THE SURFACE DIFFUSION FLOW ON ROUGH PHASE SPACES

The surface diffusion flow is the gradient flow of the surface functional of compact hypersurfaces with respect to the inner product of H−1 and leads to a nonlinear evolution equation of fourth order. This is an intrinsically difficult problem, due to the lack of an maximum principle and it is known that this flow may drive smoothly embedded uniformly convex initial surfaces in finite time into non-convex surfaces before developing a singularity [12, 13]. On the other hand it also known that singularities may occur in finite time for solutions emerging from non-convex initial data, cf. [8]. Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove wellposedness on a large (Besov) space of initial data which is in general larger than C2 (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.

1. Introduction. In this paper we study the geometric evolution equation (1.1) subject to the initial condition Here M (t) denotes an unknown surface, V n is the normal velocity of M (t), ∆ M (t) and H M (t) are the Laplace-Beltarmi operator and the mean curvature, respectively, of the surface M (t). The normal vector and the mean curvature depend on the choice of the orientation of M 0 , however we are free to choose whichever we like. Equation (1.1) was first proposed by Mullins [19] to describe thermal grooving in material sciences. It arises also in modelling viscous sintering [15] and J. W. Cahn et al. derived it as a singular limit of the Cahn-Hilliard equation with a concentration dependent mobility [3]. From the point of view of geometric analysis it appears naturally as the gradient flow of the surface functional with respect to the inner product of H −1 , cf. [21]. Local well-posedness in the classical sense, as well as intersting dynamic properties of this flow such as loss of convexity, loss of embeddedness, and formation of singularities have been discussed in [4,6,7,8,9,11,12]. In this paper we offer a further insight into the dynamic picture of (1.1) by constructing an as large as possible phase space on which weak solutions in the sense of distribution still make sense. This effort will be rewarded by the fact that we have to control a priori less than two derivatives of solutions to (1.1) in order to guarantee global existence. In order to give a precise statement of our results, let B s p,2 , with p > 1 and s > 0, denote the Besov spaces as introduced in Section 2 below. Using Marcinkiewicz's multiplier theorem [5], we first prove a priori estimates in The choice of B s p,2 spaces is required by our analysis of the nonlinear terms. In fact the particular properties of these Besov spaces allow us to control products of functions in spaces of negative order -see the Appendix. The condition p > (2n + 8)/3 implies that the field of normal vectors is Hölder continuous in space and time, hence the solutions constructed in Theorem 1.1 consists of C 1+a -surfaces. Observe furthermore that Theorem 1.1 guarantees us local in time solutions for arbitrary initial surfaces. To obtain a global result we have to restrict the set of initial data.
Using Alexandroff's characterization of embedded surfaces of constant mean curvature [1], it is not difficult to see that spheres are the only embedded steady states of (1.1). Note that this implies that steady states are not isolated, so that the principle of linearized stability cannot be used to analyze the stability properties of spheres. Instead we shall use an abstract approach which is based on the theory of maximal regularity and spectral properties of an elliptic operator defined by (1.1). Our analysis will lead us finally to a construction of a center manfold for (1.1), which consists only in equilibria. Moreover, it attracts all solutions at an exponential rate which are close enough to it. It is worthwhile to emphasize that our techniques are not based on the abstact methods from [20], but also use maximal regularity results, however in anisotropic Besov spaces. This approach is more direct and allows us to essentially improve the results of [9]. More precisely, we have Theorem 1.2. Let the initial surface M 0 be sufficiently close to a sphere in the B -topology, then the solution to system (1.1) exists globally in time and stays in a neighborhood of the sphere. Additionally it convergences exponentially fast to a sphere with a radius determined by the volume of a set bounded by M 0 .
Recall that Theorem 1.1 holds true provided p > (2n + 8)/3. Hence if n < 8 there is a p > (2n+8)/3 such that 5/2−4/p < 2, which means that C 2 (S n ) → B This implies that there is a sequence of functions over S n in B This in turn implies that there are exist arbitrarily small non-convex perturbations in B 5/2−4/p p,2 (S n ) of a given sphere. If n ≥ 8 the embedding C 2 (S n ) → B 5/2−4/p p,2 (S n ) can no longer be guaranteed. However, a different scaling argument shows, that (1.4) nevertheless is true. Combing the above reasoning with Theorem 1.2, we get Theorem 1.3. There exists a smooth non-convex surface M 0 generating a global solution to (1.1), which converges eventually to a sphere.
