On the supercritical defocusing NLW outside a ball

We study a defocusing semilinear wave equation, with a power nonlinearity $|u|^{p-1}u$, defined outside the unit ball of $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. We prove that if $p>n+4$ and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a pointwise decay estimate for the exterior linear wave equation perturbed with a large, time dependent potential.


Introduction
Consider the Cauchy problem for the defocusing wave equation on R t × R n x : u + |u| p−1 u = 0, with Sobolev initial data (u 0 , u 2 ) ∈ H s × H s−1 . The existence of global solutions to this problem has been explored in considerable detail. The critical power for global smooth solvability is p cr (n) = 1 + 4 n−2 for n ≥ 3, while p cr (1) = p cr (2) = ∞. Global existence in the energy space is known for p ≤ p cr (n) [3], [11], [5]; regularity for p = p cr has been explicitly proved up to dimension 7 [10], [5] and should hold for all dimensions. For supercritical nonlinearities p > p cr (n), very little is known, and the question of global existence or blow up is still open.
If we restrict to spherically symmetric solutions, the difficulty of the problem is essentially the same. However, if a radial solution blows up, the blow up must occur at the origin. This follows at once from the Sobolev embedding for radial functions (see [2]). Here the norm L p |x| L r ω is an L p norm in the radial direction of the L r ω norm in the angular direction. We give a precise statement of this blow-up alternative result including its short proof: Proposition 1.1. Let u 0 , u 1 be spherically symmetric test functions, and T the supremum of times τ > 0 such that a smooth solution u(t, x) of (1.1) exists on [0, τ ] × R n . Assume T < ∞. Then u blows up at x = 0 as t → T : for any R > 0, u is unbounded on the cylinder {|x| < R, t < T }.
Assume by contradiction that |u(t, x)| ≤ M for (t, x) in some cylinder {|x| < R, t < T }, then by the previous bound we get |u(t, x)| ≤ M + (CE(0)) 1/2 R 1−n/2 < ∞ for 0 ≤ t < T, x ∈ R n and by standard arguments we can extend u to a strip [0, T ′ ] × R n for some T ′ > T , against the assumptions.
In order to obtain a global radial solution, it would be sufficient to prevent blow-up at 0. This is the case if we replace the whole space with the exterior of a ball: and consider the mixed problem on R + × Ω with Dirichlet boundary conditions u + |u| p−1 u = 0, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), u(t, ·)| ∂Ω = 0. (1.4) In this situation the local solution can not blow up, due to the conservation of H 1 energy and the radial estimate (1.2). This implies the existence of a global radial solution for arbitrary powers p > 1. Indeed, we have the following result, where we use the notation C k X = C k (R + ; X), X = L 2 (R n ) or H s (R n ) or H 1 0 (Ω).
The existence of large radial solutions suggests naturally the question of stability: do nonradial perturbations of radial data give rise to global solutions?
The uniform bound (1.6) is not sufficient for a perturbation argument. However, we can prove an actual decay estimate, obtained by reduction to a mixed problem with moving boundary on S n via the Penrose transform. Define for M > 1 the quantity Then we have Theorem 1.3 (Decay of the radial solution). Let u be the solution constructed in Proposition 1.2. Assume that for some M > 1 the data satisfy Then the following decay estimate holds If the initial data are smoother then regularity propagates, and in addition higher Sobolev norms remain bounded as t → ∞. In order to state the regularity result, we introduce briefly the nonlinear compatibility conditions, discussed in greater detail in Section 3 (see Definition 3.3 and also Definition 2.1). For the mixed problem on R + × Ω we define formally the sequence of functions ψ j as follows: where in the definition of ψ j we set recursively Then we say that the data (u 0 , u 1 , f (s)) satisfy the nonlinear compatibility conditions of order N ≥ 1 if Note that in order to satisfy the compatibility conditions it is sufficient to assume that the initial data u 0 , u 1 belong to H s 0 (Ω) for a sufficiently large s. We can now state the higher regularity result for the radial solutions: Assume the radially symmetric data u 0 ∈ H N +1 (Ω), u 1 ∈ H N and f (u) = |u| p−1 u satisfy the nonlinear compatibility conditions of order N and condition (1.8).
