On blowup for the supercritical quadratic wave equation

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for all $d \geq 7$. For $d=9$, we study the stability of $u^*$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$. In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d=7$ and $d=9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.


Introduction
In this paper, we are concerned with the quadratic wave equation where (t, x) ∈ I × R d , for some interval I ⊂ R containing zero. It is well-known that in all space dimensions Eq. (1.1) admits solutions that blow up in finite time, starting from smooth and compactly supported initial data. This follows from a classical result by Levine [29], which provides an open set of such initial data. However, Levine's argument is indirect, and therefore does not give insight into the profile of blowup. A more concrete example can be produced by using the well-known ODE solution, u ODE T (t, x) := 6 (T − t) 2 , T > 0. (1.2) By truncating the initial data (u ODE T (0, ·), ∂ t u ODE T (0, ·)) outside a ball of radius larger than T , and using finite speed of propagation, one constructs smooth and compactly supported initial data that lead to blow up at t = T . What is more, invariance of Eq. (1.1) under the rescaling u(t, x) → u λ (t, x) := λ −2 u(t/λ, x/λ), λ > 0, (1.3) allows one to look for self-similar blowup solutions of the following form Key words and phrases. nonlinear wave equation, singularity formation, self-similar solution. Irfan Glogić is supported by the Austrian Science Fund FWF, Projects P 34378 and P 30076. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 258734477 -SFB 1173. 1 Note that (1.2) is a self-similar solution with trivial profile φ ≡ 6. We note that the rescaling (1.3) leaves invariant the energy normḢ 1 (R d ) × L 2 (R d ) of (u(t, ·), ∂ t u(t, ·)) precisely when d = 6, in which case Eq. (1.1) is referred to as energy critical. In this case, it can be easily shown that in addition to (1.2) no other radial and smooth self-similar solutions to Eq. (1.1) exist, see [24]. However, in the energy supercritical case, i.e., for d ≥ 7, numerics [28] indicate that in addition to (1.2) there are non-trivial, radial, globally defined, smooth, and decaying similarity profiles. In fact, for d = 7 there are infinitely many of them, all of which are positive, as recently proven by Dai and Duyckaerts [13]. A similar result is expected to hold for all 7 ≤ d ≤ 15, see [28]. From the point of view of the Cauchy problem for Eq. (1.1), the relevant similarity profiles appear to be the trivial one (1.2) and its first non-trivial "excitation". Namely, numerical work on supercritical power nonlinearity wave equations in the radial case [7,20] yields evidence that generic blowup is described by the ODE profile, while the threshold separating generic blowup from global existence is given by the stable manifold of the first excited profile, see also [4]. The first step in showing such genericity results would be to establish stability of the ODE profile, and show that its first excitation is co-dimension one stable (which indicates that the stable manifold splits the phase space locally into two connected components). The only result so far for Eq. (1.1) in this direction is by Donninger and the third author [15], who proved radial stability of u T for all odd d ≥ 7. In this paper, we exhibit in closed form what appears to be the first excitation of (1.2) for every d ≥ 7. Namely, we have the following self-similar solution to Eq. (1.1) . We note that c 3 > 0 when d ≥ 7, and thus U ∈ C ∞ [0, ∞).
To the best of our knowledge, this solution has not been known before, and with the intent of studying threshold behavior, the main object of this paper is to show a variant of co-dimension one stability of u * .
Note that U has precisely one zero at ρ * = ρ * (d) > 2. In particular, this profile is not positive and therefore not a member of the family of self-similar profiles constructed in [13]. However, it is strictly positive inside the backward light cone of the blowup point (0, 0). Hence, in this local sense u * provides a solution to the more frequently studied focusing equation What is more, as an outcome of our stability analysis we get that small perturbations of both the ODE profile and u * stay positive under the evolution of Eq. (1.1) and therefore yield solutions to Eq. (1.6) as well.

1.15)
There exist constants M > 0, δ > 0, and ω > 0, such that for all real valued (f, g) ∈ C ∞ (B 9 2 ) × C ∞ (B 9 2 ) satisfying (f, g) H 6 (B 9 2 )×H 5 (B 9 2 ) ≤ δ M spectral problem is quite delicate and only for d = 9 we are able to solve it rigorously. Nevertheless, from numerical computations, we have strong evidence that the situation is analogous in other space dimensions in the sense that the linearization has exactly one genuine unstable eigenvalue.  1) and (1.6), stability of the ODE blowup solution under small radial perturbations has been proven by Donninger and the third author [15] in all odd space dimensions d ≥ 7. By exploiting the framework of the proof of Theorem 1.1, we generalize the result from [15] to non-radial perturbations in dimensions d = 7 and d = 9.
Before we state the result, we apply the symmetry transformations (1.7) to the ODE profile (1.2) to obtain the following family of blowup solutions to both Eqns. (1.1) and (1.6), (1.19) To shorten the notation, we write C T := C T,0 for the backward light cone with vertex (T, 0).
We note that due to the invariance of u ODE 1,0,0 under spatial translations the blowup location x 0 = 0 does not change under small perturbations. 6 Remark 1.5. Stability of the ODE blowup solution for energy supercritical wave equations outside radial symmetry has been established first in d = 3 by Donninger and the third author [14]. For the cubic wave equation, the corresponding result was obtained by Chatzikaleas and Donninger [9] in d = 5, 7. Compared to these works, one important improvement in Theorem 1.2 is the regularity of the solution which allows for the classical interpretation. Furthermore, we prove Lipschitz dependence of the blowup time and the blowup point on the initial data. Finally, from a technical perspective, the adapted inner product defined in Sec. 3 is simpler than the corresponding expressions in [9] and can easily be generalized.
1.2. Related results. Wave equations with focusing power nonlinearities provide the simplest possible models for the study of nonlinear wave dynamics and have been investigated intensively in the past decades. Consequently, local well-posedness and the behavior of solutions for small initial data are by now well understood, see e.g. [30]. Concerning global dynamics for large initial data, substantial progress has been made more recently for energy critical problems. This includes fundamental works on the characterization of the threshold between finite-time blowup and dispersion in terms of the well-known stationary ground state solution, cf. [25], [26] and the references therein. In contrast, large data results for energy supercritical equations are rare. For various models, the ODE blowup is known to provide a stable blowup mechansim and Theorem 1.2 further extends these results, see Remark 1.5. In [8], non-trivial self-similar solutions are constructed for odd supercritical nonlinearities in dimension three, and [13] provides a generalization to d ≥ 4. Also, in the three dimensional case, large global solutions were obtained for a supercritical nonlinearity in [27]. Finally, for d ≥ 11 and large enough nonlinearities, manifolds of co-dimension greater or equal than two have been constructed in [10] that lead to non-selfsimilar blowup in finite time. In the description of threshold dynamics for energy supercritical wave equations, self-similar solutions appear to play the key role. This has been observed numerically for power-type nonlinearities [7,20], but also for more physically relevant models such as wave maps [6,2] or the Yang-Mills equation in equivariant symmetry [3,5]. We note that the latter reduces essentially to a radial quadratic wave equation in d ≥ 7, hence Eq. (1.1) provides a toy model. From an analytic point of view, threshold phenomena for energy supercritical wave equations are entirely unexplored. Moreover, results analogous to the energy critical case seem completely out of reach. However, very recently, the first explicit candidate for a self-similar threshold solution has been found by the second and third author in [21] for the focusing cubic wave equation in all supercritical space dimensions d ≥ 5. In d = 7, by the conformal symmetry of the linearized equation, the genuine unstable direction could be given in closed form, see also [20], which allowed for a rigorous stability analysis. Interestingly, the same effect occurs for the quadratic wave equation and the new self-similar solution (1.4) in d = 9, which explains the specific choice of the space dimension in Theorem 1.1. In view of our results, we conjecture that the self-similar profile U given in (1.5) plays an important role in the threshold dynamics for Eqns. (1.1) and (1.6).
In the proofs of Theorem 1.1 and Theorem 1.2 we build on methods developed in earlier works, in particular [21] and [14]. However, several aspects, in particular the spectral analysis, are specific to the problem and rather delicate. Furthermore, we add important generalizations such as the preservation of regularity, which improves the statements of [21] or [14]. The presentation of our results is completely self-contained and all necessary details are provided in the proofs.
1.3. Notation. Throughout the whole paper the Einstein summation convention is in force, i.e., we sum over repeated upper and lower indices, where latin indices run from 1 to d. We write N for the natural numbers {1, 2, 3, . . . }, N 0 := {0} ∪ N. Furthermore, R + := {x ∈ R : x > 0}. Also, H stands for the closed complex right half-plane. By B d R (x 0 ) we denote the open ball of radius R > 0 in R d centered at x 0 ∈ R d . The unit ball is abbreviated by B d := B d 1 (0) and S d−1 := ∂B d . The notation a b means a ≤ Cb for an absolute constant C > 0 and we write a ≃ b if a b and b a. If a ≤ C ε b for a constant C ε > 0 depending on some parameter ε, we write a ε b. By L 2 (B d R (x 0 )) and H k (B d R (x 0 )), k ∈ N 0 , we denote the Lebesgue and Sobolev spaces obtained from the completion of C ∞ (B d R (x 0 )) with respect to the usual norm . For vector-valued functions, we use boldface letters, e.g., f = (f 1 , f 2 ) and we sometime write [f] 1 := f 1 to extract a single component. Throughout the paper, W (f, g) denotes the Wronskian of two functions f, g ∈ C 1 (I), I ⊂ R, where we use the convention W (f, g) = f g ′ − f ′ g, with f ′ denoting the first derivative. On a Hilbert space H we denote by B(H) the set of bounded linear operators. For a closed linear operator (L, D(L)) on H, we define the resolvent set ρ(L) as the set of all λ ∈ C such that R L (λ) := (λ − L) −1 exists as a bounded operator on the whole underlying space. Furthermore, the spectrum of L is defined as σ(L) := C \ ρ(L) and the point spectrum is denoted by σ p (L) ⊂ σ(L).