The paper is organized as follows. First we recall the definitions of Besov spaces. In Section 3 a parametrization of solutions to (1.1) is given. It enables us to transfrom (1.1) into a system on a rigid domain. Next, we prove Theorem 1.3, based on Theorems 1.1 and 1.2. In Section 5 we introduce a linear system related to (1.1) and give a proof of the main a priori estimate (1.3). Based on the result for the linear system Theorem 1.1 is proved in Section 6. Subsequently we examine the stability of the unit sphere building a local center manifold proving finally Theorem 1.2. At the end in the Appendix we recall some facts concerning Besov spaces. Generic constants are denoted by the same letter C.
2. Besov spaces. Our analysis is described mostly in Besov spaces of B s p,2 -type [2], [22], [23]. We briefly recall the corresponding definitions. First we consider the isotropic spaces defined on R n . Given s > 0 and p ≥ 1, we introduce the Banach space B s p,2 (R n ) defined by the norm provided s ∈ (0, s). For the case s = 1 of (2.2), we refere to [2]. The above definitions can be extended on anisotropic spaces, where the time direction is distinguished. Given T > 0, we define B s,s p,2 (R n × (0, T )) by the norm where the sum of the last two terms of (2.3) is the main seminorm of B s,s p,2 , given by and F z denotes the Fourier transform with respect to z. Observe that in this definition s may be any real number.
In the case of the spacetime R n × R the norm (2.3) reads where p x k , p t k denote the Paley-Littlewood decomposition for R n and R, respectively.
The above Besov spaces can be defined in a natural way for functions on smooth submanifolds. Using an atlas of the manifold and a suitable partition of unity we define the norm as the sum taken over all maps from the atlas.
A basic tool in analysis of linear system in the presented framework is Marcinkiewicz's theorem for Fourier multipliers. A version of this result [5] reads as follows In [22], [23] one can find versions of this theorem for Besov spaces, too.
3. Parametrization. The goal of this section is to transform system (1.1) into a system on a rigid domain and to parametrize the solution by a scalar function.
For a given compact initial surface M 0 we can find a compact smooth hypersurface S satisfying the following conditions. First we fix two finite coverings The family {η k } is a partition of unity on S. Further we choose an atlas of maps (U l , Z l ) such that U l ⊂ R n and Z l : where parameter λ will be specified later. Condition (3.2) yields a bound for the derivatives of the functions Given a function φ : S × [0, T ) → R n+1 we describe a family of surface by where n is the normal vector to S. The initial datum has the following parametrization We require only that the parametrization of (3.4) is valid at least for small T > 0. The regularity of M 0 enables to choose the surface S such that where ε > 0 is an arbitrary small number. Requirement (3.6) implies that higher norms of S will depend essentially on smallness of ε, however we will overcome this obstacle by choosing T > 0 small enough. The function φ describes a diffeomorphism between the surfaces M (t) and S. Hence we are able to transform (1.1) into a problem on the fixed reference domain S in the following way where v(y, t) = V n (x(y, t), t) and v = φ t n · n t , with n t being the unit outward normal vector to M (t) in the coordinates (y, t). The Laplace-Beltrami and mean curvature operator∆ M (t) andH M (t) have the following form where g ij is the metric induced on M (t) by the coordinates y, and where g = det{g ij }, locally in each map of the atlas of the manifold S. For the explicit form of (3.8) we refer to [9], [10]. They can be determined in terms of the metric g ij . However, using similar perturbation results as presented in [9], we may concentrate on the principal parts only. The construction of the function φ allows us to reformulate Theorem 1.1 as follows.
The proof of Theorem 1.1 * will be given in section 6.
4. Proof of Theorem 1.3. We distinguish three cases: n = 1, 1 < n < 8 and n ≥ 8. We will construct a smooth surface M 0 close to the unit sphere such that M 0 will be non-convex and it will fulfill the assumptions of Theorem 1.2, namely, M 0 will be sufficiently close to the sphere in the B -topology, generating in this way a global in time solution to (3.7).