Then the radial solution constructed in Proposition 1.2 belongs to ∈ C k H N +1−k ∩ C N H 1 0 for all 0 ≤ k ≤ N + 1 and satisfies the bounds where C k = C(k, n, (u 0 , u 1 ) X ).
The main result of the paper is the following: Theorem 1.5. Let n ≥ 3, p > N > n + 3, and u 0 , u 1 radial functions satisfying the assumptions in Theorem 1.4. Then there exists ǫ = ǫ(u 0 , u 1 ) > 0 such that the following holds.
Assume v 0 ∈ H N +1 (Ω), v 1 ∈ H N (Ω) and f (u) = |u| p−1 u satisfy the nonlinear compatibility conditions of order N and In other words, nonradial perturbations of large radial initial data in a high Sobolev norm do give rise to global smooth quasiradial solutions. Note that these solutions are unique in the class of locally bounded global solutions, but one can not in principle exclude the existence of other energy class solutions. However, one can prove a weak-strong uniqueness result, which implies in particular that the solution constructed in Theorem 1.5 is the unique energy class solution satisfying an energy inequality: Theorem 1.6. Suppose all the assumptions in Theorem 1.5 are satisfied. Let I be an open interval containing [0, T ] and v ∈ C(I; H 1 0 )∩C 1 (I; L 2 )∩L ∞ (I; L p+1 (Ω)) a distributional solution for 0 ≤ t ≤ T to Problem (1.4) with the same initial data as v, which satisfies an energy inequality E(v(t)) ≤ E(v(0)) (see (1.5)). Then we have v(t) = v(t) for 0 ≤ t ≤ T . Remark 1.1. Similar results can be proved for other dispersive equations, notably for the nonlinear Schrödinger equation. This will be the topic of further work in preparation.
The proof of Theorem 1.5 is based on perturbing the radial solution u(t, x) with a small term w(t, x) satisfying the equation which can be written in the form To solve the last equation, we prove a pointwise dispersive estimate for the exterior problem with potential (see Proposition 2.6); note that this part of the argument holds also on arbitrary non trapping exterior domains. The pointwise estimate for the wave equation with potential is proved by reduction to the constant coefficient case, which is possible since from the previous results we know that the potential V has rapid decay in (t, x). In the constant coefficient case, we prove a dispersive estimate (Proposition 2.3) using the classical theory of local energy decay from [8], [15], [7], [13]. Sharper estimates can be obtained by applying the theory recently developed in [4], but the classical theory is sufficient for our purposes. On the other hand, Theorem 1.6 is an adaptation of a weak-strong uniqueness result due to M. Struwe [14].
Remark 1.2. We chose to use a pointwise estimate in the proof, instead of more efficient Strichartz estimates, since we were also interested in dispersive estimates for the wave operator outside an obstacle, which are not easy to find in the literature in closed form. See Proposition 2.3 below for an explicit statement.
The plan of the paper is the following. The linear theory and the pointwise dispersive estimates are developed in Section 2. In Section 3 the global radial solution is constructed and its regularity and decay properties are studied. In the final Sections 4, 5 we prove the main results Theorems 1.5 and Theorem 1.6.

Linear theory
We consider here the linear mixed problem on R + × Ω Given the data (u 0 , u 1 , F ) we define recursively the sequence of functions h j as follows: The function h j is obtained by applying ∂ j−2 t to the equation u tt = ∆u + F . Definition 2.1 (Linear compatibility conditions). We say that the data (u 0 , u 1 , F ) satisfy the linear compatibility conditions of order The following result is standard, and valid for general domains Ω with, say, C 1 compact boundary.