The stability problem in similarity coordinates
In this section we formulate the equation (1.1) in similarity variables. The advantage of the new setting is the fact that self-similar solutions become time-independent and stability of finite time blowup turns into asymptotic stability of static solutions. Then we state the main results in the new coordinate system.

2.1.
Stability of U a . The key to proving Theorem 1.1 is the following result, which establishes for d = 9 conditional orbital asymptotic stability of the family of static solutions {U a : a ∈ R 9 }.
There are constants C > 0 and ω > 0 such that the following holds. For all sufficiently small δ > 0 there exists a co-dimension eleven Lipschitz manifold M = M δ,C ⊂ B δ/C with 0 ∈ M such that for any Φ 0 ∈ M there are Ψ ∈ C([0, ∞), H) and a ∈ B 9 δ/ω such that

4)
and The number of co-dimensions in Proposition 2.1 is related to the number of unstable eigenvalues of the linearization around U a , and the dimension of the corresponding eigenspaces, see Sec. 5. In fact, ten of these instabilities are caused by the translation symmetries of the problem, and can be controlled by choosing appropriately the blowup parameters (T, x 0 ). There is, therefore, only one genuine unstable direction. Next, we state a persistence of regularity result for solutions to Eq. (2.4).
The proofs of Propositions 2.1 and 2.2 are provided in Sec. 7.4.
In order to derive Theorem 1.1 from the above results we prescribe in physical variables initial data of the following form for free functions (f, g) defined on a suitably large ball centered at the origin. In similarity variables, this transforms into initial data Ψ(0) = U 0 + Φ 0 for Eq. (2.3), with and The next statement asserts that for all small (f, g), there is a choice of parameters x 0 , T , and α, for which Υ((f + αh 1 , g + αh 2 ), T, x 0 ) belongs to the manifold M from Proposition 2.1.
There exists M > 0 such that for all sufficiently small δ > 0 the following holds. For any (f, g) ∈ H 6 (B 9 where M δ,C is the manifold from Proposition 2.1. In the following, we assume that a = a(τ ), a(0) = 0 and lim τ →∞ a(τ ) = a ∞ . Inserting the ansatz In the following, we define ). and study the evolution equation with initial data Φ(0) = u ∈ H. This naturally splits into three parts: First, in Sec. 3, we study the time evolution governed byL using semigroup theory. In Sec. 4, we analyze the linearized problem, where we considerL + L ′ a∞ as a (compact) perturbation of the free evolution and investigate the underlying spectral problem, restricting to d = 9. Resolvent bounds allow us to transfer the spectral information to suitable growth estimates for the linearized time evolution. The nonlinear problem will be analyzed in integral form in Sec. 7, using modulation theory and fixed-point arguments. Also, we prove Propositions 2.1 -2.3 and based on this, Theorem 1.1. In Sec. 8 we give the main arguments to prove Theorem 1.2.

