Construction of type I blowup solutions for a higher order semilinear parabolic equation

We consider the higher-order semilinear parabolic equation $$ \partial_t u = -(-\Delta)^{m} u + u|u|^{p-1}, $$ in the whole space $\mathbb{R}^N$, where $p>1$ and $m \geq 1$ is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by [Galaktionov, rspa2011], we revisit the technique developed by [Merle-Zaag, duke1997] for the classical case $m = 1$, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [Galaktionov, rspa2011].


Introduction
We are interested in the semilinear parabolic equation From Fujita [12] (m = ) and Galaktionov-Pohozaev [14] (m > ), we know that is the critical Fujita exponent for the problem in the following sense. If p > p F , for any su cient small initial data u ∈ L (R N ) ∩ L ∞ (R N ) the Cauchy problem (1.1) admits a global solution satisfying u(t) → as t → +∞ uniformly in R N . If p ∈ ( , p F ] and the initial data u ≢ with R N u dx ≥ for the nonlinearity u|u| p− replaced by |u| p or u ≥ , then the corresponding solution to problem (1.1) blows up in some nite time T > , namely that Here T is called the blowup time, and a point a ∈ R N is called a blowup point if and only if there exists a sequence (an , tn) → (a, T) such that |u(an , tn)| → +∞ as n → +∞. A solution of (1.1) is called Type I blowup if it satis es otherwise, it is of Type II blowup. In addition, we call a blowup solution self-similar if it is of the form where Φ is not identically constant. Obviously, the self-similar blowup solution is of Type I. When m = , problem (1.1) reduces to the classical semilinear heat equation which has been extensively studied in the last four decades, and no rewiew can be exhaustive. Given our interest in the construction of solutions with a prescribed blowup behavior, we only mention previous work in this direction. The rst conctructive result was given by Bricmont-Kupiainen [5] who showed the existence of type I blowup solution to equation (1.5) according to the asymptotic dynamic for some universal positive constant κ = κ(p). Note that the authors of [5] also exhibited nite time blowup solutions that verify other asymptotic behaviors which are expected to be unstable. Note also that Bressan [3,4] made a similar construction in the case of an exponential nonlinearity. Later, Merle-Zaag [23] suggested a modi cation of the argument of [5] and obtained the stability of the constructed solution verifying (1.6) under small perturbations of initial data. The stability of the asymptotic behavior (1.6) had been observed numerically by Berger-Kohn [2] (see also Nguyen [24] for other numerical analysis). In particular, Herrero-Velázquez [19] proved that the blowup dynamic (1.6) is generic in one dimensional case, and they announced the same for higher dimensional case (but never published it). The method of [5] and [23] relies on the understanding of the spectral property of the linearized operator around an expected pro le in the similarity variables setting. Roughly speaking, the linearized operator possesses a nite number of positive eigenvalues, a null eigenvalue and a negative spectrum; then they proceed in two steps: • Reduction of an in nite dimensional problem to a nite dimensional one in the sense that the control of the error reduces to the control of the components corresponding to the positive eigenvalues. • Solving the nite dimensional problem thanks to a classical topological argument based on the index theory.
This general two-step procedure has been extended to various situations such as the case of the complex Ginzgburg-Landau equation by Masmoudi-Zaag [21], Nouaili-Zaag [27] (see also Zaag [31] for an earlier work); the complex semilinear heat equation with no variational structure by Duong [7], Nouaili-Zaag [26]; non-scaling invariant semilinear heat equations by Ebde-Zaag [9], Nguyen-Zaag [25], Duong-Nguyen-Zaag [8]. We also mention the work of Tayachi-Zaag [29,30] and Ghoul-Nguyen-Zaag [16] dealing with a nonlinear heat equation with a double source depending on the solution and its gradient in some critical setting. In [17,18], we successfully adapted the method to construct a stable blowup solution for a non variational semilinear parabolic system.
As for the present paper, we aim at extending the above mentioned method to construct for problem (1.1) nite time blowup solutions satisfying some prescribed asymptotic behavior. Although the general idea is the same as for the classical case (1.5), we would like to emphasis that the above mentioned strategy is heavy and its implementation never being straightforward, the context and di culties are di erent for each speci c problem. As a step forward to better understanding the blowup dynamics for (1.1), we obtain the following result. . (1.10)

