Stable Singularity Formation for the Keller–Segel System in Three Dimensions

We consider the parabolic–elliptic Keller–Segel system in dimensions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \geqq 3$$\end{document}d≧3, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=3$$\end{document}d=3. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887–912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller–Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.


Introduction
We consider the following system of equations ∂ t u(t, x) = ∆u(t, x) + ∇•(u(t, x)∇v(t, x)), ∆v(t, x) = u(t, x), equipped with an initial condition u(0, •) = u 0 , for u, v : [0, T ) × R d → R and some T > 0. This model is frequently referred to as the parabolic-elliptic Keller-Segel system, named after the authors of [36], who introduced a system of coupled parabolic equations to describe chemotactic aggregation phenomena in biology.The parabolic-elliptic version (1.1) was derived later by Jäger and Luckhaus [34].System (1.1) arises also as a simplified model for self-gravitating matter in stellar dynamics, with u representing the gas density and v the corresponding gravitational potential, see e.g.[51,1].The equation for v in (1.1) can be solved explicitly in terms of u, which reduces the system to a single (non-local) parabolic equation where v u = G * u, with G denoting the fundamental solution of the Laplace equation.This equation is invariant under the scaling transformation u → u λ , u λ (t, x) := λ −2 u(t/λ 2 , x/λ), λ > 0.
Furthermore, assuming sufficient decay of u at infinity, the total mass ), the model is mass critical for d = 2 and mass supercritical for d ≥ 3.
Irfan Glogić is supported by the Austrian Science Fund FWF, Projects P 30076 and P 34378. 1 It is well known that Eq. (1.1) admits finite-time blowup solutions in all space dimensions d ≥ 2, for which in particular lim for some T > 0. This is natural in view of the phenomena that the model is supposed to describe, and there is a strong interest in understanding the structure of singularities.Consequently, there is a huge body of literature addressing this question for (1.1) and variants thereof, for a review see e.g.[32,33].
Being the natural setting for biological applications, a lot of attention has centred around the mass critical case d = 2. There, the L 1 -norm of the stationary ground state solution Q, defined in (1.3) below, represents the threshold for singularity formation, see e.g.[7,6,19,15].Particular solutions that blow up in finite time via dynamical rescaling of Q, with λ(t) → 0 for t → T − , have been constructed for different blowup rates λ, see [12,44,49,31].
In particular, in [12] it is shown that the blowup solution corresponding to for certain explicit constant κ > 0, is stable outside of radial symmetry.In addition to this, Mizogouchi [37] recently proved that for solutions with non-negative and radial initial data, (1.3)-(1.4)describes the universal blowup mechanism.
In comparison, the dynamics in the supercritical case d ≥ 3 are more complex; in particular, multiple blowup profiles are known to exist.In a recent work, Collot, Ghoul, Masmoudi and Nguyen [13] proved for all d ≥ 3 the existence of a blowup solution that concentrates in a thin layer outside the origin and implodes towards the center.Other known examples of singular behavior are provided by self-similar solutions, which are proven to exist in all dimensions d ≥ 3, see [30,8,47].A particular example was found in closed form in [8], and is given by (1.5) 1.1.The main result.To understand the role of the solution (1.5) for generic evolutions of (1.1), the authors of [8] performed numerical experiments and conjectured as a consequence that u T is nonlinearly radially stable.In spite of a number of results on the nature of blowup, this conjecture has remained open for more than two decades now.In the main result of this paper we prove this conjecture for d = 3.More precisely, we show that there is an open set of radial initial initial data around u 1 (0, •) = U for which the Cauchy evolution of (1.1) forms a singularity in finite time T > 0 by converging to u T , i.e., to the profile U after self-similar rescaling.The formal statement is as follows.
Theorem 1.1.Let d = 3.There exists ε > 0 such that for any initial datum where ϕ 0 is a radial Schwartz function for which there exists T > 0 and a classical solution u ∈ C ∞ ([0, T ) × R 3 ) to (1.1), which blows up at the origin as t → T − .Furthermore, the following profile decomposition holds where ϕ(t, •) H 3 (R 3 ) → 0 as t → T − .Remark 1.2.As it will be apparent from the proof, the extension of this result to any higher dimension is straightforward.This involves developing the analogous well-posedness theory and solving the underlying spectral problem for a particular choice of d ≥ 4. We therefore restrict ourselves to the lowest dimension, and the physically most relevant case, d = 3.
Remark 1.3.Due to the embedding H 3 (R 3 ) ֒→ L ∞ (R 3 ), the conclusion of the theorem implies that the evolution of the perturbation (1.6), when dynamically self-similarly rescaled, converges back to U in L ∞ (R 3 ).In other words, Related results for d ≥ 3.There are many works that treat the system (1.1) in higher dimensions.Here we give a short and noninclusive overview of some of the important developments.