The free wave evolution in similarity variables
In this section we prove well-posedness of the linear version of Eq. (2.3) in H. In other words, we show that the (closure of the) operatorL generates a strongly continuous one-parameter semigroup of bounded operators on H. What is more, in view of the regularity result Proposition 2.2, we consider the evolution in Sobolev spaces of arbitrarily high integer order. In Sec. 4 we then restrict the problem again to H.
and consider the densely defined operatorL : D(L) ⊂ H k → H k . We now state the central result of this section.
for all u ∈ H k , all τ ≥ 0, and some M k > 1. Furthermore, the following holds for the spectrum of L k σ(L k ) ⊂ z ∈ C : Re z ≤ − 1 2 , (3.2) and the resolvent has the following bound 12 for λ ∈ C with Re λ > − 1 2 , and f ∈ H k . Remark 3.1. We prove Proposition 3.1 via the Lumer-Phillips Theorem. By using the standard inner product on H k , one can easily prove existence of the semigroup (S k (τ )) τ ≥0 , but in order to show that it decays exponentially, and to prove the growth bound (3.1) in particular, we need to introduce an appropriate equivalent inner product. Necessity for such approach will become apparent in the proof of Lemma 3.3 below. We note that for d = 9 the restriction on k is optimal within the class of integer Sobolev spaces. In particular, for scaling reasons exponential decay cannot be expected at lower integer regularities. For d = 7, a similar statement can be obtained for k = 2.
For d ∈ {7, 9} and k ≥ 3 we define the following sesquilinear form and for j ≥ 4 we use the standardḢ j (B d ) ×Ḣ j−1 (B d ) inner product We then set u H k := (u|u) H k .
In particular, · H k defines an equivalent norm on H k .
Proof. Note that it suffices to prove the following The first estimate in (3.4) follows from the fact that 13 for all u ∈ C ∞ (B d ), which is a simple consequence of the identity (3.5) Using this, it is easy to see that . Similar bounds imply the first inequality in (3.4). Another consequence of Eq. (3.5) is the trace theorem, which asserts that for all u ∈ C ∞ (B d ); using this, it is straightforward to obtain the second inequality in (3.4). Hence, we obtain the claimed estimates in Lemma 3.2 for all u ∈ C ∞ (B d ) × C ∞ (B d ) and by density, we extend this to all of H k .
Now we turn to proving Proposition 3.1. As the first auxiliary result, we have the following dissipation property ofL.
The proof is provided in Section A of the appendix. To apply the Lumer-Phillips theorem, we also need the following density property ofL.
Proof. Let d ∈ {7, 9} and k ≥ 3. We prove the statement by showing that there is exists a λ such that given f ∈ H k and ε > 0 there is some where P ℓ are the projection operators defined in (1.22). Furthermore, according to (1.23) It is therefore sufficient to consider and produce a solution u ∈ D(L). First, we rewrite Eq. (3.6) as a system of equations in u 1 and u 2 We now treat the case d = 9, for which we choose λ = 5 2 . With this choice, Eq. (3.7) reads as Note that g N is a finite linear combination of spherical harmonics, and this allows us to decompose the PDE (3.9) which is posed on B 9 into a finite number of ODEs posed on the interval (0, 1). To this end, we switch to spherical coordinates ρ = |ξ| and ω = ξ |ξ| . In particular, the relevant differential expressions transform in the following way Consequently, Eq. (3.9) becomes Now we decompose the right hand side of (3.10) into spherical harmonics for some g ℓ,m ∈ C ∞ [0, 1]. Then by inserting the ansatz into Eq. (3.10), we obtain a system of ODEs for ℓ = 0, . . . , N and m ∈ Ω ℓ . For later convenience, we first set v ℓ,m (ρ) = ρ 3 u ℓ,m (ρ) and thereby transform (3.12) into Then, by means of further change of variables v ℓ,m (ρ) = ρ ℓ+3 w ℓ,m (ρ 2 ) we turn the homogeneous version of (3.13) into a hypergeometric equation in its canonical form The second and the third term above are obviously smooth up to ρ = 1; for the first term, the square root factors in fact cancel out, as can easily be seen via substitution s = ρ + (1 − ρ)t, and smoothness of v ℓ,m up to ρ = 1 follows. Consequently, the function u 1 defined in (3.11) belongs to C ∞ (B 9 \ {0}) and it solves Eq. (3.9) in the classical sense away from zero. Furthermore, from (3.15) one can check that u ℓ,m and u ′ ℓ,m are bounded near zero, and hence u 1 ∈ H 1 (B 9 ). In particular, u 1 solves Eq. (3.9) in the weak sense on B 9 and since the right hand side is a smooth function, we conclude that u 1 ∈ C ∞ (B 9 ) by elliptic regularity. Consequently, u 1 ∈ C ∞ (B 9 ), and therefore u 2 ∈ C ∞ (B 9 ) according to Eq. (3.8).
In conclusion, u := (u 1 , u 2 ) ∈ D(L) solves Eq. (3.6). For d = 7, the same proof can be repeated by choosing λ = 3 2 . Namely, by decomposing the functions into spherical harmonics and by introducing the new variableṽ ℓ,m (ρ) = ρ 2 u ℓ,m (ρ), the problem is reduced to which the same as Eq. (3.13) up to a shift in ℓ and the weight on the right hand side. Hence, the same reasoning applies.
We conclude this section with proving certain restriction properties of the semigroups (S k (τ )) τ ≥0 . This will be crucial in showing persistence of regularity for the nonlinear equation.
Proof. Let d ∈ {7, 9} and k ≥ 3. We prove the claim only for j = 1, as the general case follows from the arbitrariness of k. The crucial ingredients of the proof are continuity of the embedding H k+1 ֒→ H k , and the fact that D(L) is a core for both L k and L k+1 . First, we prove that L k+1 is a restriction of L k ; more precisely we show that for all u ∈ D(L k+1 ). For u ∈ D(L), from the definition of L k+1 and L k , it follows that From the embedding H k+1 ֒→ H k we infer that and by the closedness of L k it follows that u ∈ D(L k ) and we get by the Post-Widder inversion formula (see [17], p. 223, Corollary 5.5) and the embedding H k+1 ֒→ H k , that for every τ > 0, This proves that (S k+1 (τ )) τ ≥0 is the restriction of (S k (τ )) τ ≥0 to H k+1 . Consequently, from Proposition 3.1 we have that , for all u ∈ H k+1 and all τ ≥ 0.