Remark 1.2.
We believe that such a blowup pro le (1.8) exists for all m ∈ N * . We note that the constant Bm,p < when m is even (see (1.9)), so the pro le Φ blows up on the nite interface |ξ | = ξ * = − Bm,p − m . This says that the case when m is even would lead to type II blowup solutions in the sense of (1.3). Although main ideas for a full justi cation of such a blowup behavior with m even remains the same, the proof would be very delicate and will be addressed in a separate work. Remark 1.3. The blowup solution described in Theorem 1.1 is not self-similar in the sense of (1.4). Note that in contrast to blowup solutions of the classical second order semilinear heat equation (1.5), Budd-Galaktionov-Williams [6] through numerical and asymptotic calculations conjectured that there are at least m nontrivial self-similar blowup solutions to (1.1), and that pro les having a single maximum correspond to stable (generic) self-similar blowup solutions. Remark 1.4. The proof of Theorem 1.1 (in dimension N = for simplicity) involves a detailed description of the set of initial data leading to the asymptotic dynamic (1.7). In particular, our initial data is roughly of the form (see formula (3.13) below) where A and s are large xed constants, Φ is the pro le de ned by (1.8), χ is some smooth cut-o function, d , · · · , d m− ∈ R m are free parameters, ψ k , ≤ k ≤ m − are the eigenfunctions of the linearized operator (see Proposition 2.1 for a precise de nition) corresponding to the positive eigenvalue λ k = − k m . Through a topological argument, we show that there exists a suitable choice of parameters d , · · · , d m− such that the solution to equation (1.1) with the initial datum u satis es the conclusion of Theorem 1.1. In some sense, our constructed solution is (m − )-codimension stable in the following sense. The m components of the linearized solution corresponding to λ k = − k m have the exponential growth e λ k s . However, the rst two modes corresponding to λ and λ can be eliminated by means of the time and space translation invariance of the problem. Hence, by xing (m − ) directions ψ , · · · , ψ m− and perturbing the remaining components (in L ∞ ), we still obtain the same asymptotic dynamic (1.7) of the perturbed solution. The proof of (m − )-codimenison stability would require some Lipschitz regularity of the considered initial data set and it would be addressed separately in another work. Remark 1.5. According to our construction, the asymptotic dynamic (1.7) lies on the center manifold generated by eigenfunctions corresponding to the null eigenvalue λ m = . Our analysis can be extended to construct for equation (1.1) a nite time blowup solution having a di erent asymptotic dynamic from (1.7). Such solutions particularly have asymptotic dynamics laying on the stable manifold generated by eigenfunctions corresponding to the negative eigenvalue λ k = − k m < with k ≥ m + . As explained in Remark 1.4, the corresponding initial data leading to such solutions would involve k parameters with k ≥ m + (consider N = ), so that a topological argument is assigned in order to control the rst k components corresponding to the eigenvectors ψ j for ≤ j ≤ k − . Although the constructive method are similar for all cases, we decide to only deal with the case of the center manifold, since the proof is the most delicate in the sense that it requires a more re ned analysis in the blowup region leading to some logarithmic correction of the blowup variable as shown in (1.7).

Strategy of the proof.
Let us sketch the main ideas of the proof of Theorem 1.1.
-Similarity variables and linearized problem. According to the scaling invariance of the problem (1.1), we introduce the similarity variables (1.11) which leads to the new equation where the linear operator Lm is given by It is then natural to linearize equation (1.12) around the expected pro le Φ by introducing which leads to the equation of the form ∂s q = Lm + V(y, s) q + B(q) + R(y, s), (1.15) where B(q) is built to be quadratic, R measures the error generated by Φ and is uniformly bounded by O(s − ), and V is the potential de ned as Our goal is to construct for equation (1.15) a solution q de ned for all (y, s) ∈ R N × [s , +∞) such that sup y∈R N |q(y, s)| → as s → +∞.
-Properties of the linearized operator. In view of equation (1.15), we see that the nonlinear quadratic and the error term are small and can be negligible in comparison with the linear term. Roughly speaking, the linear part will play an important role in the dynamic of the solution. It is essential to determine the spectrum and corresponding eigenfunctions of both Lm and its adjoint L * m . According to [15], the spectrum of the linear operator Lm comprises real simple eigenvalues only, , β = (β , · · · , βn) ∈ N n , |β| = β + · · · + βn , and the corresponding eigenfunction ψ β with |β| = n is polynomial of order n (see Proposition 2.1 below). Moreover, the family of the eigenfunctions {ψ β } β∈N n forms a complete subset in L ρ (R N ) where ρ is some exponentially decaying weight function.
Depending on the asymptotic behavior of the potential V, we observe that • Inside the blowup region, |y| ≤ Ks m for some K large, the e ect of V is regarded as a perturbation of Lm.