Local existence and uniqueness of radial solutions for (1.1) holds in L ∞ (R d ) as well as in other function spaces, see e.g.[23,2].Concerning global existence, various criteria are given in terms of critical (i.e.scaling invariant) norms.For example, it is known that initial data of small L d/2 (R d )-norm lead to global (weak) solutions [14].This result was later extended by Calvez, Corrias, and Ebde [9] to all data of norm less than a certain constant coming from the Gagliardo-Nierenberg inequality.For results in terms of the critical Morey norms, see e.g.[3,2].Concerning the existence of finite time blowup, the aforementioned works [14,9] give sufficient conditions in terms of the size of the second moment of the initial data.For an earlier result of that type see the work of Nagai [38].For other, more recent results see [4,42,48,5,2,39].We point out, however, that in contrast to the d = 2 case, for d ≥ 3 still no simple characterization of threshold for blowup in terms of a critical norm is known.
Concerning the structure of singularities, not much is known.It is straightforward to conclude that blowup solutions of (1.1) satisfy lim inf t→T − (T − t) u(t, •) L ∞ (R d ) > 0, see e.g.[40].Accordingly, singular solutions are classified as type I if lim sup and type II otherwise.The first formal construction of type II blowup was performed by Herrero, Medina and Velázquez for d = 3 in [29]; the singularity they construct consists of a smoothedout shock wave concentrated in a ring that collapses into a Dirac mass the origin.This blowup mechanism was later observed numerically in [8] for higher dimensions as well, and is furthermore conjectured to be radially stable.A rigorous construction of this solution for all d ≥ 3 came only recently in the work of Collot, Ghoul, Masmoudi and Nguyen [13], who also prove its radial stability.In contrast to these results, if the initial profile of a blowup solution is radially nonincreasing and of finite mass then the limiting spatial profile is very much unlike the Dirac mass, since it satisfies near the origin, as proven by Souplet and Winkler [48].This is in particular the case for self-similar solutions, for which lim t→T − u(t, x) = C|x| −2 at the blowup time.To add to the importance of self-similar solutions for understanding the structure of singularities, Giga, Mizoguchi and Senba [23] showed that any radial, non-negative type I blowup solution of (1.1) is asymptotically self-similar.For 3 ≤ d ≤ 9, it is known that there are infinitely many similarity profiles, while for d ≥ 10 there is at least one, see [47].However, a full classification of the set of self-similar (blowup) solutions, even in radial symmetry, is not available so far.Finally, we note that in the two-dimensional case, any blowup solution is necessarily of type II, see e.g.[41].In the mass subcritical case d = 1 blowup has been excluded in [11].The advantage of this change of variable lies in the fact that it reduces the system (1.1) to a single local semilinear heat equation on R 5 for w(t, x) := w(t, |x|) where Λf (x) := x • ∇f (x), and the initial datum is radial w 0 = w(0, | • |).Additionally, the self-similar solution (1.5) turns into The bulk of our proof consists of showing stability of w T .Then, by using the equivalence of norms of u and w we turn the obtained stability result into Theorem 1.1.In Section 2, as is customary in the study of self-similar solutions, we pass to similarity variables We remark that the application of similarity variables in the study of blowup for nonlinear parabolic equations goes back to 1980's and the early works of Giga and Kohn [20,21,22].
Furthermore, by rescaling the dependent variable (T − t)w(t, x) =: Ψ(τ, ξ), the solution w T becomes τ -independent, ξ → φ(ξ).Consequently, the problem of stability of finite time blowup via w T is turned into the problem of the asymptotic stability of the static solution φ.To study evolutions near φ, we consider the pertubation ansatz Ψ(τ, •) = φ+ψ(τ ), which yields the central evolution equation of the paper Here, the linear operator L and the remaining nonlinearity N are explicitly given as (1.9) The next step is to establish a well-posedness theory for the Cauchy problem (1.8).To that end, in Section 3 we introduce the principal function space of the paper To construct solutions to (1.8) in X k , we take up the abstract semigroup approach.In Section 4 we concentrate on the linear version of (1.8).First, we prove that L, being initially defined on test functions, is closable, and its closure L generates a strongly continuous semigroup S(τ ) on X k .Our proof is based on the Lumer-Phillips theorem, and it involves a delicate construction of global, radial and decaying solutions to the Poisson type equation Lf = g.Thereby we establish existence of the linear flow near φ.This flow exhibits growth in general, due to the existence of the unstable eigenvalue λ = 1 of the generator L. This instability is not a genuine one though, as it arises naturally, due to the time translation invariance of the problem.By combining the analysis of the linear evolution in X k with a thorough spectral analysis of L in a suitably weighted L 2 -space, we prove the existence of a (non-orthogonal) rank-one projection P : X k → X k relative to λ = 1, such that the linear evolution in X k decays exponentially on ker P.This is expressed formally in the central result of the linear theory, Theorem 4.1.
