precise.
Definition 1.1 (Solution). A function u : I × Rd→ R on an open time interval
0 ∈ I ⊂ R is a (strong) solution to (1.1) if (u,ut) ∈ C0
u ∈ L(d+1)p/2
?u(t)
ut(t) −|∇|sin(t|∇|)
?t
t(K;˙Hsc
x×˙Hsc−1
x
) and
t,x
(K × Rd) for all compact K ⊂ I, and obeys the Duhamel formula
?
cos(t|∇|)
?|∇|−1sin?(t − s)|∇|?
cos?(t − s)|∇|?
Date: February 9, 2010.
=
?
cos(t|∇|)|∇|−1sin(t|∇|)
??u(0)
ut(0)
?
−
0
?
F(u(s))ds
(1.4)
1

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2ROWAN KILLIP AND MONICA VISAN
for all t ∈ I. We refer to the interval I as the lifespan of u. We say that u is a
maximal-lifespan solution if the solution cannot be extended to any strictly larger
interval. We say that u is a global solution if I = R.
We define the scattering size of a solution to (1.1) on a time interval I by
?u?S(I):=
??
I
?
Rd|u(t,x)|
(d+1)p
2
dxdt
?
2
(d+1)p. (1.5)
Associated to the notion of solution is a corresponding notion of blowup. By
the standard local theory, the following precisely corresponds to the impossibility
of continuing the solution in a manner consistent with Definition 1.1.
Definition 1.2 (Blowup). We say that a solution u to (1.1) blows up forward in
time if there exists a time t1∈ I such that
?u?S([t1,supI))= ∞
and that u blows up backward in time if there exists a time t1∈ I such that
?u?S((inf I,t1])= ∞.
Our purpose here is to give a short proof of the following result:
Theorem 1.3 (Spacetime bounds). Assume that
d(d−1)−√
4
d−2< p <
if d ≥ 7. Let u : I × Rd→ R be
t(I;˙Hsc
4
d−3for 3 ≤ d ≤ 6
and that
a spherically-symmetric solution to (1.1) such that (u,ut) ∈ L∞
Then
?u?S(I)≤ C??(u,ut)?L∞
For d = 3 this was proved by Kenig and Merle [12]. In [18] we proved this
result for non-radial data. While preparing [18], we realized that it is possible to
give a short proof of Theorem 1.3 that works uniformly in all dimensions. That
is the topic of this paper. In keeping with our goal of a simple presentation, we
have restricted ourselves to the specific values of p stated in the theorem. These
hypotheses represent the combination of two restrictions, one related to the local
theory (which dominates in high dimensions) and another dictated by the Morawetz
inequality (sc< 3/2). The former restriction stems from our desire to present as
simple and uniform a local theory as possible. While one may certainly obtain a
larger range of p in this setting by some piecemeal approach, it is not clear to us
how to obtain the full range dictated by our principal hypothesis sc< 3/2. Indeed,
note that one natural restriction is the smoothness condition sc< p+1; this allows
us to take scderivatives of the nonlinearity. Our condition for d ≥ 7 is equivalent
to sc< p + 1 − (
In low dimensions, the sole restriction on p is p <
sc< 3/2, which nevertheless covers all values of p when d = 3. This condition is
dictated by the Morawetz inequality. The methods presented here do not immedi-
ately extend to higher values of p. The problem arises in Section 4 and could be
circumvented by proving that the almost periodic solutions discussed below actu-
ally lie in L∞
x
) for some s < 3/2. Arguments showing how this can be
done (even in the non-radial setting) may be found in [14, 16, 18]; however, this is
significantly more involved than what we chose to present here.
4
d−2< p <
d2(d−1)2−16(d+1)2
2(d+1)
x×˙Hsc−1
x
).
t(I;˙Hsc
x ×˙Hsc−1
x
)
?.
1
d+1+p
2).
4
d−3and corresponds to
t(˙Hs
x×˙Hs−1

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THE RADIAL DEFOCUSING ENERGY-SUPERCRITICAL NLW3
As mentioned above, finite-time blowup of a solution to (1.1) must be accom-
panied by divergence of the scattering size defined in (1.5). Thus, Theorem 1.3
immediately implies
Corollary 1.4 (Spacetime bounds). If u : I × Rd→ R is a maximal-lifespan
spherically-symmetric solution to (1.1) with (u,ut) ∈ L∞
is global,
?u?S(R)≤ C??(u,ut)?L∞
and u scatters in the sense that
t(I;˙Hsc
x×˙Hsc−1
x
), then u
t(R;˙Hsc
x ×˙Hsc−1
x
)
?,
??(u,ut) − (u±,u±
t)??˙Hsc
x ×˙Hsc−1
x
→ 0ast → ±∞,
for two solutions u±of the linear wave equation.