•
Outside the blowup region, |y| ≥ Ks m , the full linear part Lm + V behaves like Lm − p p− , which has a purely negative spectrum. Hence, the control of the solution in this region is simple.
-Decomposition of the solution and reduction to a nite dimensional problem. According to the spectrum of Lm, we decompose where q− is the projection of q on the subspace of Lm corresponding to strictly negative eigenvalues. Since the spectrum of the linear part of the equation satis ed by q− is negative, it is controllable to zero. We would like to notice that we do not use the Feymann-Kac representation¹ as for the case m = treated in [23], because of its complicated implementation for higher order cases m ≥ . To avoid such a formula, we further decompose In [5] and [23], the kernel K of the heat semigroup associated to the linear operator L + V is de ned through the Feymann-Kac where e tL (y, x) is given by Mehler's formula and dµ t y,x is the oscillator measure on the continuous path: ω : from which we obtain the rough bound |θ β (s)| = O(s − |β|+ m ) for |β| = m + , · · · , M. For the in nite part q M,⊥ , we explore the properties of the semigroup e sLm and a standard Gronwall inequality to close the estimate for this part.
The control of the null mode q m is delicate since the potential has in some sense a critical size in our analysis. In particular, we need a careful re nement of the asymptotic behavior of V to derive the sharp ODE which shows a negative spectrum after changing the variable τ = ln s, hence, the rough bound Here the precise value of Bm,p given in (1.9) is crucial for many algebraic identities to derive this sharp ODE. At this stage, we reduce the in nite dimensional problem to a nite dimensional one in the sense that it remains to control a nite number of positive modes q β for |β| ≤ m − . This is done through a classical topological argument based on the index theory.
The rest of the paper is organized as follows: In Section 2 we recall basic spectral properties of the linearized operator L and its adjoint L * , then we perform a formal spectral analysis to derive an approximate blowup pro le served for our analysis later. In Section 3 we give all arguments for the proof of Theorem 1.1 without going to technical details (the reader who is not interested in the technical details can stop at this section). Section 4 is the hearth of our analysis: it is devoted to the study of the dynamic of the linearized problem from which we reduce the problem to a nite dimensional one.

Acknowledgments:
The authors would like to thank the anonymous referees for their careful reading and suggestions to improve the presentation of the paper.

A formal approach via spectral analysis.
In this section we rst recall basic spectral properties of the linearized operator, then present a formal approach based on a spectral analysis to derive the blowup pro le given in Theorem 1.1.

Spectral properties of the linearized operator.
In this subsection, we recall from [15] the basic spectral properties of the linear operator Lm and its adjoint L * m . The case m = is well known, since we can rewrite which is a self-adjoint operator in the weighted Hilbert space L (e −|y| / dy) with the domain D(L) = H (e −|y| / dy). It has a real discrete spectrum and the corresponding eigenfunctions are derived from Hermite polynomials. For m ≥ , the operator Lm is not symmetric and does not admit a self-adjoint extension. Then we denote L * m the formal adjoint of Lm as From [10] and [11], we know that the following elliptic equation has a unique radial solution given by the explicit formula with Jν being the Bessel function. In particular, F satis es the estimate for some positive constants D and d depending on m and N. We introduce the weight functions where a = a(m, N) ∈ ( , d] is a small constant, d and ν are introduced in (2.5).
Proposition 2.1 (Spectral properties of Lm and L * m , [15]). Let m ∈ N * , we have is a bounded linear operator with the spectrum

Remark 2.2.
We note from the orthogonality (2.11) and the de nition of ψ k that for all polynomials Pn(y) of degree n < |γ|, we have Pn , ψ * γ = .
For N = , we have We end this subsection by recalling basic properties of the semigroup e sLm for s > .

Lemma 2.3 (Properties of the semigroup e sLm
). The kernel of the semigroup e sLm is given by where F is de ned as in (2.4). The action of e sLm is de ned by We also have the following estimates: where Π M,⊥ is de ned as in (3.10).
Proof. The formula (2.12) can be veri ed by a direct computation thanks to equation (2.3). The estimates (i)-(iii) are straightforward from the de nitions (2.12) and (2.13).