The existence of S(τ ) allows us to express the nonlinear equation (1.8) in the integral from As is customary, we employ a fixed point argument to construct solutions to (1.10) in X k .For this, we need a suitable Lipschitz continuity property of the nonlinear operator N in X k .However, the space X k is not invariant under the action of N , due to the presence of the derivative nonlinearity, recall (1.9).Nevertheless, by exploiting the smoothing properties of S(τ ), we show that the operator f → S(τ )N (f ) is locally Lipschitz continuous in X k , which will suffice for setting up a contraction scheme for (1.10).The proofs of the smoothing properties of S(τ ) and those of the accompanying nonlinear Lipschitz estimates comprise the main content of Section 5.With these technical results at hand, in Section 6 we use a fixed point argument to construct for (1.10) global, exponentially decaying strong X 4 -solutions for small data.To deal with the growth stemming from the presence of rg P in the initial data, we employ a Lyapunov-Perron type argument, by means of which we also extract the blowup time T .In Section 7, we use regularity arguments to show that the constructed strong solutions are in fact classical.By translating the obtained result back to physical coordinates (t, x), we get stability of w T .Finally, by means of the equivalence of norms of w and u, from this we derive Theorem 1.1.
We finalize with a remark that our methods can straightforwardly be generalized so as to treat (1.7) in any higher dimension n ≥ 6.This involves carrying out the analogous well-posedness theory for (1.8) in the high-dimensional counterpart of the space X k 1, along with using the same techniques to treat the underlying spectral problem.
1.4.Notation and Conventions.We write N for the natural numbers {1, 2, 3, . . .}, N 0 := {0} ∪ N. Furthermore, R + := {x ∈ R : x > 0}.By C ∞ c (R d ) we denote the space of smooth functions with compact support.In addition, we define Also, S rad (R d ) stands for the space of radial Schwartz functions.By L p (Ω) for Ω ⊆ R d , we denote the standard Lebesgue space.For the Fourier transform we use the following convention We use the common notation x := 1 + |x| 2 also known as the Japanese bracket.

Equation in similarity variables
To restrict to radial solutions of (1.1) we assume that u(t, Furthermore, the self-similar solution (1.5) turns into 2.1.Similarity variables.We pass to similarity variables Note that transformation (2.3) maps the time slab S T := [0, T ) × R n into the upper half-space H + := [0, +∞) × R n .Also, we define the rescaled dependent variable Consequently, the evolution of w inside S T corresponds to the evolution of Ψ inside H + .Furthermore, since we get that the nonlinear heat equation (2.1) transforms into with the initial datum Ψ(0, •) = T w 0 ( √ T •).For convenience, we denote Now, we have that for all n ≥ 5 the function is a static solution to (2.6).To analyze stability properties of φ n , we study evolutions of initial data near φ n , and for that we consider the perturbation ansatz This leads to the central evolution equation of the paper where (2.9) Furthermore, we write the initial datum as where, for convenience, we denoted v = w 0 − φ n . (2.11) Now we fix n = 5, and study the Cauchy problem (2.8)-(2.10).For this we need a convenient functional setup.

Functional setup
This section is devoted to defining the function spaces in which we study the Cauchy evolution of (2.8).Furthermore, we gather the basic embedding properties that will be used later on.
3.1.The space where ∆ rad := r −4 ∂ r (r 4 ∂ r ) denotes the radial Laplace operator on R 5 .We also define an inner product on The central space of our analysis is the completion of (C ∞ c,rad (R 5 ), • X k ), and we denote it by X k .Throughout the paper we frequently use the equivalence Now, we list several properties of X k that will be used later on.First, a simple application of the Fourier transform yields the following interpolation inequality.
), and all α ∈ N 5 0 with 1 ≤ |α| ≤ k.Also, it is straightforward to see that the following result holds.
In particular, elements of X k can be identified with functions in The following result follows from an elementary approximation argument.
Throughout the paper, we frequently use the commutator relation for α ∈ N 0 , |α| = k, as well as its radial analogue where We then define the following weighted L 2 -spaces of radial functions, We note that both H 0 and H have C ∞ c,rad (R 5 ) as a dense subset.An immediate consequence of the exponential decay of the weight functions is the following result.
).Consequently, we have the following continuous embeddings 3.3.Operators.For convenience, we copy here the linear operators defined in (2.7) and (2.9) and we formally let Note that by Lemma 3.3, φ ∈ X k for any k ∈ N, k ≥ 3, and the same holds for Λφ.

Linear theory
In this section we concentrate on the linear version of (2.8), and show that it is well-posed in X k for k ≥ 3. To accomplish this, we use the semigroup theory.Before we state the central result of the section, we make some technical preparations.First, we note that due to the underlying time-translation symmetry, the linear operator L has a formal unstable eigenvalue, λ = 1, with an explicit eigenfunction According to Lemma 3.3, the function ν belongs to X k , and this allows us to define a projection operator P : This whole section is devoted to proving the following theorem, which, in short, states that the linear flow of (2.8) decays exponentially in time on the kernel of P.
is closable, and its closure generates a strongly continuous semigroup (S(τ )) τ ≥0 of bounded operators on X k .Furthermore, there exists ω k ∈ (0, 1  4 ) such that S(τ for all f ∈ X k and all τ ≥ 0.