Proposition 2.2. Let N ≥ 1, and assume u 0 ∈ H N +1 (Ω), u 1 ∈ H N (Ω) and F ∈ C k H N −k (Ω) for 0 ≤ k ≤ N satisfy the linear compatibility conditions of order N . Then Problem (2.1) has a unique solution belonging to The solution satisfies the energy estimates, for 0 ≤ k ≤ N , t ≥ 0, with a constant depending only on k.
We can also represent the solution via spectral calculus. We denote by −∆ D the Dirichlet Laplacian, i.e., the nonnegative selfadjoint operator with domain H 2 (Ω) ∩ H 1 0 (Ω), and by Λ its (nonnegative selfadjoint) square root The solution of the full problem (2.1) can be represented as where ∂ t S(t)u 0 = cos(tΛ)u 0 . As byproducts of the spectral theory we have the estimates .

(2.6)
Here and in the following we use the equivalence where T r denotes the trace operator at ∂Ω, the energy estimate (2.3) gives We now turn to the proof of pointwise decay estimates for S(t). First of all we recall the corresponding estimates for the wave flow on R n . The operator represents the solution u = S 0 (t)g of the free wave equation on R n and satisfies the well known pointwise decay estimate where for odd n the Besov space norm can be replaced by the W n−1 2 ,1 norm. This estimate is sharp but singular as t → 0. A weaker, non-singular estimate can be obtained by combining (2.8) for t > 1 with the energy estimate (here C(t) may grow linearly in time due to the L 2 term). If we pick N 0 = ⌊n/2⌋ + 1 we obtain and in conclusion t (2.9) The following result holds for any exterior domain Ω which is non trapping in the sense e.g. of [13]. In order not to introduce unnecessary complications, we state it here for the special class of strictly convex obstacles. As mentioned in the introduction, for n = 3 the logarithmic loss in t can be avoided, but it is harmless in the present context. Proposition 2.3 (Basic dispersive estimate). Let m be a non negative integer and assume (2.11) The operator ∂ t S(t) satisfies similar estimates with m replaced by m + 1 in the norms at the right hand side of (2.10), (2.11).
Proof. Since m is integer, the estimates follow from the case m = 0 on which we focus in the following. Let χ(x) be a smooth function vanishing in a nbd of the obstacle R n \Ω. By Duhamel's principle we obtain the operator identity where χ denotes the multiplication operator with the function χ(x) and [∆, χ] is its commutator with ∆; note that [∆, χ] is a first order operator. In a similar way, one has the dual identity Now, assume the obstacle is contained in the ball {|x| ≤ R}. Following [13] and [6], we fix three smooth functions ψ 1 , ψ 2 , ψ 3 such that Then, using (2.12) and (2.13), we cen write Note that in all terms the operator S(t) is localized on a bounded set. We can thus apply the classical results on local energy decay for the exterior wave equation [8], [15], [7]. The following formulation is a special case of Lemma 4.3 in [13]; also in this case a nontrapping assumption on Ω would be sufficient.
with a constant depending only on n, m, χ.
Combining (2.15) and (2.9), we can estimate each term in (2.14) as follows (we use freely the standard Sobolev embeddings; recall in particular that H N0 ֒→ L ∞ since N 0 = ⌊n/2⌋ + 1): On the other hand, one has for n ≥ 4 Applying these estimates, we obtain (2.10), (2.11) for S(t). The estimates for ∂ t S(t) are proved in a similar way.
We shall also need estimates for the exterior wave equation with a time dependent potential. We denote by u = S V (t, t 0 )g the solution of the mixed problem with initial data at time The existence and uniqueness of a solution is standard (the potential V is smooth and with good behaviour at infinity). Note that the solution of the full problem can be represented by Duhamel as for n ≥ 3 and some k > n/2, ν > 1. Then for all t, t 0 we have Proof. Take the L 2 norm in (2.18); using (2.6) and assumption . By Gronwall's Lemma we obtain the first estimate in (2.21).