Approximate blowup pro le.
In this subsection we recall the formal approach of [13] (see also [6]) to gure out an appropriate blowup pro le for our analysis later. This approach had been used in several problems involving the second order Laplacian, see for example [5], [30], [16][17][18]. The argument relies on the basis of the known spectral properties of the rescaled operator Lm and its adjoint L * m given in the previous subsection. For simplicity, we consider the one dimensional case and symmetric positive solutions.
where κ = (p − ) − p− is the constant equilibrium to equation (1.12). This yields the following perturbed equation where |R(w)| ≤ C|w| for |w| . From Proposition 2.1, we know that ψn with n ≥ m + correspond to negative eigenvalues of Lm. Therefore, we may considerw where m i= |w i (s)| → as s → +∞. Plugging this ansatz to equation (2.15), taking the scalar product with ψ * j and using Proposition 2.1, we nd that for ≤ j ≤ m, Assuming thatw m is dominant, i.e.
Solving this system yields which is in agreement with the assumption (2.18). Hence, from (2.16) and the de nition of ψ m , we derive the following asymptotic behavior where the convergence takes place in L ρ (R) as well as uniformly in compact sets by standard parabolic regularity.
Plugging this ansatz to equation (1.12) and comparing the leading order terms, we arrive at Solving this ODE yields Φ(z) = κ + Bm,p z m − p− for some Bm,p ∈ R.
By matching expansions (2.19) and (2.20), we nd that Let us computeμ m in the following. Using (2.8), we write Using the orthogonality relation (2.11), the de nition (2.10) of ψ * m and the fact that R F(y)dy = , we compute by an integration by parts, . (2.22) In conclusion, we have derived the following candidate for the blowup pro le in the similarity variables: where Since we want the pro le φ bounded, this requests Bm,p > orμ m > , which only happens for m odd.

Proof of Theorem 1.1 without technical details.
In this section we give all arguments of the proof of Theorem 1.1. We only deal with the one dimensional case N = for simplicity since the analysis for the higher dimensional cases N ≥ is exactly the same up to some complicated calculation of the projection of (1.15) on the eigenspaces of Lm. The proof of Theorem 1.1 is completed in three parts: • In the rst part we formulate the problem by linearizing the rescaled equation (1.12) around the approximate pro le φ given by (2.23). We also introduce a shrinking set in which the constructed solution of the linearized equation is trapped. • In the second part we exhibit an explicit formula of the initial data and show that the corresponding solution belongs to the shrinking set. In particular, the reader can nd how to reduce the problem to a nite dimensional one (all technical details will be left to the next section) and the use of a topological argument based on index theory to conclude. • The last part is devoted to the proof of items (i) and (iii) of Theorem 1.1. Although the argument of the proof is almost the same as for the classical case m = , we would like to sketch the main ideas for the reader convenience.

Formulation of the problem.
According to the formal analysis given in Section 2.2, we introduce q(y, s) = w(y, s) − φ(y, s), (3.1) and write from (1.12) the equation driving by q, where Lm is the linearized operator de ned by (1.13) and where q k , q M,⊥ and qe are de ned as in the decomposition (3.8) and (3.7).

Existence of solutions trapped in V A .
In this step we aim at proving that there actually exists initial data ϕ(y) = q(y, s ) such that the corresponding solution q(y, s) to (1.15) belongs to the shrinking set V A (s). Given A ≥ and s ≥ e, we consider initial data of the form where d = (d , d · · · , d m− ) ∈ R m are real parameters to be xed later, χ is introduced in (3.6). In particular, the initial data (3.13) belongs to V A (s ) as shown in the following proposition.

Proposition 3.3 (Properties of initial data (3.13)). For each A , there exist s = s (A)
and a cuboid Ds ⊂ [−A, A] m such that for all d , · · · , d m− ∈ Ds , the following properties hold: The initial data ϕ A,s ,d de ned in (3.13) belongs to V A (s ) with strict inequalities except for the positive The map Γ : Ds → R m , de ned as Γ(d , · · · , d m− ) = ϕ , · · · , ϕ m− ), is linear, one to one from Ds to − As − , As − m , and maps ∂Ds into ∂ − As − , As − m . Moreover, the degree of Γ on the boundary is di erent from zero.
Proof. The proof directly follows from the de nition of the projection Π k (ϕ A,s ,d ) and it is straightforward, so we omit it here.
Starting with initial data ϕ A,s ,d belonging to V A (s ), we claim that we can ne-tune the parameters d , · · · , d m− ∈ Ds such that the corresponding solution q(s) to equation (1.15) stays in V A (s) for all s ≥ s . More precisely, we claim the following.