In the first case n = 1, we introduce where ε > 0, m ∈ N and n is the normal vector to S 1 , and where S 1 is the unit circle on the plane. Then we see that the "distance" between M 0 and the sphere is measured by the function ε sin my. A simple calculation leads us to Since n = 1, we find p > 10 3 (= 2n+8 3 with n = 1) and δ > 0 such that On the other hand the curvature of M 0 is dominated by the second derivative of this function, for which we have The simple form of M 0 allows us even to compute the curvature of M 0 explicitly. Parametrization by angle gives the following formula for the curvature with R(t) = 1 + ε sin mt. Choosing ε = m −2+δ , we get from (4.5): Now relation (4.5) shows that M 0 as given by (4.1) becomes non-convex provided m is large enough. This proves Theorem 1.3 in the first case. In the second case, i.e. for 1 < n < ∞, we proceed similarly. First we introduce spherical coordinates on S n , i.e. we represent x ∈ S n as where s(·) depends on two parameters ε and m in the following way where l is a smooth function such that supp l ⊂⊂ (− π 2 , π 2 ) n−1 . Because of n < 8 we then find δ > 0 and p > 2n+8 3 such that 5/2 − 4/p + δ < 2. Next putting ε = m −2+δ in (4.9), we conclude as for (4.3) that (4.10) Repeating the considerations for (4.4), we also get that So we are able to find m > 0 so large that M 0 , defined by (4.8), will be non-convex. In the case n ≥ 8 we proceed in a different way. Consider 1] such that supp f ⊂ B(0, 1) and f (0) = 1, and with constants ε > 0 and m ∈ N. To compute the semi-norm of s given by (4.12), we observe that for n ≥ 8 we have and changing coordinates z = mx we get: So we find the following estimate for the semi-norm of the function s Now observe that we are able to find p > 2n+8 3 and δ > 0 such that (4.14) Inserting ε = m −2+δ into (4.12), we see that (4.10) and (4.11) hold true also for n ≥ 8. This completes the proof of Theorem 1.3.
5. The linear system. Our next task is to find a suitable linearization of system (3.7) and to prove an existence result with an appropriate Schauder-type estimates. We propose the following linear problem with ∆ being the Laplacian on R n , and {η l }, {Z l } are given by (3.1) -see Section 3. The parabolic character of the system of order four implies that (5.1) should be solved in B 4s,s p,2 -type spaces. From the analysis of the nonlinear system we obtain that the optimal factor is 4s = 5/2. In this part we show the key result in proving Theorem 1.1.
(S × (0, T ))) * , then there exists a unique solution to the system (5.1) such that First we prove a model version of Theorem 3.1 in the case S = R n .
(R n ) be given and assume that these data are compactly supported. Then the problem has a unique solution in the class B ) and the following estimate holds true, where C may depend on diameter of supports of the data.
Proof. System (5.4) can be split into the following two problems Clearly the solution to (5.4) is then the sum of φ 1 and φ 2 . Take the first system of (5.6). Since , and since the restriction p > 8/3 implies that q < 8/5, we can use Proposition 8.2 from the Appendix. It allows us to extend H on the whole R n+1 simple by zero. LetH denotes this extension. Hence the first system of (5.6) can be extended in the whole R n+1 tõ (R n × R), which follows from (8.2). Solving (5.7), we get The form of the multiplier in (5.8), via Marcinkiewicz's theorem, yields the following estimate Thus we conclude that the solution to (5.6) 1 satisfies φ 1 ∈ B ). Now we want to prove uniqueness of solutions to (5.6) 1 . By properties of the trivial extention of H, applying the standard energy approach, we get (5.10) and by the choice of extension of H and (5.9) the r.h.s. of (5.10) is well defined. Hence we obtain an estimate in L 2 . Moreover sinceφ 1 → 0 as t → −∞ inequality (5.10) for H ≡ 0 proves uniqueness of solutions to (5.7) in the considered class. Hence the obtained solutions must solve the first system of (5.6). In particular the initial condition φ 1 | t=0 ≡ 0 is fulfilled.