Linearization around a self-similar solution -Preliminaries on the structure of the spectrum
From now on, for fixed d ∈ {7, 9}, we will work solely in the Sobolev space H To abbreviate the notation, we write We also denote by (S(τ )) τ ≥0 and L : D(L) ⊂ H → H, the corresponding semigroup (S k (τ )) τ ≥0 and its generator L k for k = d+1 2 .
With an eye towards studying the flow near the orbit {U a : a ∈ R d }, see Sec. 2.1.1, in this section we describe some general properties of the underlying linear operator with U a given in Eq. (1.12).
Remark 4.1. We emphasize that the results of this section apply to any smooth V a : B d → R that depends smoothly on the parameter a. Obviously, such potentials arise in the linearization around smooth self-similar profiles.
Proof. The compactness of L ′ a follows from the smoothness of V a and the compactness of the embedding H d+1 The fact that L a generates a semigroup is a consequence of the Bounded Perturbation Theorem, see e.g. [17], p. 158. For the Lipschitz dependence on the parameter a, we first note that by the fundamental theorem of calculus we have that . This implies that given δ > 0 we have that and we thus have that δ , which implies the claim. Next, we show that the unstable spectrum of L a : D(L a ) ⊂ H → H consists of isolated eigenvalues and is confined to a compact region. This is achieved by proving bounds on the resolvent and using compactness of the perturbation.
Then Proposition 3.1 implies that given ε > 0 for all λ ∈ C with Re λ ≥ − 1 2 + ε and all f ∈ H. Together with (4.6), this gives L ′ a R L (λ)f |λ| −1 f , and the uniform bound (4.4) holds for some c > 0 when we restrict to |λ| ≥ κ for suitably large κ. The second statement follows from the compactness of L ′ a . Indeed, if Re λ > − 1 2 then λ ∈ ρ(L), and according to (4.5) we have that λ ∈ σ(L a ) only if the operator 1 − L ′ a R L (λ) is not bounded invertible, which is equivalent to 1 being an eigenvalue of the compact operator L ′ a R L (λ), which according to Eq. (4.5) implies that λ is an eigenvalue of L a . The fact that λ is isolated follows from the Analytic Fredholm Theorem (see [34], Theorem 3.14. 3, p. 194 The previous proposition implies that there are finitely many unstable spectral points of L a , i.e., the ones belonging to H := {λ ∈ C : Re λ ≥ 0}, all of which are eigenvalues. This can actually be abstractly shown just by using the compactness of L ′ a , see [18], Theorem B.1. We nonetheless need Proposition 4.2 as it allows us later on to reduce the spectral analysis of L a for all small a to the case a = 0, see Sec. 5.3. Note that the eventual presence of unstable spectral points of L a prevents decay of the associated semigroup (S a (τ )) τ ≥0 on the whole space H. What is more, since L ′ a is compact, a spectral mapping theorem for the unstable spectrum holds (see [18], Theorem B.1), and hence eventual growing modes of (S a (τ )) τ ≥0 are completely determined by the unstable spectrum of L a and the associated eigenspaces. Therefore, in what follows we turn to spectral analysis of L a . First, we show an important result which relates solvability of the spectral equation (λ − L a )u = 0 for a = 0, λ ∈ H, to the existence of smooth solutions to a certain ordinary differential equation. We note that for a = 0, the potential V a is radial, more precisely V 0 (ξ) = 2U 0 (ξ) = 2U(|ξ|) =: V (|ξ|) with U given in (1.5).
Proof. Let λ ∈ H ∩ σ(L 0 ). By Proposition 4.2, λ is an eigenvalue, and hence there is a nontrivial u ∈ D(L 0 ) satisfying (λ − L 0 )u = 0. By a straightforward calculation we get that the components u 1 and u 2 satisfy the following two equations and Furthermore, we may decompose u 1 into spherical harmonics where ρ = |ξ| and ω = ξ/|ξ|. To be precise, the expansion above holds in H k (B d 1−ǫ ) for any k ∈ N and ǫ > 0, see Eqns. (1.22) and (1.23). Since the potential V 0 is radially symmetric, Eq. (4.8) decouples by means of (4.10) into a system of infinitely many ODEs posed on the interval (0, 1), where the operator T (d) ℓ (λ) is given by (4.7). Since u 1 is nontrivial, there are indices ℓ ∈ N 0 and m ∈ Ω ℓ , such that u ℓ,m is non-zero and satisfies (4.11).
. Now we prove that u ℓ,m is smooth up to ρ = 1. Note that ρ = 1 is a regular singular point of equation (4.11), and the corresponding set of Frobenius indices is {0, 2 − λ} when d = 9, and {0, 1 − λ} when d = 7. In the first case, if λ / ∈ {0, 1, 2}, then u ℓ,m is either analytic or behaves like (1 − ρ) 2−λ near ρ = 1. If λ ∈ {0, 1, 2}, then the non-analytic behavior can be described by ( In each case, singularity can be excluded by the requirement that u ℓ,m ∈ H 5 ( 1 2 , 1). This implies that u ℓ,m belongs to C ∞ [0, 1] and solves Eq. (4.7) on (0, 1). The same reasoning applies to the case d = 7. Implication in the other direction is now obvious. Consequently, determining the unstable spectrum of L 0 amounts to solving the connection problem for a family of ODEs. We note that the connection problem is so far completely resolved only for hypergeometric equations, i.e., the ones with three regular singular points, while the ODE (4.7) has six of them. In fact, their number can, by a suitable change of variables, be reduced to four, but this nonetheless renders the standard ODE theory useless. Nevertheless, by building on the techniques developed recently to treat such problems, see [12,11,19,21], for d = 9 we are able to solve the connection problem for (4.7) and we thereby provide in the following section a complete characterization of the unstable spectrum of L 0 . 5. Spectral analysis for perturbations around U a -The case d = 9 From now on we restrict ourselves to d = 9.
5.1. Analysis of the spectral ODE. In this section we investigate the ODE (4.7) for d = 9, and for convenience we shorten the notation by letting T ℓ (λ) := T where the potential is given by V (ρ) = 480(7 − ρ 2 ) (7 + 5ρ 2 ) 2 . Now, in view of Proposition 4.3, given ℓ ∈ N 0 , we define the following set The central result of our spectral analysis is the following proposition.
Proposition 5.1. The structure of Σ ℓ is as follows.
(2) For ℓ = 1, Σ 1 = {0, 1}, and the corresponding solutions are To prove this proposition, we use an adaptation of the ODE techniques devised in [12,11,19] and [21]. We will therefore occasionally refer to these works throughout the proof. Also, we found it convenient to split the proof into two cases, ℓ ∈ {0, 1} and ℓ ≥ 2.
To explore this further, we treat cases ℓ = 0 and ℓ = 1 separately.
Proof of Lemma 5.2. First we show that for n ≥ 6 the functions δ 6 (0, ·), ε n (0, ·), and C n (0, ·) are analytic in H. This, based on (5.10) and (5.12), follows from the fact that the zeros ofr n (0, ·) and the poles of r 6 (0, ·) are all contained in the (open) left half plane. This is immediate forr n (0, ·) as it is a quadratic polynomial with two negative zeros. As for the zeros od the denominator of r 6 (0, λ), which is a polynomial of degree 10, this, although it can be proven by elementary means, can be straightforwardly checked by the Routh-Hurwitz stability criterion, see [21], Sec. A.2. Furthermore, being rational functions, δ 6 (0, ·), ε n (0, ·), and C n (0, ·) are all polynomially bounded in H. Therefore, to prove the lemma, it is enough to establish the estimates (5.13) on the imaginary axis only as they can be then extended to all of H by the Phragmen-Lindelöf principle (in its sectorial form), see e.g. [35], p. 177. In the following we prove only the third estimate in (5.13), as the rest of two are shown similarly. We proceed with writing C n+6 (0, λ) (note the shift in the index) as the ratio of two polynomials P 1 (n, λ) and P 2 (n, λ), both of which belong to Z[n, λ]. Then for t ∈ R we have the following representation on the imaginary line which is in turn equivalent to Finally, the last inequality trivially holds as the polynomial on the left (when expanded) has manifestly positive coefficients.

5.1.2.
Proof of Proposition 5.1 for ℓ ≥ 2. Since the parameter ℓ is now free, the analysis is more complicated. Namely, in addition to having to emulate the global behavior in ℓ as well, a quasi-solution also has to approximate the actual solution well enough so as to, with an additional parameter ℓ, obey the estimates analogous to (5.15). We note that a similar problem was treated by the second and the third author in [21], Secs. 4.2.1 and 4.2.2, and we closely follow their approach. First, we introduce the following change of variable x = 12ρ 2 5ρ 2 +7 , by means of which the singular points ρ = 0 and ρ = 1 remain fixed, while the remaining finite singularity (which corresponds to ρ = ∞) is now further away from the unit disk, at x = 12 5 . Furthermore, by applying also the following transformation to T ℓ (λ)f (ρ) = 0 we arrive at a Heun equation forỹ The Frobenius indices of Eq. (5.16) at x = 0 are s 1 = 0 and s 2 = − 7+2ℓ 2 . Therefore, we consider the (normalized) analytic solution at x = 0 y(x) = ∞ n=0ã n (ℓ, λ)x n ,ã 0 (ℓ, λ) = 1. (5.17)