Proposition 3.4 (Existence of solutions trapped in V A (s)). There exist A , s = s (A) and d , · · · , d m− ∈ Ds such that if q(s) is the solution to equation (1.15) with initial data (3.13), then q(s) ∈ V A (s) for all s ≥ s (A).
Proof. From the local Cauchy problem of (1.1) in L ∞ (R), we see that for each initial data ϕ A,s ,d ∈ V A (s ), equation (1.15) has a unique solution q(s) ∈ V A (s) for all s ∈ [s , s * ) with s * = s * (d). If s * = +∞ for some d = (d , · · · , d m− ) ∈ Ds , we are done. Otherwise, we proceed by contradiction and assume that s * (d) < +∞ for all d ∈ Ds . By continuity and the de nition of s * , we remark that q(s * ) ∈ ∂V A (s * ). We claim the following. (Transversality) There exists µ > such that q(s + µ) ∉ V A (s + µ) for all µ ∈ ( , µ ).
Let us postpone the proof of Proposition 3.5 to the next section and continue our argument. From (i) of Proposition 3.5, we obtain q , · · · , q m− (s * ) ∈ ∂ [−As − * , As − * ] m , and the following mapping is well de ned From the transversality given in item (ii) of Proposition 3.5, q , · · · , q m− actually crosses its boundary at s = s * , resulting in the continuity of s * and Θ. Applying again the transversality, we see that if (d , · · · , d m− ) ∈ ∂Ds , then q(s) leaves V A (s) at s = s , thus, s * = s and Θ |∂Ds = Γ, the map de ned in item (ii) of Proposition 3.3. Using that item, we see that the degree of Θ is not zero. Since Θ is continuous, this is a contradiction. This concludes the proof of Proposition 3.4, assuming that Proposition 3.5 holds. The existence of the nal blowup pro le u * ∈ C ∞ (R \ { }) follows from the technique of Merle [22]. Here we focus on a precise description of the nal blowup pro le u * in a neighborhood of the singularity. To do so, we follow the technique of Herrero-Velázquez [20] (see also , Zaag [31] for a similar approach) by introducing the auxiliary function , t), (3.14) where

Conclusion of Theorem 1.1.
and t (x ) is uniquely determined by We note that h(ξ , τ; x ) is also a solution to (1.1) because of the invariance of (1.1) under dilations. From (1.7), we have sup Letĥ K (τ) be the solution to (1.1) with the constant initial datum Φ(K), de ned aŝ By the continuity with respect to initial data for equation (1.1), one can show that Passing to the limit τ → yields From the de nition (3.15), we compute from which we obtain the asymptotic behavior This concludes the proof of Theorem 1.1, assuming that Proposition 3.5 holds.

Reduction to a nite dimensional problem.
This section is the central part in our analysis where we give all details of the proof of Proposition 3.5, completing hence the proof of Theorem 1.1. The essential idea is to project equation (1.15) onto di erent components according to the decomposition (3.8). In particular, we claim that Proposition 3.5 is a direct consequence of the following.
(ii) (Control of the in nite dimensional and outer part) q M,⊥ (y, s) for A large enough. Hence, estimates (4.7) and (4.8) hold for s − s ≤ λ.
Part (ii) of Proposition 3.5 is a direct consequence of the dynamics of q k given in (4.1) and (4.6). Indeed, from part (i) of Proposition 4.1, we know that q k (s ) = ϵ A s for some k ∈ { , · · · , m − } and ϵ = ± . Using estimate (4.1) yields Thus, for ≤ k ≤ k m and A large enough, we have ϵq k (s ) > . Therefore, q k is traversal outgoing to the boundary curve s → ϵAs − at s = s . This concludes the proof of Proposition 3.5, assuming that Proposition 4.1 holds.
We now give the proof of Proposition 4.1 to complete the proof of Proposition 3.5. We divide the proof into two subsections according to the two parts of Proposition 4.1.