From (2.18) we get also We next apply (2.7) to estimate By the algebra property of H n/2+ we can write and by assumption (2.20) we get Summing up, we have proved and by Gronwall's Lemma we obtain the estimate for ∇S V . The proof of the estimate for S ′ V is similar (actually simpler).
satisfies the same estimates, with the order of the norms at the right increased by one.
Proof. Consider first space dimension n ≥ 4. We use the representation (2.18) and the dispersive estimate (2.10) to get n+2 ∇ x u H 2N 0 ; on the other hand, separating the terms with no derivative on u, and by (2.21) we get we conclude the proof. The case n = 3 is similar, using instead in the final steṕ

The global radial solution
This section is devoted to the proof of Proposition 1.2 and Theorems 1.3, 1.4. We begin with a few preliminary results on the mixed problem For data of low regularity, a solution to (3.1) is intended to be a solution of the integral equation with S(t) as in (2.4). We will give only sketchy proofs of standard results, which are virtually identical to the corresponding ones for semilinear wave equations on R n (for which we refer e.g. to Chapter 6 of [12]).
Then for any initial data The solution satisfies the energy bound for all t > 0 (Ω) and solves (3.1) in both distributional and a.e. sense.
If we further assume that 0 ≤ f (s)s F (s) for s ∈ R, where F (s) =´s 0 f (σ)dσ, then the solution satisfies for all times the energy identity Proof. Differentiate the equation once w.r.to space variables, noting that S(t) commutes with spatial derivatives. The nonlinear term produces a term f ′ (u)∂u where f ′ (u) is uniformly bounded; then the (linear) energy estimate gives D 2 x u ∈ CL 2 (Ω). The estimate for u tt is deduced from the equation itself.
Note that the assumption 0 ≤ sf (s) F (s) is sufficient to prove the existence of a global weak solution for data in H 1 0 (Ω) × L 2 (Ω), even if f is not Lipschitz (Segal's Theorem). This can be proved like in the case of the whole space R n by approximating f with a sequence of truncated Lipschitz functions and using weak compactness. The weak solution thus constructed satisfies then a weaker energy inequality E(t) ≤ E(0) (proved using Fatou's Lemma). We shall not need this variant in the sequel.
For smoother data, one can prove a local existence theorem which does not require a global Lipschitz condition, similarly to the case Ω = R n . However, one must assume suitable compatibility conditions, analogous to the linear ones from Definition 2.1. Define formally a sequence of functions ψ j , j ≥ 0 as follows: differentiating the equation u tt = ∆u + f (u) with respect to time, set where values of ∂ k t u(0, x) for 0 ≤ k ≤ j − 2, required to compute ψ j , are set recursively equal to ψ k . For instance, ψ 2 = ∆u(0, x) + f (u(0, x)) = ∆ψ 0 + f (ψ 0 ),  Proof. The existence part is standard; it is usually proved for more general quasilinear equations, which require higher smoothness of the data; see e.g. Theorem 3.5 in [13], where local existence is proved for a nonlinear term of the form f (t, x, ∂ j t ∂ α x u) with j + |α| ≤ 2, j ≤ 1 (and a regularity of order ⌊ n 2 ⌋+ 8 is imposed on the data). The proof is based on a contraction mapping argument, combined with Moser type estimates of the nonlinear term. The final blow up alternative in the statement is a byproduct of the proof. for some universal constant C 0 . Since |x| ≥ 1 on Ω, this gives The same argument guarantees also uniqueness of radially symmetric solutions. More generally, a solution in H 2 ∩ L ∞ loc with the same initial data must coincide with the radial one, as it follows by a localization argument and finite speed of propagation. The proof of Proposition 1.2 is concluded.