The mere fact that the closure of L generates a semigroup on X k can be proved by an application of the Lumer-Phillips Theorem, see Section 4.2.However, determining the precise growth of the semigroup is highly non-trivial, in view of the non-self-adjoint nature of the problem and the lack of an abstract spectral mapping theorem that would apply to this situation.To get around this issue, we combine the analysis in X k with the self-adjoint theory for L in H.

4.1.
The linearized evolution on H.We equip L 0 and L with domains The following result for the operator L 0 is well-known, and we refer the reader to [25], Lemma 3.1, for a detailed proof.
Lemma 4.2.The operator L 0 : D(L 0 ) ⊂ H 0 → H 0 is closable, and its closure (L 0 , D(L 0 )) generates a strongly continuous semigroup (S 0 (τ )) τ ≥0 of bounded operators on H 0 .Explicitly, where The operator L, on the other hand, has a self-adjoint realization in the space H; we have the following fundamental result.
Proposition 4.3.The operator L : D(L) ⊂ H → H is closable, and its closure (L, D(L)) generates a strongly continuous semigroup (S(τ )) τ ≥0 of bounded operators on H.The spectrum of L consists of a discrete set of eigenvalues, and moreover where λ = 1 is a simple eigenvalue with the normalized eigenfunction g from (4.2).Furthermore, for the orthogonal projection P : H → H defined in (4.2) there exists ω 0 > 0 such that for all f ∈ H and all τ ≥ 0.
Proof.We define the unitary map Using standard results, we infer that A c is limit-point at both endpoints of the interval (0, ∞) (see, e.g., [50], Theorem 6.6, p. 96, and Theorem 6.4, p. 91).Hence, the unique self-adjoint extension of A c is given by its closure, which is the maximal operator A : and Au = Au for u ∈ D(A).The inclusion A c ⊂ A ⊂ A = A c implies A = A. Now, since q is bounded from below, the same is true for the operator A, i.e, there is a constant µ > 0 such that Re Au, u L 2 (R + ) ≥ −µ u L 2 (R + ) for all u ∈ D(A).Since q(r) → +∞ when r → ∞, A has compact resolvent, and its spectrum therefore consists of a discrete set of eigenvalues.The analogous properties of L follow by the unitary equivalence.In particular, L : D(L) ⊂ H → H is essentially self-adjoint and its closure is given by L = −U AU −1 with D(L) = U D(A).Moreover, we have that Re Lf, f H ≤ µ f H for all f ∈ D(L).This implies that L, being self-adjoint, generates a strongly continuous semigroup on H.

r) .
We define A S := A + A − − 1. Explicitly, we have This gives rise to the maximally defined self-adjoint operator A S : which is, by construction, isospectral to A except for λ = −1.For a detailed discussion on the above process of "removing" an eigenvalue, see [26], Section B.1.Now, to prove (4.5), it is enough to show that A S does not have any eigenvalues in (−∞, 0].For this, we use an adaptation of the so-called GGMT criterion; see [25], Theorem A.1.In particular, we show that for p = 2, where α = 6 and Q − S (r) = min{Q S (r), 0}.By calculating explicitly the root of By noting that we can calculate the integral explicitly and find that where the right hand side of this inequality is the value of the right hand side of Eq. (4.7) for p = 2 and α = 6.Theorem A.1 in [25] now implies that σ(A S ) ⊂ (0, +∞), and (4.5) follows.Furthermore, a simple ODE analysis yields that the geometric eigenspace of λ = 1 is equal to g .Consequently, P is the orthogonal projection onto rg P in H, and we readily get (4.6).
Next statement shows that the growth bounds are preserved when the linear evolution is measured in graph norms associated to fractional powers of the operator 1 − L. This result will be crucial in Section 4.2, in particular, in conjunction with Lemma 4.6 below.
Proof.By standard results (see, e.g., [35], p. 281, Theorem 3.35), the square root (1 − L) 1 2 exists and commutes with any bounded operator that commutes with L. We show that The growth bounds for the semigroup can be proved by induction using the fact that (1 − L) 1 2 commutes with the projection and the semigroup.
We conclude this section by proving two technical results that will be crucial in the sequel.Lemma 4.5.Let k ∈ N 0 and R > 0. Then ) and all α ∈ N 5 0 with |α| = k.
Proof.First, for f ∈ C ∞ c,rad (R 5 ) and µ(x) := |x| 4 e −|x| 2 /4 we define By inspection, it follows that 1 − L = B * B and thus, , where the fact that B and B * are formally adjoint follows from a straightforward calculation.Hence, . With this at hand, we prove the lemma by induction.For k = 0 the inequality is immediate, and for k = 1 we have that Assume that the claim holds up to some k ≥ 1.Then we have that for all 1 ≤ j ≤ k Lemma A.1 then implies the claim for |α| = k + 1.