5.2.
The spectrum of L 0 . With the results from above at hand, we can provide a complete description of the unstable spectrum of L 0 .
In what follows, we prove that for each unstable eigenvalue the geometric and the algebraic eigenspaces are the same. To this end, we define the associated Riesz projections. Namely, we set 28 where γ j (s) = λ j + ω 0 2 e 2iπs for s ∈ [0, 1], and j = 0, 1, 2. Lemma 5.6. We have that dim ran H 0 = 1, dim ran P 0 = 10, dim ran Q 0 = 9.
Proof. We start with the observation that the ranges of the projections are finite-dimensional. Indeed, λ j would otherwise belong to the essential spectrum of L 0 (see [23] Theorem 5.28 and Theorem 5.33) which coincides with the essential spectrum of L (since L 0 is a compact perturbation of L), but this is in contradiction with (3.2). Now we show that dim ran P 0 = 10. We know from properties of the Riesz integral that ker(L 0 − λ 1 ) ⊂ ran P 0 . We therefore only need to prove the reversed inclusion. First, note that the space ran P 0 reduces the operator L 0 , and we have that σ(L 0 | ran P 0 ) = {1}, see e.g. [22], Proposition 6.9. Consequently, since P 0 is finite-rank, the operator 1 − L 0 | ran P 0 is nilpotent, i.e., there is m ∈ N, such that (1 − L 0 | ran P 0 ) m = 0. Note that it suffices to show that m = 1. We argue by contradiction, and hence assume that m ≥ 2. Then there is u ∈ D(L 0 ) such that (1 − L 0 )u = v, for a nontrivial v ∈ ker(1 − L 0 ). This yields for u 1 the following elliptic equation Since 0 ), we have that v = 9 k=0 α k g (k) 0 for some α 0 , . . . , α 9 ∈ C, not all of which are zero. To avoid cumbersome notation we let g k = g (k) 0,1 . In the new notation, based on (5.25) and (5.26) we have that Furthermore, according to Proposition 5.1 we can rewrite F in polar coordinates where we denoted f 0 = f 0 (·; 1) and f 1 = f 1 (·; 1). By decomposition of u 1 into spherical harmonics as in (4.10), equation (5.28) can be written as a system of ODEs posed on the interval (0, 1), where G i (ρ) = 2ρf ′ i (ρ) + 7f i (ρ) for i = 0, 1. Moreover, from the properties of u 1 , we infer that u ℓ,m ∈ C ∞ [0, 1) ∩H 5 ( 1 2 , 1), and by Sobolev embedding we have that u ℓ,m ∈ C 2 [0, 1]. To obtain a contradiction, we show that if some α k is non-zero then the corresponding ODE in (5.29) does not admit a C 2 [0, 1] solution. To start, we assume that α 0 = 0. For convenience, we can without loss of generality assume that α 0 = −1. Then u 0,1 solves the following ODE 29 where G 0 (ρ) = 5ρ 4 − 102ρ 2 + 49 (7 + 5ρ 2 ) 4 .
Note that is a solution to the homogeneous version of Eq. (5.30), and by reduction of order we find a second one Furthermore, simple calculation yields that With the fundamental system {u 1 , u 2 } at hand, we can solve Eq. (5.30) by the variation of parameters formula. Namely, we have that for some constants c 1 , c 2 ∈ C. If u ∈ C 2 [0, 1], then c 2 must be equal to zero in the above expression, owing to the singular behavior of u 2 (ρ) near ρ = 0. Then by differentiation we obtain for ρ ∈ (0, 1) that Now we inspect the asymptotic behavior of u ′ as ρ → 1 − . We first note that u ′ 1 is bounded near ρ = 1. Furthermore, note that Finally, we infer that the two integral terms cannot cancel and thus In conclusion, there is no choice of c 1 , c 2 for which u belongs to C 2 [0, 1].
We similarly treat α j for j ∈ {1, . . . , 9}. It is enough to consider just α 1 , and without loss of generality assume that α 1 = −1. Then (5.29) yields the following ODE Note that is a solution for the homogeneous problem. Similarly as above, we obtain another solution by the reduction formula and by inspection of the integral we get that u 2 (ρ) ≃ ρ −8 near the origin and u 2 (ρ) ≃ 1 − ρ near ρ = 1. Now, the general solution of (5.32) on (0, 1) is given by Assumption that u belongs to C 2 [0, 1] forces c 2 = 0 above, due to the singular behavior of u 2 at ρ = 0. Furthermore, from the last term in (5.33) we see that u ′ (ρ) ≃ ln(1 − ρ) as ρ → 1 − . In conclusion, Eq. (5.32) admits no C 2 [0, 1] solutions, and this finishes the proof for P 0 .
The remaining two projections are treated similarly, so we omit some details. For H 0 we obtain the analogue of (5.28) with F (ξ) = 2ξ i ∂ i h 0,1 (ξ)+11h 0,1 (ξ). This leads to the following ODE Note that u 2 is singular at both ρ = 0 and ρ = 1; more precisely u 2 (ρ) ≃ ρ −7 as ρ → 0 + , and u 2 (ρ) ≃ (1 − ρ) −1 as ρ → 1 − . With u 1 and u 2 at hand, the general solution of (5.34) on the interval (0, 1) can be written as where the parameters c 1 , c 2 ∈ C are free. Assumption that u is bounded near ρ = 0 forces c 2 = 0. Note that the first and the third term in (5.35) are bounded near ρ = 1. However, due to the singular behavior of u 2 , the last term is unbounded near ρ = 1, owing to the integrand being strictly positive on (0, 1). In conclusion, the general solution u in (5.35) is unbounded on (0, 1).
Finally, for Q 0 , we have that and the accompanying analogue of (5.30) is A fundamental solution set to the homogeneous version of the above ODE is given by and therefore any solution to it on (0, 1) can be written as for a choice of c 1 , c 2 ∈ C. Again, by similar asymptotic considerations as above, we infer that u ′′ is necessarily unbounded on (0, 1), and this concludes the proof.

5.3.
The spectrum of L a for a = 0. We now investigate the spectrum of L a . In particular, by a perturbative argument we show that, for small a, an analogue of Proposition 5.5 holds for L a as well.
Lemma 5.7. There exists δ * > 0 such that for all a ∈ B 9 δ * the following holds.
Moreover, the geometric eigenspaces of λ 0 and λ 1 are spanned by {g (k) a,2 )} 9 k=0 , and {q (j) a,2 )} 9 j=1 respectively, where a,1 (ξ). Additionally, the eigenfunctions depend Lipschitz continuously on the parameter a, i.e., for all a, b ∈ B 9 δ * . Proof. Let ε = − ω 0 2 + 1 2 and δ > 0. Then take κ defined by Proposition 4.2, and introduce the following two sets Note that Proposition 4.2 implies thatΩ ⊂ ρ(L a ) for all a ∈ B 9 δ . Hence, we only need to investigate the spectrum in the compact set Ω. First, note that by Proposition 4.2, the set Ω contains a finite number of eigenvalues. By a direct calculation it can be checked that q a , and h a are eigefunctions that correspond to λ 0 = 0, λ 1 = 1, and λ 2 = 3 respectively. Note that we get the explicit expression above by just Lorentz transforming the corresponding eigenfunctions for a = 0. We now show that there are no other eigenvalues in Ω. For this, we utilize the Riesz projection onto the spectrum contained in Ω, see (5.38) below. This, however, necessitates that ∂Ω ⊂ ρ(L a ), and we now show that this holds for small enough a. First, note that for λ ∈ ∂Ω we have the following identity Then, from Proposition 4.1, we have that for all a ∈ B 9 δ . Therefore, there is small enough δ * > 0 such that for all λ ∈ ∂Ω, and all a ∈ B 9 δ * . Now from (5.37) and (5.36) we infer that ∂Ω ⊂ ρ(L a ) for all a ∈ B 9 δ * . Thereupon we define the projectioñ For a = 0, by Lemma 5.6 the rank of the operatorT a is 20. Furthermore, continuity of a → R La (λ) (which follows from Eq. (5.36)) implies continuity of a →T a on B 9 δ * . Thus, we conclude that dim ranT a = 20 for all a ∈ B 9 δ * , see e.g. [23], p. 34, Lemma 4.10. By this, we exclude and further eigenvalues. Lipschitz continuity for the eigenfunctions follows from the fact that they depend smoothly on a, c.f. (4.2) and (4.3).