Next, we examine the second system of (5.6). To simplify the notation we write φ instead of φ 2 . The Fourier transform applied to space directions yields the explicit solution (5.11) First, we study the regularity with respect to time. By the definition (2.4) we have whereφ k h = p k (ξ)φ 0 (e −|ξ| 4 t − e −|ξ| 4 (t+h) ), where p k are the partition of unity from the Paley-Littlewood decomposition (2.5)-(2.7). So Marcinkiewicz's theorem (Theorem 2.1) gives . Taking the norm with respect to time, we get Inserting (5.14) into (5.12) yields To estimate the last integral we note that, taking l = k + 1, we have So putting w = 2 4l h, we obtain: (5.16) Hence inserting this estimate into (5.15) (I 2 to I 1 ), recalling definition (2.7), we conclude (5.17) Next, we consider regularity with respect to spatial variables. We have where hereφ k = p kφ -see (2.5)-(2.7). Applying Marcinkiewicz's theorem, remembering the form (5.11), we conclude Returning to (5.18) we have (see (2.7)) Therefore it follows that Summing up the estimates (5.9), (5.17) and (5.21), we obtain with a time independent constant C. But to estimate the whole norm, which is necessary in order to take the trace, we are required to use (5.10). Here extra conditions on the support of H and for φ 0 , and finiteness of T are needed. Then we easily conclude (5.5) and the proof of Lemma 5.1 is completed. Now we are prepared to consider the problem in on the manifold S. Let The method of proving Theorem 5.1 via Lemma 5.1 is adapted from the classical approach to parabolic systems introduced in [14]. It is based on the regularizer operator, see the definition below, which somehow gives the solution by a procedure related to the Banach fixed point theorem. This technique have been effectively applied for similar problems to (5.23) in [16], [17].
By Lemma 3.1 and the properties of the atlas (U l , Z l ) of S we are able to construct an extensionφ 0 of the initial data such that it belongs to B Thus we obtain (5.23) with the initial condition φ 0 ≡ 0 by a subtraction putting φ old =φ 0 + φ new and H old = Lφ 0 + H new into (5.1). By (5.24) the regularity of the data in (5.23) is preserved.
is the solution to the following problem Assume that supp η l ∩ supp π k = ∅. Then we conclude from (5.25) that where the term r lk has the following structure (in the highest order of derivatives): r lk ∼ λ −1 ∇φ k supported on supp η l π k . Hence taking into account (5.28) and recalling the properties of η l , we get Moreover the second term on the r.h.s. of (5.27) has a structure similar to r kl , namely To estimate these remainders, we apply the imbedding theorem (8.1) (S×(0,T )) , where we used the fact that Observe that C is independent of T , since φ k | t=0 ≡ 0 -see also (5.10). Thus by (5.30) and the form of the remainder there exists T * > 0 such that the norm of the operator T is less than 1/2 on the time interval [0, T ] for T ≤ T * . This completes the proof of Lemma 5.2 Note that the constant in (5.30) must be small enough in order to absorb the terms of order λ −1 . This is possible by choosing T sufficiently small. Lemma 5.3. Assume that T > 0 is small enough. Then RLφ = φ + Wφ and ||W|| ≤ 1 2 . Proof. Repeating the considerations from the proof of Lemma 3.3, we get with a suitable estimate for the norm of operator W for small T ≤ T * . Thus we claim that Lemma 5.3 is true. Similar results with detailed proofs can be found in [17].
To prove Theorem 5.1 we first prove the existence of solutions to system (5. Remark. It is worthwhile to underline that the magnitude of the constant in (5.35) is independent of λ and T * . It depends only on the geometry of S. However we have λ → 0 as T * → 0. On the other hand fixing C in (5.35), we can choose λ and then consider T ∈ (0, T * ]. This remark is crucial in the treating of the nonlinear problem in the next section -see in particular (6.15). The simplest solution to avoid the mentioned difficulty is just to use this 'universal' constant from (5.35), and then prescribe first λ and then T .

Local in time existence.
Here we prove Theorem 1.1 * which implies unique solvability of system (1.1) in the meaning of Theorem 1.1. The system (1.1) will be considered in the following form where L φ is defined by n and n t are normal vectors to S and M (t), respectively. Additionally 3) The goal of this section is to prove existence and uniqueness of weak solutions to (6.1) in the following sense: Definition 6.1. We say that a function φ : and φ| t=0 = φ 0 and the following identity is valid The proof of Theorem 1.1 * will follow from a standard application of the Banach fixed point theorem and Theorem 5. , but this form gives us directly the trace theorem (in time) and an inequality with the constant independent of T . In other words we control all constants in following considerations uniformly in time. This is of particular importance, since T has to be small. In order to underline the domain we shall write Ξ(S × (0, T )).
Choosing at the very beginning a sufficiently small ε in (3.6), our goal to show the existence of δ > ε > 0 such that the mapping is the solution to the following problem in the distributional sense on [0, T ) for each π ∈ B 3/2,3/8 q,2 (S × (0, T )) with 1/p + 1/q = 1. In (6.7) N (t) is the surface generated by ψ via definition (3.4), and L ψ is the version of (6.3) for N (t). Let us explain the notation in (6.7). Here we use (Λφ, Λπ) L2(S) = (Λ * Λφ, π) L2(S) in the sense of distributions -see (5.2). Λ denotes a second order elliptic operator, being a part of the operator Λ * Λ, which is a consequence of integration by parts.