Finally, we show that graph norms can be controlled by X k -norms.
Proof.We prove the statement for f ∈ C ∞ c,rad (R 5 ).The claim then follows by density and the closedness of (1 − L) κ/2 .First, it is easy to see that for smooth, radial and polynomially bounded functions p α .Using the exponential decay of the weight function σ along with interpolation, see Lemma 3.1, and Hardy's inequality, we get 4.2.The linearized evolution on X k .With the technical results from above, we now show that the linear evolution of (2.8) is well-posed in denoting the closure we have that C ⊂ D(L k ).Furthermore, the operator L k generates a strongly continuous semigroup (S k (τ )) τ ≥0 of bounded operators on X k , which coincides with the restriction of S(τ ) to X k , i.e, Proof.We prove the first part of the statement by an application of the Lumer-Phillips theorem.For this, we show that for some ωk > 0 and all f ∈ D(L).In fact, we prove a more general estimate, which will be instrumental in proving Theorem 4.1 later on.More precisely, we show that for R ≥ 1, there are constants An application of Lemma 4.6 to Eq. (4.9) immediately implies Eq. (4.8).Eq. (4.9) will be proved in several steps.First, recall that L = L 0 + L ′ .By partial integration, Based on the identity Re Λf, , which easily follows from integration by parts and the commutator relation Eq. (3.4), we get Consequently, and thus Re Lf, f To obtain Eq. (4.8), one can estimate the part containing L ′ in X k in a straightforward manner.However, for the refined bound (4.9), the argument is more involved.We write ) and the properties of φ imply that for α ∈ N 5 0 .First, we prove that for suitable constants C R , C > 0. For this, we use the fact that ).Hence, we can apply the standard Leibniz rule to estimate products.More precisely, for α, β, γ ∈ N 5 0 for which |α| = |β + γ| = k, we estimate . By Eq. (4.14) and Lemmas 4.5 and 3.1 the last term can be bounded by Similarly, we have that For γ = 0, we estimate the last term by Hardy's inequality, For γ = 0, by interpolation and equivalence of norms, see Lemma (3.1), we obtain which implies Eq. (4.15).To estimate the second term in Eq. (4.13), we use the following relation with certain smooth functions |ϕ β (x)| x −2 .Note that by partial integration we have the following identity . Therefore, the first term in (4.16) can be estimated . For the second term we have ) .Based on this, similarly to above we infer that for suitably chosen C R , C > 0. Using the same arguments, we arrive at (4.15) and (4.17) for D instead of D k , and we hence get (4.9), and thereby (4.8) as well.From this, we infer that the operator L is closable (see, e.g., [18], p. 82, Proposition 3.14-(iv)), and that the closure for all f ∈ D(L k ) and some suitable ωk > 0.
Next, we prove that C ⊂ D(L k ) by showing that for f ∈ C there is a sequence , where χ a smooth, radial cut-off function equal to one for |x| ≤ 1 and zero for |x| ≥ 2. It is easy to see that (f n ) ⊂ C ∞ c,rad (R 5 ) converges to f in X k .More precisely, by exploiting the decay of f , We have and similarly and (f n ) converges in X k to some limiting function which must be equal to f by the L ∞ −embedding and the uniqueness of limits.Now, Λf ∈ C for f ∈ C. Since Λf n = f Λχ n + χ n Λf and ).By similar considerations as above one finds that (Λf n ) is Cauchy in X k and thus Λf n → Λf in X k for n → ∞.Now, Arguments as above and the Banach algebra property of ) and by convergence of the individual terms, v = ∆f − 1 2 Λf − f + 2Λ(φf ) + 12φf.This shows in particular, that L k acts as a classical differential operator on C.
For the invocation of the Lumer-Phillips theorem, it is left to prove the density of the range of λ k − L k for some λ k > ωk .This crucial property is established by an ODE argument, the proof of which is rather technical and therefore provided in Appendix C.More precisely, let An application of the Lumer-Phillips Theorem now proves that (L k , D(L k )) generates a strongly continuous semigroup (S k (τ )) τ ≥0 on X k .In view of the embedding X k ֒→ H and the fact that C ∞ c,rad (R 5 ) is a core for L and L k for any k, an application of Lemma C.1 in [24] proves the claimed restriction properties.
In view of the restriction properties stated in Proposition 4.7, we can safely omit the index k in the notation of the semigroup.
Before turning to the proof of Theorem 4.1, we state a result for the free evolution, which follows in a straightforward manner analogous to the proof of Proposition 4.7, using in particular Eq.(4.12) and setting L ′ = 0 in the subsequent arguments.
generates a strongly continuous semigroup on X k , which coincides with the restriction of the S 0 (τ ) to X k for τ ≥ 0. Furthermore, for all f ∈ X k and τ ≥ 0.