Perturbations around U a -Bounds for the linearized time-evolution
We fix δ * > 0 as in Lemma 5.7 for the rest of this paper. In this section we propagate Lemma 5.6 to L a . For that, given a ∈ B 9 δ * we define the Riesz projections where γ j (s) = λ j + ω 0 4 e 2πis for s ∈ [0, 1]. Lemma 6.1. We have that that ran H a = span (h a ), ran P a = span (g (0) a , . . . , g (9) a ), ran Q a = span (q (1) a , . . . , q (9) a ), for all a ∈ B 9 δ * . Moreover, the projections are mutually transversal, H a P a = P a H a = H a Q a = Q a H a = Q a P a = P a Q a = 0, and depend Lipschitz continuously on the parameter a, i.e., for all a, b ∈ B 9 δ * . Proof. The Riesz projections depend continuously on a, hence the dimensions of the ranges remain the same. Transversality follows from the definition of Riesz projections. The Lipschitz bounds follow from the second resolvent identity and Proposition 4.1.
Since P a and Q a are finite-rank, for every f ∈ H there are α k ∈ C and β j ∈ C, such that a , and Q a f = 9 j=1 β j q (j) a .
We thereby define the projections P (k) a f := α k g (k) a , and Q (j) a f := β j q (j) a . Clearly, the projections satisfy the following identities, and We also define By Lemma 6.1, we have that T a is Lipschitz continuous with respect to a, and the projections T a , H a , P (k) a , and Q (j) a are mutually transversal. Moreover, the Lipschitz continuity of Q a and P a with respect to a implies that |a − b|, j = 1, . . . , 9, and for all a, b ∈ B 9 δ * . In the following proposition we describe the interaction of the semigroup (S a (τ )) τ ≥0 with these projections. for j = 1, . . . , 9, k = 0, . . . 9, and τ ≥ 0. Furthermore,

2)
and there exists ω > 0 such that for all u ∈ H, a ∈ B 9 δ * and τ ≥ 0. Moreover, we have that for all a, b ∈ B 9 δ * and τ ≥ 0. Proof. Eq. (6.1) follows from the properties of the Riesz projections H a , P a and Q a . In particular, they commute with S a (τ ), and this yields, for example, that Eq. (6.2) follows from the correspondence between point spectra of a semigroup and its generator. Eq. (6.3) follows from Gearhart-Prüss Theorem. More precisely, we have that ran T a reduces both L a and S a (τ ), and furthermore R La| ran Ta (λ) exists in {z ∈ C : Re z ≥ − ω 0 2 } and is uniformly bounded there, i.e., according to Proposition 4.2 there exists c > 0 such that R La| ran Ta (λ) ≤ c for all Re λ ≥ − ω 0 2 and all a ∈ B 9 δ * . Hence, by Gearhart-Prüss theorem (see [17], page 302, Theorem 1.11), for every ε > 0 we have that for all a ∈ B 9 δ * and τ ≥ 0. From here Eq. (6.3) holds for any ω < ω 0 2 . We remark in passing that Eq. (6.3) also follows from purely abstract considerations, see [18], Theorem B.1. Finally, to obtain the estimate (6.4) we do the following. First, for u ∈ D(L a ) we define the function Note that this function satisfies the evolution equation 35 with the initial condition and therefore by Duhamel's principle we have Now, from Proposition 4.1 and Lemma (6.1) we get that and from this and Eq. (6.5) we obtain By choosing ε > 0 such that ω = ω 0 2 − 2ε > 0, we conclude the proof.