We define a new function U (T, α, θ) via Since u, U are independent of θ, we shall write simply U (T, α). Commuting with Π gives on the subset E Ω of R × S n given by the conditions which is the image of R + t × Ω via Π. We introduce also the notation D T = {(α, θ) ∈ S n : Γ(T ) ≤ α < π − T }, 0 ≤ T < π for the slice of Π(R + × Ω) at time T . Note that in coordinates, equation (3.13) reads We plan to extend the solution beyond the line T + α = π, i.e., in the region where ω < 0. Thus we consider the following extended equation on (T, α) ∈ [0, π] 2 : where we have replaced ω by The solution U satisfies the identity We can now extend U to a larger domain in the cylinder R T × S n . Recall that the data u 0 , u 1 satisfy C(M ) < ∞ with C(M ) as in (1.7). Thus if we fix a smooth cutoff function χ(x) equal to 0 for |x| ≤ M + 1 and equal to 1 for |x| ≥ M + 2, we have Denote by U 0 , U 1 the functions obtained by applying the transformation (3.12) to χu 0 , χu 1 respectively (with t = 0). By the first Lemma in Section 4 of [1] we have then In order to solve (3.13) locally via the energy method we require that the coefficient ω ν be sufficiently smooth i.e. ω ν ∈ C N0 . This is true as soon as ν = n−1 2 p − n+3 2 > N 0 which is implied by p > 11 2 . Then a standard local existence result guarantees the existence of a local solution U to equation (3.13) with data U 0 , U 1 on some strip [0, δ) × S n . The lifespan δ, which can be assumed ≪ 1, depends only on δ = δ(C(M ), n, p) (3.16) where C(M ) was defined in (1.7). Comparing U with the solution U constructed above, and noting that equation (3.13) has finite speed of propagation equal to 1, by local uniqueness we see that U, U must coincide on the forward dependence domain emanating from the set T = 0, 2 arctan(M + 2) < α < π.
Thus we can glue the two solutions, at least for 0 < T < δ/2, and we obtain an extended solution of (3.13), which we denote again U (T, α), defined on the larger domain . We next prove that the energy of U , defined as remains bounded for 0 ≤ T < δ/2. To this end we integrate identity (3.15) on a slice for arbitrary times 0 ≤ T 1 ≤ T 2 < δ/2. Dropping negative terms at the RHS, we are left with the inequality where ν α , ν T are the components of the exterior normal to the curve T = γ(α), i.e., The Dirichlet condition U (γ(α), α) = 0 along the curve implies Thus the RHS of (3.17) is equal to Recalling (3.11), we see that the RHS of (3.17) is negative, and we conclude that the energy is nonincreasing as claimed: Now, consider the set, which is a dependence domain for (3.13) (keep in mind that the speed of propagation for (3.13) is exactly 1): We have already extended U to the part of D in the time strip δ/3 < T < δ/2, and our next goal is to prove that U can be extended to a bounded solution of (3.13) on the whole set D. Clearly, it is sufficient to prove an a priori L ∞ bound of the solution on this domain in order to achieve the result via a continuation argument.
To this end we prove an energy estimate similar to the previous one, but now we integrate identity (3.15) over the slice T 1 < T < T 2 , Γ(T ) < α < π − T + δ/4, where δ/3 ≤ T 1 < T 2 < π are fixed, and we denote by F (T ) the energy After integration of (3.15), the terms on the side α = π − T + δ/4 give a negative contribution at the RHS which can be dropped, as in the standard energy estimate, since the speed of propagation is 1. Proceeding as before, we are left with the inequality and again the RHS here is negative thanks to (3.11). We conclude that the energy F (T ) is nonincreasing: Since F (δ/3) ≤ E(δ/3), by (3.18) we conclude To proceed, we need a Lemma: with a constant independent of I and V .