4.3.
Proof of Theorem 4.1.First, we show that the operator P from (4.2) induces a nonorthogonal rank-one projection on X k .To indicate the dependence on k, we write for f ∈ X k .By its decay properties, the function g is an element of C ⊂ X k for any k ∈ N. In view of the embedding X k ֒→ H, the inner product makes sense for f ∈ X k and by definition P 2 X k = P X k .The fact that the projection commutes with S k (τ ) follows from the respective properties on H.
Let f ∈ C ∞ c,rad (R 5 ).By Proposition 4.7, f := (1 − P X k )f ∈ C ⊂ D(L k ).Using Eq. (4.9), Lemma 4.4 and Lemma 4.6, we infer that for R ≥ 1 sufficiently large, and integration yields X k , for some suitably chosen ω k > 0 and all τ ≥ 0. For f ∈ X k , the same bound follows by density.Again, for simplicity, we write Pf = P X k f for f ∈ X k .

Nonlinear estimates
Now we turn to the analysis of the full nonlinear equation (2.8).In this section, we establish for the operator N a series of estimates which will be necessary later on for constructing solutions to (2.8).Recall that for k ≥ 3 the space X k embeds into C k−3 rad (R 5 ).Therefore, multiplication and taking derivatives of order at most k − 3 is well defined for functions in X k .With this in mind, we formulate and prove the following important lemma.
for all f, g ∈ X k .
Proof.In this proof we crucially rely on a recently established inequality for the weighted L ∞norms of derivatives of radial Sobolev functions; see [24], Proposition B.1.For convenience, we copy here the version of this result in five dimensions.Namely, given s ∈ ( 1 2 , 5 2 ) and α ∈ N 5 0 , we have that ).Now we turn to proving (5.1).Due to the W 1,∞ -embedding of X k for k ≥ 4 and the fact that (f, g) → Λ(f g) is bilinear, it is enough to show (5.1) for f, g ∈ C ∞ c,rad (R 5 ).
To estimate the Ḣk−1 part, we do the following.If k is odd, then where the last estimate follows from a combination of the L ∞ -embedding of X k , Hardy's inequality, and the inequality (5.2).To illustrate this, we estimate the first sum above.Without loss of generality we assume that |α| ≤ |β|.We then separately treat the integrals corresponding to the unit ball and its complement.For the unit ball, we first assume that α = 0. Then by the L ∞ -embedding.If α = 0, we have that by ( 5.2) and Hardy's inequality.For the complement of the unit ball, we have 2) only.The second sum is estimated similarly.If k is even, then we have that and the desired estimate follows similarly to the previous case.The Ḣ1 part of the norm is treated in the same fashion.
Now we establish the crucial smoothing properties of S 0 (τ ).maps (0, ∞) continuously into X k+1 .Furthermore, denoting β(τ for all τ > 0 and all f, g ∈ X k . Proof.Similarly to above, by the embedding (3.2) and the underlying linearity, it is enough to show the proposition for f, g ∈ C ∞ c,rad (R 5 ).First, note that for u ∈ C ∞ c,rad (R 5 ) and v ∈ S rad (R 5 ) the following relation holds Accordingly, we have that where S0 (τ ) is given by the (scaled) convolution with the radial Schwartz function Gα(τ) := ΛG α(τ ) .To prove the estimate (5.5) we do the following.First, we note that the L 1 (R 5 )-norm of Gα(τ) does not depend on τ , so by Young's inequality we have that For the Ḣk+1 -norm, again by Young's inequality we have that In summary S0 (τ )u X k+1 β(τ )e − τ 4 u X k .In the same way we get the estimate for S 0 (τ )u.According to (5.7), it remains to treat ΛS 0 (τ )u.According to the commutator relation (3.4), to bound the Ḣ1 -norm, it is enough to estimate ΛDS 0 (τ )u in L 2 .To that end, we have for all τ > 0 and u ∈ C ∞ c,rad (R 5 ).To estimate the Ḣk+1 -norm, it is enough to bound ΛD k+1 S 0 (τ )u in L 2 .For this, we similarly get for all τ > 0 and u ∈ C ∞ c,rad (R 5 ).In summary, we have that Finally, by putting f g instead of u, the estimate (5.5) follows from the Banach algebra property of X k and Lemma 5.1.According to the above estimates, the continuity of the map τ → S 0 (τ )Λ(f g) : (0, ∞) → X k+1 follows from the continuity of the kernel maps D Gα(•) , DG α(•) : (0, ∞) → L 1 (R 5 ), and ).The estimate (5.6) is obtained similarly.