Nonlinear theory
With the linear theory at hand, in this section we turn to studying the Cauchy problem for the nonlinear equation (2.9). Following the usual approach of first constructing strong solutions, we recast Eq. (2.9) in an integral formà la Duhamel (where (S a∞ (τ )) τ ≥0 is the semigroup generated by L a∞ ), and resort to fixed point arguments.
Our aim is to construct global and decaying solutions to (7.1). An obvious obstruction to that is the presence of growing modes of S a∞ (τ ), see (6.2), and we deal with them in the following way. First, we note that the instabilities coming from Q a∞ and P a∞ are not genuine, as they are given rise to by the Lorenz and space-time translation symmetries of Eq. (1.1). We take care of the Lorenz instability by modulation. Namely, the presence of the unstable space ran Q a∞ is related to the freedom of choice of the function a : [0, ∞) → R 9 in the ansatz (2.8), and, roughly speaking, we prove that given small enough initial data Φ(0), there is a way to choose a such that it leads to a solution Φ of Eq. (7.1) which eventually (in τ ) gets stripped off of any remnant of the unstable space ran Q a∞ brought about by initial data.
With the rest of the instabilities, which cause exponential growth, we deal differently. Namely, we introduce to the initial data suitable correction terms which serve to suppress the growth. Also, as mentioned, the unstable space ran P a∞ is another apparent instability as it is an artifact of the space-time translation symmetries, and we use it to prove that the corrections corresponding to P a∞ can be annihilated by a proper choice of the parameters x 0 and T , which appear in the initial data Φ(0), see (2.6). The remaining instability, coming from H a∞ , is the only genuine one, and the correction corresponding to it is reflected in the modification of the initial data in the main result, see (1.16).
To formalize the process described above, we first make some technical preparations. For the rest of this paper, we fix ω > 0 from Proposition 6.2. Then, we introduce the following function spaces For a ∈ X, we define a ∞ := lim τ →∞ a(τ ).
Furthermore, for δ > 0 we set To ensure that all terms in Eq. (7.1) are defined, we must impose some size restriction on the function a. Note that is enough to consider a ∈ X δ for δ < δ * ω, as then |a(τ )| ≤ δ/ω < δ * for all τ ≥ 0. We will also frequently make use of the inequality Furthermore, note that for a, b ∈ X δ and τ ≥ 0 we have |a 7.1. Estimates of the nonlinear terms. With an eye toward setting up a fixed point scheme for Eq. (7.1), we now establish necessary bounds for the nonlinear terms. Namely, we treat for all Φ, Ψ ∈ X δ , a, b ∈ X δ , and τ ≥ 0, where the implicit constants in the above estimates are absolute.
Proof. First, since H 5 (B 9 ) is a Banach algebra we have that and hence for all u, v ∈ H. Next, we prove the second estimate in Lemma 7.1, as the first one follows from it. From Eq. (7.4), Proposition 4.1, and inequality (7.2) we obtain for Φ, Ψ ∈ X δ and a ∈ X δ . Furthermore, using the fact that with ϕ a,k (ξ) = ∂ a k V a (ξ), together with the smoothness of ϕ a,k we infer that for a, b ∈ X δ and Ψ ∈ X δ , and this, together with (7.5) concludes the proof.
7.2. Suppressing the instabilities. In this section we formalize the process of taming the instabilities. In particular, by introducing correction terms to the initial data we arrive at a modified equation, to which we prove existence of global and decaying solutions.
We first derive the so-called modulation equation for the parameter a.
Lemma 7.2. For all sufficiently small δ > 0 and all sufficiently large C > 0 the following holds. For every u ∈ H satisfying u ≤ δ C and every Φ ∈ X δ , there exists a unique a = a(Φ, u) ∈ X δ such that (7.8) holds for τ ≥ 0. Moreover, for all Φ, Ψ ∈ X δ and u, v ∈ B δ/C .
Proof. We use a fixed point argument. Using the bounds from Lemma 7.1, one can show that given u and Φ that satisfy the above assumptions, the following estimates hold for all a, b ∈ X δ . From here, according to the definition in (7.8) we have that for all small enough δ > 0 and all large enough C > 0, given Φ ∈ X δ and u ∈ B δ/C the ball X δ is invariant under the action of the operator A(·, Φ, u), which is furthermore a contraction on X δ . Hence, the equation (7.8) has a unique solution in X δ . The Lipschitz continuity of the solution map follows from the following estimate by taking small enough δ > 0.
For the remaining instabilities, we introduce the following correction terms and set C := C 1 + C 2 . Consequently, we investigate the modified integral equation Proposition 7.3. For all sufficiently small δ > 0 and all sufficiently large C > 0 the following holds. For every u ∈ H with u ≤ δ C there exist functions Φ ∈ X δ and a ∈ X δ such that (7.10) holds for τ ≥ 0. Furthermore, the solution map u → (Φ(u), a(u)) is Lipschitz continuous, i.e., for all u, v ∈ B δ/C .
Proof. We choose C > 0 and δ > 0 such that Lemma 7.2 holds. Then for fixed u ∈ B δ/C there is a unique a = a(Φ, u) ∈ X δ associated to every Φ ∈ X δ , such that the modulation equation (7.8) is satisfied. Hence we can define K u (Φ) := K(Φ, a, u). We intend to show that for small enough δ > 0 the operator K u is a contraction on X δ . To show the necessary bounds, we first split K u (Φ) according to projections P a∞ , Q a∞ , H a∞ , and T a∞ , and then estimate each part separately.
First, note that the transversality of the projections implies that for all a ∈ X δ . This, together with Lemma 7.1 and the fact that (see Eq. (7.7)) yields the following bounds δ C e −2ωτ (7.14) for all Φ ∈ X δ . On the other hand, for the stable subspace we have and by Lemma 7.1, Proposition 6.2, and Eq. (7.12), we get that (7.15) for all Φ ∈ X δ . Now, from (7.14) and (7.15) we see that K u maps X δ into itself for all δ > 0 sufficiently small and all C > 0 sufficiently large. The contraction property of K u is established similarly. Namely, there is the analogue of Eq. (7.12) for all a, b ∈ X δ . Furthermore, by Lemma 7.1, Eq. (7.13), and Lemma 7.2 we get the analogous estimates to (7.14), namely, we have that for all Φ, Ψ ∈ X δ , where a = a(Φ, u) and b = a(Ψ, u). Also in line with (7.15) we have that for all Φ, Ψ ∈ X δ . By combining these estimates we get that for all Φ, Ψ ∈ X δ , and contractivity follows by taking small enough δ > 0. For the Lipschitz continuity, similarly to proving Eq. (7.9), we use the integral equation (7.10) to show that given sufficiently small δ > 0, for all u ∈ B δ/C , and then Eq. (7.9) implies (7.11).
7.3. Conditional stability in similarity variables. According to Proposition 7.3 and Eq. (2.8) there exists a family of initial data close to U 0 which lead to global (strong) solutions to Eq. (2.3), which furthermore converge to U a∞ , for some a ∞ close to a = 0; with minimal modifications, the same argument can be carried out for U a for any a = 0. In conclusion, we have conditional asymptotic orbital stability of the family {U a : a ∈ R 9 }, the condition being that the initial data belong to the set which ensures global existence and convergence. In this section we show that this set represents a Lipschitz manifold of co-dimension eleven.
Let δ > 0 and C > 0 be as in Proposition 7.3, and let u ∈ B δ/C . Also, let us denote where the mapping u → (Φ(u), a(u)) is defined in Proposition 7.3. Moreover, we denote the projection corresponding to all unstable directions by Note that by definition J a∞ C(u) = C(u), and we have the Lipschitz estimate Furthermore, for every u ∈ M there exists (Φ, a) = (Φ u , a u ) ∈ X δ × X δ satisfying the equation for all τ ≥ 0. Moreover, there exists K > C such that u ∈ B δ/K C(u) = 0 ⊂ M δ,C .
Proof. First, we show that for small enough δ > 0, C(u) = 0 if and only if J 0 C(u) = 0. Assume that J 0 C(u) = 0. Then we obtain the estimate Since a u,∞ = O(δ), we get C(u) = 0. The other direction is obvious. Now we construct the mapping M. Let u ∈ H and decompose it as u = v + w ∈ ker J 0 ⊕ ran J 0 . Fix v ∈ ker J 0 and defineC v : ran J 0 → ran J 0 ,C v (w) = J 0 C(v + w). We establish that this mapping is invertible at zero, provided that v is small enough, and we obtain w =C −1 v (0). This defines a mapping M : To show this, we use a fixed point argument. Recall the definition of the correction terms C = C 1 + C 2 , C 1 = 9 k=0 C k with C k 1 (Φ, a, u) = P (k) a∞ u + P (k) a∞ I 1 (Φ, a), and where We denote and F 2 (u) := H a∞ I 2 (Φ u , a u ).
By Lemma 7.1 and Eq. (7.12) we infer that Introducing the notation we rewrite equationC v (w) = 0 as w = Ω v (w). (7.19) Now, for δ > 0 and C > 0 from Proposition 7.3, we set We show that Ω v :B δ/C (v) →B δ/C (v) is a contraction mapping for sufficiently small v.
Let v ∈ H with v ≤ δ 2C , and let w ∈B δ/C (v). Using the (7.18), we estimate Hence, by fixing C > 0 we have that v+Ω v (w) ≤ δ C for all small enough δ > 0. So the ball B δ/C (v) is invariant under the action of Ω v . To prove contractivity, first for w,w ∈B δ/C (v), we associtate to v+w and v+w the functions (Φ, a) and (Ψ, b) in X δ ×X δ by Proposition 7.3. Then we obtain and writing we get by Proposition 7.3 the following estimate On the other hand, by Lemma 7.1 and Eq. (7.12) we obtain for k = 0, . . . , 9 that and Thus we get the following Lipschitz estimate: and we conclude that for all small enough δ > 0 the operator Ω v :B δ/C (v) →B δ/C (v) is contractive, with the contraction constant 1 2 . Consequently, by the contraction map principle we get that for every v ∈ ker J 0 ∩ B δ/2C there exists a unique w ∈B δ/C (v) that solves (7.19), hence C(v + w) =C v (w) = 0. Next, we establish the Lipschitz-continuity of the mapping v → M(v). Let v,ṽ ∈ ker J 0 ∩ B δ/2C , and w,w ∈B δ/C be the corresponding solutions to (7.19). We get The second term we estimate with Thereby we obtain the claimed Lipschitz estimate We note that for u = 0, the associated (Φ, a) is trivial, i.e., Φ = 0 and a = 0. Thus, we have C(0) = F 1 (0) + F 2 (0) = 0. Moreover, u = v + w = 0 if and only if v = w = 0. Since in this case v satisfies the smallness condition, w solving C(0 + w) = 0 is unique, hence M(0) = 0. Finally, let u ∈ H satisfying C(u) = 0. Then, since 1 − J 0 is a bounded operator on H, We obtain v u := (1 − J 0 )u ∈ ker J 0 and v u ≤ δ 2C for u ≤ δ K for K > C large enough. Uniqueness yields w u := J 0 u = M(v u ), hence u ∈ M δ,C .
Remark 7.1. For each correction term, the same argument yields the existence of Lipschitz manifolds M 1 , M 2 ⊂ H of co-dimension ten, respectively, one, characterized by the vanishing of C 1 and C 2 . In particular, in a small neighborhood around zero, M can be characterized as a subset of the intersection M 1 ∩ M 2 . Proof of Proposition 2.1. Let Φ 0 ∈ M δ,C , where M δ,C is the manifold defined in Proposition 7.4. In particular, Φ 0 ≤ δ C and C(Φ 0 ) = 0. By Proposition 7.4 there is a pair (Φ, a) ∈ X δ × X δ which solves equation (7.17) with initial data u = Φ 0 . Furthermore, after substituting the variation of constants formula into Eq. (7.17), a straightforward calculation yields that Ψ(τ ) := U a(τ ) + Φ(τ ) satisfies for all τ ≥ 0. Then, based on (4.3) and (7.2) we infer that for all τ ≥ 0, as claimed.
Let v ∈ C ∞ (B 9 2 ). Let T ∈ 1 2 , 3 3 and x 0 , y 0 ∈ B 9 1/2 . Then we get by the fundamental theorem of calculus that This implies that v(T · +x 0 ) − v(T · +y 0 ) L 2 (B 9 ) v H 1 (B 9 2 ) |x 0 − y 0 |. The same argument yields for all k ∈ N that is obtained, most results are purely abstract and the proofs can be adapted from previous sections.
Since L ′ κa is compact and depends Lipschitz continuously on a, the results of Sec. 4 apply. In particular, for small enough a, the spectrum of L + L ′ κa in the right half plane consists of isolated eigenvalues confined to a compact region. Furthermore, an analogous result to Proposition 4.3 holds with V replaced a constant. This substantially simplifies the spectral analysis and with the above prerequisites it is easy to derive the following statement. For all of the ensuing statements d ∈ {7, 9}.
Proof. We only sketch the main steps of the proof, since many parts are abstract operator theory and can be copied verbatim from previous sections.
This can only be the case if a or b are a poles of the gamma function, i.e., −a ∈ N 0 or −b ∈ N 0 . In particular, this implies that λ ∈ R. Since Re λ ≥ 0, −b ∈ N 0 can be excluded. On the other hand, −a ∈ N 0 is possible only if λ ∈ {0, 1}, which contradicts our assumption and proves (8.5). For λ = 1 and λ = 0, one can easily check that explicit solutions to the eigenvalue equation are given byg 0 andq (k) 0 , respectively, wherẽ for k = 1, . . . , d. Similar to the above reasoning one shows that these functions indeed span the eigenspaces for the corresponding eigenvalues, i.e., the geometric multiplicities of λ = 1 and λ = 0 are 1 and d, respectively. The algebraic multiplicities are determined by the dimension of the ranges of the corresponding Riesz projections where for j ∈ {0, 1}, γ j (s) = λ j + ω 0 4 e 2πis for s ∈ [0, 1]. An ODE argument analogous to the proof of Lemma 5.6 shows that ranP 0 = span(g 0 ), ranQ 0 = span(q