This corollary takes on a more appealing form if we rephrase it in the contra-
positive:
Corollary 1.5 (Nature of blowup). A spherically-symmetric solution u : I ×Rd→
R to (1.1) can only blow up in finite time or be global but fail to scatter if its
˙Hsc
x
norm diverges.
x×˙Hsc−1
When p =
bounded in time by virtue of the conservation of energy. This energy-critical case
of (1.1) has received particular attention because of this property. Global well-
posedness was proved in a series of works [6, 7, 8, 23, 27, 24, 25] with finiteness of
the scattering size being added later; see [1, 5, 21, 22, 29]. Certain monotonicity
formulae, namely the Morawetz and energy flux identities, play an important role
in all these results. It is important that these monotonicity formulae also have
critical scaling.
In the energy-supercritical case discussed in this paper, all conservation laws
and monotonicity formulae have scaling below the critical regularity. At the present
moment, there is no technology for treating large-data dispersive equations without
some a priori control of a critical norm. This is the purpose of the L∞
˙Hsc−1
x
) assumption in Theorem 1.3; it plays the role of the missing conservation
law at the critical regularity. To deal with the fact that the basic monotonicity
formula scales like the energy (rather than the critical regularity), we employ a
space truncation in the manner of [2]; see also [17, 28].
4
d−2, or equivalently, sc= 1, the critical Sobolev norm is automatically
t(I;˙Hsc
x×
1.1. Outline of the proof. We argue by contradiction. By the fundamental obser-
vations of Keraani [13] and Kenig–Merle [10], we know that failure of Theorem 1.3
guarantees the existence of certain minimal counterexamples and moreover, such
solutions have good compactness properties. These properties are best described
in terms of the following notion:
Definition 1.6 (Almost periodicity modulo scaling). A solution u to (1.1) with
lifespan I is said to be almost periodic modulo scaling if (u,ut) is bounded in
˙Hsc
x
and there exist functions N : I → R+and C : R+→ R+such that
for all t ∈ I and η > 0,
?
x×˙Hsc−1
|x|≥C(η)/N(t)
??|∇|scu(t,x)??2dx +
?
|x|≥C(η)/N(t)
??|∇|sc−1ut(t,x)??2dx ≤ η

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4ROWAN KILLIP AND MONICA VISAN
and
?
|ξ|≥C(η)N(t)
|ξ|2sc|ˆ u(t,ξ)|2dξ +
?
|ξ|≥C(η)N(t)
|ξ|2(sc−1)|ˆ ut(t,ξ)|2dξ ≤ η.
We refer to the function N(t) as the frequency scale function for the solution u and
to C(η) as the compactness modulus function.
Remarks. 1. Spherical symmetry forces the bulk of the solution to concentrate
around the spatial origin. This is the reason for the absence of a spatial center
function x(t).
2. The continuous image of a compact set is compact. Thus, by Sobolev em-
bedding, almost periodic (modulo scaling) solutions obey the following: For each
η > 0 there exists C(η) > 0 so that
??u(t,x)??
where ∇t,xu = (ut,∇u) denotes the space-time gradient of u.
With these preliminaries out of the way, we can now describe the first major
milestone in the proof of Theorem 1.3.
L∞
tL
dp
2
x
({|x|≥C(η)/N(t)})+??∇t,xu(t,x)??
L∞
tL
dp
p+2
x
({|x|≥C(η)/N(t)})
≤ η, (1.6)
Theorem 1.7 (Three special scenarios for blowup). Suppose that Theorem 1.3
failed. Then there exists a maximal-lifespan spherically-symmetric solution u : I ×
Rd→ R, which obeys (u,ut) ∈ L∞
scaling, and ?u?S(I)= ∞. Moreover, we can also ensure that the lifespan I and the
frequency scale function N : I → R+match one of the following three scenarios:
I. (Finite-time blowup) We have that either supI < ∞ or |inf I| < ∞.
II. (Soliton-like solution) We have I = R and N(t) = 1 for all t ∈ R.
III. (Low-to-high frequency cascade) We have I = R,
t(I;˙Hsc
x ×˙Hsc−1
x
), is almost periodic modulo
inf
t∈RN(t) ≥ 1, andlimsup
t→+∞
N(t) = ∞.