Proof.Define ξ f : [0, τ ] → X k by s → S(τ − s)S 0 (s)f .We prove that ξ f is continuously differentiable.More precisely, we show that which is a continuous function from [0, τ ] into X k .To show this, we first write and then by letting h → 0 we get (5.12).For the first term above, this follows from the fact that S 0 (s)f ∈ C ⊂ D(L k ) and that L k f = Lf for f ∈ C. The conclusion for the second term follows by similar reasoning for S 0 (τ ), together with the strong continuity of S(τ ) in X k .Now, continuity of ξ ′ f follows from the continuity of the map and the strong continuity of S(τ ) in X k .We note that, according to the definition of L ′ , the continuity of (5.13) follows from the strong continuity of S 0 (τ ) on X k+1 and the estimate (5.1).Finally, by integrating (5.12), we get (5.10).To prove (5.11), we do the analogous thing.Namely, we consider the function which is also continuously differentiable, with To establish differentiability, it is important to note that according to the definition of the operator domain, by Lemma 5.1 we have that that D(L k+1 ) ⊂ D(L 0,k ), and therefore S(s)f ∈ D(L 0,k ) for every k ≥ 3. Continuity of η ′ f , similarly to above, follows from the continuity of s → L ′ S(s)f : [0, τ ] → X k and the strong continuity of S 0 (τ ) in X k .
Recall the operator N from (3.5).According to Lemma 5.1 we have that N : X k → X k−1 for k ≥ 4. Also, recall the projection operator P = P X k from (4.21).Now we prove the central result of this section.
is a continuous map.Furthermore, there exist ω, ω > 0 such that for all τ > 0 and all f, g ∈ X 4 .In addition, for every k ≥ 4 there exists ωk > 0 such that for all τ > 0 and all f, g ∈ X k .
As the last result of this section, we prove the local Lipschitz continuity in X 4 of the composition of P and N .Lemma 5.6.We have that for all f, g ∈ X 4 .
Proof.By definition, for u, v ∈ X 4 we have Therefore, by Cauchy-Schwarz, the embedding X 3 ֒→ H, and Lemma 5.1, we get that for all u, v ∈ X 4 .The estimate (5.18) then follows by letting u = f + g and v = f − g.

Construction of strong solutions
For simplicity, from now on we will drop the subscript in • X 4 , and assume that an unspecified norm corresponds to X 4 .With the linear theory and the nonlinear estimates from the previous section at hand, we turn to constructing solutions to (2.8).For convenience, we copy here the underlying Cauchy problem To solve (6.1), we utilize the standard techniques from dynamical systems theory.First, we use the fact that L generates the semigroup S(τ ), to rewrite (6.1) into the integral form Then, as S(τ ) decays exponentially on the stable subspace, we employ a fixed point argument to show existence of global solutions for small initial data.Obstruction to this is, of course, the presence of the linear instability λ = 1.Nevertheless, as this eigenvalue is an artifact of the time translation symmetry, we use a Lyapunov-Perron type argument to suppress it by appropriately choosing the blowup time.Before stating the first result, we make some technical preparations.First, we introduce the Banach space where ω is from Proposition 5.5.Then, we denote Now, we define a correction function C : X 4 × X → X 4 by C(u, ψ) and a map K u : X → C([0, ∞), X 4 ) by The fact that K u (ψ)(τ ) is a well-defined element of X 4 for every τ ≥ 0, follows from (5.15).Then, the continuity of K u (ψ) : [0, ∞) → X 4 follows from the continuity of ψ and that of τ → S(τ )N (f ) : (0, ∞) → X 4 given f ∈ X 4 .Proposition 6.1.For all sufficiently small δ > 0 and all sufficiently large C > 0 the following holds.If u ∈ B δ/C then there exits a unique ψ = ψ(u) ∈ X δ for which ψ = K u (ψ).(6.4) Furthermore, the map u → ψ(u) : B δ/C → X is Lipschitz continuous.
Proof.To utilize the decay of S(τ ) on the stable subspace, we write K u in the following way Then, according to Proposition 5.5 we get that if ψ(s) ∈ B δ for all s ≥ 0 then Furthermore, if u ∈ B δ/C and ψ ∈ X δ then the above estimate implies the bound Also, we similarly get that for all ψ, ϕ ∈ X δ .Now, the last two displayed equations imply that for all small enough δ and for all large enough C, given u ∈ B δ/C the operator K u is contractive on X δ , with the contraction constant 1 2 .Consequently, the existence and uniqueness of solutions to (6.4) follows from the Banach fixed point theorem.The show continuity of the map u → ψ(u) we utilize the contractivity of K u .Namely, we have the estimate , wherefrom the Lipschitz continuity follows.Lemma 6.2.For δ ∈ (0, 1  2 ] and v ∈ X 4 the map is continuous.In addition, we have that Let ε > 0. By density, we know that there exists w0 ∈ C ∞ c,rad (R 5 ) for which w 0 − w0 < ε.Now, by writing and using the fact that lim S→T w0 ( Then, continuity follows by letting ε → 0. For the second part of the lemma, we write U (v, T ) in the following way ) From here, the estimate (6.5) follows.