Control of the nite dimensional part.
We prove item (i) of Proposition 4.1 in this part. A direct projection of equation (1.15) on the eigenfunction ψ k Estimate of Π k (Vq).
We claim the following. where V j 's are even polynomial of degree mj. More precisely, we have Proof. Estimate (4.10) is trivial. Estimate (4.11) follows from a Taylor expansion in the variable z = |y| m s . Formula (4.12) comes from the de nitions (2.23) and (2.8) of φ and ψ m . From Lemma 4.3, we obtain the following estimate for Π k (Vq).
Proof. By De nition 3.1, we have Using this, the estimate (4.10) and noting from the de nition (2.10) and (2.5) that ψ * k is exponentially decaying, we obtain for ≤ k ≤ m − , For m + ≤ k ≤ M, we write from the decomposition (3.8), Using (4.10) and the bound of q M,⊥ given in De nition 3.1 yields From the orthogonality (2.11), we note that Pn , ψ * k = for all polynomial Pn of degree n ≤ k − . We then use (4.11) and (4.13) to estimate The last term is bounded by O(e −cs ) because of the exponential decay of ψ * k . By taking n ∈ N * such that n + + m ≥ k+ m , the second term is bounded by O A M+ s − k+ m . Using the fact that V j ψ i is polynomial of degree mj + i, we see that V j ψ i , ψ * k = for mj + i ≤ k − . Combining this with the bounds of q i given in De nition 3.1 of V A , we obtain the rough estimate As for k = m, we need to use the precise de nition (4.12) of V and process similarly as for k ≥ m + . Indeed, from decomposition (3.8) and expansion (4.11) with n = , we have Using the de nition (4.12) of V , the bound of q i given in De nition 3.1 and the fact coming from the de nition (2.8) of ψ β and the orthogonality (2.11) that This concludes the proof of Lemma 4.4.
We now turn to the estimate of the main contribution of the nonlinear term B(q) under the projection Π k . We begin with the following expansion. With Lemma 4.5 at hand, we estimate the main contribution of B(q) under the projection Π k . We claim the following. Π k B(q) ). Under the assumption of Proposition 4.1, we have

Lemma 4.6 (Estimate for
Proof. From (4.14), the exponential decay of ψ * k and (4.13), we have We now deal with the generated error term R. We begin with the following expansion. where R j 's are polynomials of degree mj of the form R j (y) = j i= d i y mi . Moreover, the coe cient of degree m of R is identically zero, hence, R , ψ * m = .
Proof. Let z = ys − m and note that Φ(z) satis es equation (2.21). Therefore, we rewrite (3.5) as follows: where Q(h) = h p . Since c ≤ Φ(z) ≤ C for all |z| ≤ with c , C some positive constants, we have the following expansion of Q, Then, we expand Q j and all the remaining terms in (4.17) in power series of Z = z m to obtain the desired result. Note that the coe cient of s in the expansion of R (after an elementary computation) is given by where we used (2.24). Moreover, R (y) = C y m + C , where C = C (p, m) and Again, the precise values of Bm,p and Am,p given in (2.24) are crucial in deriving that C is identically zero. Therefore, the orthogonality relation (2.11) yields R , ψ * m = . This concludes the proof of Lemma 4.7.
As a direct consequence of Lemma 4.7, we have the following.
Proof. The proof directly follows from the expansion (4.16) and the fact that For k ∈ N with (k mod m) ≠ , we use the expansion (4.16) with n = M − (we can replace M by any positive integer L ) and write For (k mod m) ≠ , i.e. k = mi for some i ∈ N, we use (4.16) with n = i + to get the conclusion. This ends the proof of Lemma 4.8.
A collection of all estimates given in Lemmas 4.4, 4.6, 4.8 and equation (4.9) yields the conclusion of part (i) of Proposition 4.1.

Control of the in nite dimensional and the outer part.
We prove item (ii) of Proposition in this part. We rst deal with the in nite dimensional part q M,⊥ , then the outer part qe.

Control of q M,⊥ :
Applying Π M,⊥ to equation (1.15) and using the fact that Π M,⊥ ψn = for all n ≤ M (see (3.10)), we obtain As for the estimate of Π M,⊥ (Vq), we claim the following. Concerning the control of Π M,⊥ (B(q)), we claim the following. from which we conclude the proof of (4.4).

Control of qe.
We write from (1.15) the equation satis ed by qe = ( − χ(y, s))q, where R is de ned by (3.5), For K large enough, we have Recall from Lemma 4.7 that R(s ) L ∞ ≤ C s . As for E , we use the de nition of χ given in (3.6) and the bounds given in De nition 3.1 to obtain the estimate