53
The perturbative characterization of the spectrum of L + L ′ κa for a ∈ B d δ * is purely abstract. Along the lines of the the proof of Lemma 5.7, one shows that σ(L + L ′ κa ) ⊂ λ ∈ C : Re λ < − 2πi γ 0 R L+L ′ κa (λ)dλ follows again from the same abstract arguments as provided in the proof of Lemma 6.1. The same holds for the Lipschitz dependence of the projections on the parameter a. The growth bounds for the semigroup follow from the structure of the spectrum, resolvent bounds and the Gearhart-Prüss theorem analogous to the proof of Theorem 6.2. Finally, the proof for the Lipschitz continuity (8.2) can be copied verbatim.
The analysis of the integral equation is completely analogous to Sec. 7. In particular, to derive the modulation equation for a, one uses the fact that ∂ τ κ a(τ ) =ȧ k (τ )q (k) 0 . By introducing the correctioñ it is straightforward to prove the following result.
We note that similarly to the manifold M one can construct a manifold N ⊂ kerP 0 ⊕ ranP 0 of co-dimensions (1+d) characterized by the vanishing of the correction termC. However, in the context of stable blowup this is not of much interest, since the existence of this manifold is solely caused by the translation instability. In particular, no correction of the physical 54 initial data is necessary, if blowup time and point are chosen appropriately, i.e., for suitably small (f, g), there are T , x 0 , such that Υ(v, T, x 0 ) ∈ N . This is contained in the following result, where Y = H d+3 2 (B d ) × H d+1 2 (B d ).
Lemma 8.3. There exists C > 0 such that for all sufficiently small δ > 0 the following holds. For every v ∈ Y satisfying v Y ≤ δ C 2 , there is a choice of parameters T ∈ 1 − δ C , 1 + δ C and x 0 ∈ B d δ/C in Proposition 8.2 such that C(Φ, a, Υ(v, T, x 0 )) = 0.
Moreover, the parameters depend Lipschitz continuously on the data, i.e., for all v, w ∈ Y satisfying the above smallness assumption.
The proof is along the lines of the proof of Lemma 7.6 above with obvious simplifications. With these results, in combination with persistence of regularity that is completely analogous to Proposition 2.2, Theorem 1.2 is obtained by the same arguments as in the above Sec. 7.6.
Analogous calculations for d = 7 and k ≥ 3 yield an even better bound, namely for all u ∈ D(L) from which we obtain in particular the claimed estimate. Another way to see that (A.8) holds is by Lemma 3.2 of [21], which is formulated in terms of above inner product for the specific case d = 7, k = 3. The operator considered there corresponds toL shifted by a constant, which immediately gives the inequality (A.8).