Remark. Here we shall explain the properties of (6.11). This highest term in (6.11) describes us the optimal regularity of the whole considerations. In order to estimate it, we need five derivative applied to ψ and π. Hence the space of solutions requires at least 5/2 derivative. Next, inclusions (6.13) and restrictions coming from the multiplication results for the Besov spaces imply that p > 2n+8 3 -see Proposition 8.3. In this way we find our space B The regularity given by Ξ implies ∇ψ ∈ L ∞ -even the Hölder continuity in time and space. Remembering that ∇ψ ∈ C a,a/4 (S ×(0, T )) which follows from (8.1), lower terms (in derivatives of functions ψ and π) from (6.8) can be estimated by (S×(0,T )) . (6.14) We emphasize that the estimations of the lower terms in local coordinates are connected to estimations of terms with derivatives of the localizers ζ k , π k and η k . From (3.3) we know that these quantities are bounded only by 1 λ . However thanks to (5.31) we estimate them with a constant T a for some a > 0. Thus a suitable small T will absorb the influence of 1 λ -see (6.14) and the Remark at the end of the previous Section 5.
Our considerations lead to the following bound of the l.h.s. of (6.8) Thanks to the smallness of ε and λ, we find δ > 0 such that K : Ξ → Ξ -see (6.5).
The proof of the contraction property of K on Ξ is almost the same as for (6.15). So repeating all steps done for (6.7), we get provided T sufficiently small. This shows the unique solvability of system (3.7) in the meaning of definition (6.4). Theorems 1.1 * and 1.1 are therefore proved.
7. Stability of spheres. In this part we establish stability of spheres in the B 5/2−4/p p,2 -topology. For simplicity we concentrate our analysis on the behavior of solutions to (1.1) in a neighborhood of the unit sphere. Since the required regularity guarantees that the normal vector is at least Hölder continuous we can choose S n as the reference surface. So the surface M (t) is parametrized by φ via formula (3.4) with S n as S and the system (6.1) can be restated as follows Our analysis is concerned with the stability of the zero solution, so we need the Fréchet derivative of G at 0. From [9] -Lemmas 3.1 and 3.2 -we have:

2)
where ∆ S n is the Laplace-Beltrami operator on the unit sphere S n , then Furthermore the spectrum A consists of a sequence of real eigenvalues. 0 = λ 0 < λ 1 < λ 2 < ..., (7.4) where λ 0 is of multiplicity n + 2. The properties of operator A allow us to introduce the following decomposition of the phase space .., Y n+1 } and Y 0 , ..., Y n+1 are eigenvectors for the eigenvalue λ 0 = 0, Y 0 = 1 and Y 1 , ..., Y n+1 are homogeneous functions of degree one. In the canonical setting they are just Y l = x l . Furthermore we define projectors P c and P s on subspaces X c and X s , respectively. The space X s called the stable subspace since the spectrum of A here is strictly positive, i.e. spec A| X s ⊂ [λ 1 , +∞). Since X c is finite dimensional, the decomposition (7.5) is well defined.
To apply the properties of A, we restate (7.1) in the following way: In order to analyze (7.6), we have a closer look at g. Thanks to (7.3) we have This point of view on the r.h.s. of (7.6) is helpful in the following considerations. Our next goal is to "parametrize" all small static solutions to (7.6). Eventually it will appear that they build a local unique center manifold for (7.1). Let S z be a sphere close to S n and let (z 1 , ..., z n+1 ) be the coordinates of its center. Then z 0 is chosen such that 1 + z 0 is the radius of S z . By the properties of Y l we find that where ψ is the distance to S n . Straightforward calculations lead us to the following representation of the distance to the original sphere We restrict our attention to a neighborhood of zero in R n+2 . Then we get So the set of all S z sufficiently close to S n is represented in X by The set M is a smooth (n+2)-dimensional manifold in X, tangent to X c at zero. Also by the properties of m, P c and (7.9) we find Summarizing the above analysis, we get that the set of all small solutions to (7.6) is described in the following way (7.14) Now we want to apply (7.14) for our investigations of (7.1). Using (7.5) the system (7.6) can be decomposed in the following way: Additionally from Lemma 7.1 we get It follows from (7.15) that This means that any small solution to (7.6) has the following form i.e. the evolution of our system acts only on X s . This particular feature allows us to describe precisely the behavior of solutions on X s . In order to reduce equations (7.15), we note that the set of all small equilibria (7.14) is the graph of the function σ : P c M → X s in the sense that z + σ(z) ∈ M with z ∈ P c M. Consequently, any solution has the form φ(t) = P s φ(t) + z, with some z ∈ P c M, where z is obtained from (7.18), i.e. z = P c φ 0 .