Next, split I in thirds I = I 1 ∪ I 2 ∪ I 3 (with I 1 at the left and I 3 at the right). If α ∈ I 1 ∪ I 2 , pick β ∈ I 3 arbitrary and apply (3.21) to get 2 ) our claim (3.20) is proved for the points in I 1 ∪ I 2 . On the other hand, if α ∈ I 3 , we pick β ∈ I 2 arbitrary, we apply (3.21), and we get We apply (3.20) to U (T, α) at a fixed T ∈ (δ/3, π) on the interval I = [Γ(T ), π − T + δ/4] ⊆ (0, K] = (0, π − δ 12 ]; note that |I| ≥ δ/4. We get thus substituting in the previous inequality we get, for all α ∈ [Γ(T ), π − T + δ/4], Recalling the definition of F (T ) and (3.19) we conclude where S n denotes the image of {t = 0} × Ω via the Penrose transform, and This gives the estimate Since the data are radial, v is also radial, and by the uniqueness part of Proposition 1.2 we see that u ≡ v. In particular, v is bounded as t ↑ T * , hence T * = +∞ and the first claim is proved. It remains to prove the H k bounds (1.11). The L 2 norm of u can be estimated as follows by Hölder and Sobolev embedding, and we note the consequence of (1.9) (also valid only if n ≥ 3) (3.24) and the conservation of energy (1.5). Summing up, we obtain where the integral converges since p > 1 + n+2 n−2 = 2n n−2 . This implies (1.11) for k = 0, since Estimate (1.11) for k = 1 is a consequence of the sharper energy conservation E(t) = E(0). To prove (1.11) for higher orders we use the energy estimate X´t 0 s 1−p u(s, ·) H k−1 ds and Gronwall's Lemma gives the desired conclusion.

Global quasiradial solutions
This section is devoted to the proof of Theorem 1.5. Let u be the global radial solution with initial data (u 0 , u 1 ) given by Proposition 1.2 and Theorem 1.4, and let v be the local solution with data (v 0 , v 1 ) given by Lemma 3.4. Denote by w = v − u the difference of the two solutions, which satisfies the equation Using Gagliardo-Nirenberg estimates one obtains, for all p ≥ 3 with p ≥ k + 1, H k and this implies, recalling (1.9) and (1.11), with a constant depending on (Note that Gagliardo-Nirenberg estimates are valid also on the unbounded set Ω, since we can define a bounded extension operator E : W k,p (Ω) → W k,p (R n ) for all k, p). In a similar way, for all p ≥ 2 with p ≥ k + 1, and this implies V (t, ·) W k,∞ t 2−p (4.3) with a constant depending on (u 0 , u 1 ) X + u 0 H k+ n 2 +1 + u 1 H k+ n 2 . Thus we see that if n ≥ 3, p > n + 5 2 (4.4) the assumption of Propositions 2.5 and 2.6 are verified, and the flows S V , S ′ V associated to + V (t, x) satisfy the corresponding energy and dispersive estimates (2.21), (2.23), (2.24). Let now n ≥ 4 (we will examine the case n = 3 later) and consider the problem where w j = v j − u j , j = 0, 1, and V (t, x) = p|u| p−1 . Define the spacetime norm where N will be precised later. Our goal is to prove the a priori estimate where ǫ is a suitable norm of the data (w 0 , w 1 ). First we estimate the H N norm. Represent the solution via (2.17) and apply (2.21): We have and we conclude´t provided p > 3. On the other hand, we have if k > n/2 and p − 2 > k, by the algebra property of H k and Moser estimates. Now we recall that u(s, ·) L ∞ s −1 while the H k norms of u are uniformly bounded, thus we get (if N ≥ k) (4.8) Next we estimate the L ∞ part. From (2.17) we have, using (2.23) and the corresponding one We already estimated the H k norms and we can use again (4.8) provided N ≥ k = 2N 0 + 1. We now estimate the W 2N0,1 norm. We can write For the first term we have . For the second term we use the estimate, valid if p − 2 ≥ k = 2N 0 , provided N ≥ 2N 0 . Thus choosing N = 2N 0 + 1 and combining the previous estimate, we obtain our claim (4.6), with and with an implicit constant depending on It is clear that if ǫ is sufficiently small, relative to C(u 0 , u 1 ), then a simple continuation argument based on (4.6) gives the conclusion of the Theorem. Finally, we consider the case n = 3, when the dispersive estimate (2.24) has a logarithmic loss. In this case we replace the definition of M (t) with It is then easy to check that all the steps in the previous proof can be carried through and give again (4.6).