The proof of Theorem 1.7 follows a well-travelled path. See [15] for an introduc-
tion to these techniques including two worked examples and references up to that
time. Let us briefly review the ingredients: (i) A concentration compactness prin-
ciple (= profile decomposition) for the linear propagator. The very first example
of this was worked out for the wave equation (with d = 3) in [1]. The extension
to all dimensions can be found in [3]. (ii) A perturbation theory for the nonlinear
equation. While the basic framework is standard, each equation has its peculiari-
ties, particularly when small-power nonlinearities are involved. We discuss this at
some length in Section 3, in part because our arguments unify and simplify existing
results for certain special cases. (iii) A decoupling argument. This is usually fairly
direct; however, some subtleties arise in the model discussed in this paper due to
the fact that sc> 1 and p is small. The requisite technology can be found in [16].
With Theorem 1.7 in hand, the proof of Theorem 1.3 reduces to showing that
none of the three special scenarios can occur. In Section 5 we show that the first
scenario, a finite-time blowup solution, cannot exist because it is inconsistent with
the conservation of energy. In Section 4 we show that neither of the other two
scenarios can occur, since they are inconsistent with the (truncated) Morawetz
identity (cf. Lemma 2.5) when p <
d−3.
4

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THE RADIAL DEFOCUSING ENERGY-SUPERCRITICAL NLW5
Acknowledgements. The first author was supported by NSF grant DMS-0701085.
The second author was supported by NSF grant DMS-0901166.
2. NLW background
We write X ? Y to indicate that X ≤ CY for some dimension-dependent con-
stant C, which may change from line to line. Other dependencies will be indicated
with subscripts, for example, X ?uY .
2.1. Strichartz estimates. One of the most fundamental tools in the modern
analysis of nonlinear wave equations is the Strichartz estimate. We record some
particular instances of this estimate below. For further information, see [9, 22, 26]
and the references therein.
Definition 2.1 (Admissible pairs). We say that the pair (q,r) is wave-admissible
if
1
q+d−1
2 ≤ q ≤ ∞,
Lemma 2.2 (Strichartz estimates). Fix d ≥ 3. Let I be a compact time interval
and let u : I × Rd→ R be a solution to the forced wave equation
utt− ∆u + F = 0.
Then for any t0∈ I and any wave-admissible pair (q,r),
??∇t,xu??
?
??∇t,xu(t0)??˙Hsc−1
provided
2r≤d−1
4, and2 ≤ r < ∞.
L∞
t
˙Hsc−1
x
+ ?u?S(I)+??|∇|sc−1
2u??
+??|∇|sc−1
L
2(d+1)
d−1
t,x
+??|∇|γ−1∇t,xu?Lq
L
t,x
tLr
x
x
2F??
2(d+1)
d+3
,
1
q+d
r=2
p+ γ. All spacetime norms in the formula above are on I × Rd.
We will use the notation
?u?Ssc(I):= sup??|∇|γ−1∇t,xu?Lq
the scaling condition
r=
exponent appearing in the arguments below.
The following result will be needed in Section 3; its proof requires only minor
modifications to the proof of the Christ–Weinstein fractional chain rule presented
in [30, §2.5].
Lemma 2.3 (Derivatives of differences). Let F(u) = |u|pu with p > 0 and let
0 < s < 1. Then for 1 < q,q1,q2< ∞ such that1
??|∇|s[F(u + v) − F(u)]??
Proof. By the Fundamental Theorem of Calculus,
tLr
x, (2.1)
where the supremum is taken over all admissible pairs (q,r) and numbers γ obeying
1
q+d2
p+ γ, with r ≤ r∗(d,p) dictated by the largest
q=
1
q1+
p
q2, we have
q???|∇|su??
q1?v?p
q2+ ?|∇|sv??
q1?u + v?p
q2.
??F(u(x)) − F(u(y))??= |u(x) − u(y)|
and similarly for F(u + v). Combining these gives
????
?1
0
F′?u(y) + t[u(x) − u(y)]?dt
????
?p|u(x) − u(y)|?|u(x)|p+ |u(y)|p?,
??[F(u + v) − F(u)](x) − [F(u + v) − F(u)](y)??