(7.16) yields P s g(z + σ(z)) = P s Aσ(z), too. Hence we can reformulate the second equation of (7.15) as follows (7.19) The quantity P s φ − σ(z) measures the distance to M in X s . Let us describe the structure of the r.h.s. of (7.19). By (7.8) (7.20) which implies that Now, we want to analyze (7.19). Because of (7.21) we are required to return to (7.6), to control the norm of the whole φ. The form of (7.8) enables us to prove a so-called almost global in time existence result. We have Proposition 7.1. Let T > 0, then there exists M > 0, ε 0 > 0, such that there exists a unique solution to (7.6) on time interval [0, T ] fulfilling the following a priori estimate for all ε ≤ ε 0 . In fact, Proposition follows from the results and methods presented in Sections 5 and 6 as well as from Lemma 7.1 and expression (7.7). In order to avoid too much technicalities we omit the proof here.
Based on Proposition 7.1 we may express the solution to (7.19) in the in the following way (7.23) Remembering that spec P s A ⊂ [λ 1 , +∞), using the results for the nonlinear system from Section 6, (7.21) and Proposition 7.1, we conclude where B 1 is independent of t, but B 2 may depend on t. The inequality holds as long as the solution exists. Let us explain (7.24). First we prove existence of almost global solutions to (7.19). The estimate for the linear system implies For a sufficiently small (depending on T ) initial data, we get B 3 (T )||φ|| Ξ(S n ×(0,T )) ≤ 1/2, thanks to Proposition 7.1. Hence we obtain On the other hand the last term in (7.24) is bounded by 27) see our considerations in Section 6 (6.8)-(6.15) So (7.26) and (7.27) give the form of the last term in (7.24). The first term in the r.h.s. of (7.24) follows from standard theory of semigroups. Inequality (7.24) is thus proved. Proposition 7.1 implies that we can choose the initial datum φ 0 sufficiently small in order to guarantee that B 2 (T * )||φ|| Ξ(S n ×(0,T * )) ≤ B 1 e −λ1T * , (7.28) where T * taken so small that B 1 e −λ 1 T * ≤ 1 4 . Then (7.24) yields Without loss of generality we can require that the projection of the initial datum P c φ 0 = z is so small that ||σ(z)|| B 5/2−4/p p,2 (S n ) < ε/2. Note that his quantity is fixed in time for each considered solution. Then by (7.29) we obtain ||φ(T * )|| B 5/2−4/p p,2 (S n ) ≤ ε/2 + ε/2 = ε. (7.30) Hence we restart our considerations for the initial time t = T * with φ| t=T * = φ(T * ), eventually getting for t ∈ [0, ∞). (7.31) Estimate (7.31) shows that each small solution tends exponentially to an element of M, i.e. to P c φ 0 + σ(P c φ 0 ). As a result of (7.31) and the proved properties of M we obtain that M is a local unique center manifold for the dynamical system (1.1), with a basin of attraction being a neighborhood of the sphere in the B 5/2−4/p p,2 -topology. The proof of Theorem 1.2 is thus completed.
Proposition 8.1. We have: The next result concerns B 3/2,3/8 q,2 (R n × (0, T )) for q < 8/5. One can check elementary that these spaces can be obtained as the closure of compactly supported functions: Proposition 8.2. We have The above fact allows to extend any function from this class trivially by zero. It follows that the constant in the imbedding inequality from (8.1) does not depend on smallness of T which will be important in our considerations.
Next we prove a result which controls the regularity of the product of two functions from Besov spaces. It is an important tool for analysis of the nonlinear terms appeared in Section 6. Proof. All calculations are done in local coordinate of the surface S. By (8.2) we are able to assume that the function g is defined on the whole S × R. This property implies that constants in (8.3) and (8.4) are independent of T .