Finally, by using the results above, we prove that given initial datum v that is small in X 4 , there exists a time T and an exponentially decaying solution ψ ∈ C([0, ∞), X 4 ) to (6.2).Theorem 6.3.There exist δ, N > 0 such that the following holds.If then there exist T ∈ [1 − δ N , 1 + δ N ] and ψ ∈ X δ such that (6.2) holds for all τ ≥ 0. Proof.Lemma 6.2 and Proposition 6.1 imply that for all small enough δ and all large enough N we have that if v satisfies (6.8) and T ∈ (6.9) We remark that ψ(τ ) is real-valued for all τ ≥ 0, since the set of real-valued functions in X 4 is invariant under the action of both S(τ ) and P. Now, to construct solutions to (6.2), we prove that there is a choice of δ and N such that for any v that satisfies (6.8) there is for which the correction term in (6.9) vanishes.As C takes values in rg P = g , it is enough to show existence of T for which C(U (v, T ), ψ(v, T )), g X 4 = 0. (6.10) We therefore consider the real function T → C(U (v, T ), ψ(v, T )), g X 4 and employ the Brouwer fixed point theorem to prove that it vanishes on [1 − δ N , 1 + δ N ].The central observation to this end is that, according to (4.1), we have for some c > 0. Based on this, by Taylor's formula, from (6.7) we get that where R 1 (v, T ) is continuous in T and R 1 (v, T ) δ/N 2 .Furthermore, based on the definition of the correction function C, we similarly conclude that where Therefore, there is a choice of sufficiently large N and sufficiently small δ such that Based on this, we get that (6.10) is equivalent to T = F (T ) (6.11) for some function F which maps the interval [1− δ N , 1+ δ N ] continuously into itself.Consequently, by the Brouwer fixed point theorem we infer the existence of T ∈ [1 − δ N , 1 + δ N ] for which (6.11), and therefore (6.10), holds.The claim of the theorem follows.

Upgrade to classical solutions
In this section we show that if the initial datum v is smooth and rapidly decaying, then the corresponding strong solution to (6.2) is in fact smooth, and satisfies (6.1) classically.To accomplish this, we first use abstract results of the semigroup theory to upgrade strong solutions to classical ones in the semigroup sense.Then we use repeated differentiation together with Schwarz's theorem on mixed partials to upgrade these to smooth solutions that solve (6.1) classically.
in the classical sense.
Proof.Recall from the linear theory that By this, from (6.2) and (5.17) we have that there is α ∈ R such that Consequently, ψ(τ ) ∈ X 5 for all τ ≥ 0. Since U (v, T ) ∈ X k for all k ≥ 5, we proceed inductively to get that ψ(τ ) ∈ X k for every k ≥ 5.Then, by the embedding X k ֒→ C k−3 rad (R 5 ) we conclude that ψ(τ ) ∈ C ∞ (R 5 ) for all τ ≥ 0.
Proof.First, we construct a fundamental system for the homogeneous equation We note that the origin ρ = 0 is a regular singular point.Hence, by the Frobenius method, there is a fundamental system {u 0 , u 1 } on (0, ∞), where u 0 is analytic at ρ = 0 with u 0 (0) = 1, u ′ 0 (0) = 0, and u 1 (ρ) ∼ ρ −3 near ρ = 0. To analyze the behavior of solutions at infinity we write the equation in normal form.With ω(r) := e Obviously, v ∈ C ∞ (0, ∞).Since f has bounded support, the first integral vanishes for large r and therefore there is a constant c such that v(r) = cv ∞ (r) for all large enough r.For r → 0, the first integral converges, hence the behavior of the first term is governed by v 0 .The second integral is of order O(r 5 ) which compensates the singular behavior of v ∞ at the origin.In particular, there is a constant C such that r −2 v(r) → C and r −1 v ′ (r) → 2C when r → 0 + .By transforming back, we obtain a solution u ∈ C 1 [0, ∞) ∩ C ∞ (0, ∞).By inspection, u ′ (0) = 0 and u (k) (ρ) = O(ρ −2−2λ−k ) for ρ → ∞ and k ∈ N 0 .

1. 3 .
Outline of the proof of the main result.Since we assume radial symmetry, i.e. u(t, x) = ũ(t, |x|), we first reformulate (1.1) in terms of the reduced mass w(t, r) s)s 2 ds.

. 11 )
For a closed linear operator (L, D(L)), we denote by ρ(L) the resolvent set, and for λ ∈ ρ(L) we use the following convention for the resolvent operator R L (λ) := (λ − L) −1 .The spectrum is defined as σ(L) := C \ ρ(L).The notation a b means a ≤ Cb for some C > 0, and we write a ≃ b if a b and b a.

5 )
via the Fourier transform.Since we are concerned with radial functions only, i.e., f = f (| • |), we straightforwardly get that