Abstract
We consider the parabolic–elliptic Keller–Segel system in dimensions \(d \geqq 3\), 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 \(d=3\). 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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We consider the system of equations
equipped with an initial condition \(u(0,\cdot )=u_0\), for \(u, v: [0,T) \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) and some \(T>0\). This model is frequently referred to as the parabolic–elliptic Keller–Segel system, named after the authors of [39], 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 [37]. 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. [1, 55].
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 \(u \mapsto u_{\lambda }\),
Furthermore, assuming sufficient decay of u at infinity, the total mass
is conserved. Since \({\mathcal {M}}(u_{\lambda }) = \lambda ^{d-2} {\mathcal {M}}(u)(\cdot /\lambda ^2)\), the model is mass critical for \(d=2\) and mass supercritical for \(d \geqq 3\).
It is well known that Equation (1.1) admits finite-time blowup solutions in all space dimensions \(d \geqq 2\), for which, in particular,
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. [35, 36].
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, 8, 17, 21]. Particular solutions that blow up in finite time via dynamical rescaling of Q,
with \(\lambda (t) \rightarrow 0\) for \(t \rightarrow T^-\), have been constructed for different blowup rates \(\lambda \), see [13, 34, 48, 53]. In particular, in [13] it is shown that the blowup solution corresponding to
for a certain explicit constant \(\kappa > 0\), is stable outside of radial symmetry. In addition to this, Mizoguchi [41] 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 \(d \geqq 3\) are more complex; in particular, multiple blowup profiles are known to exist. In a recent work, Collot, Ghoul, Masmoudi and Nguyen [14] proved for all \(d \geqq 3\) 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 \(d \geqq 3\), see [9, 33, 51]. A particular example was found in closed form in [9], and is given by
1.1 The Main Result
To understand the role of the solution (1.5) for generic evolutions of (1.1), the authors of [9] 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 data around \(u_1(0,\cdot )=U\) 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 \(\varepsilon >0\) such that, for any initial datum
where \(\varphi _0\) is a radial Schwartz function for which
there exists \(T>0\) and a classical solution \(u \in C^\infty ([0,T) \times {\mathbb {R}}^3)\) to (1.1), which blows up at the origin as \(t \rightarrow T^{-}\). Furthermore, the following profile decomposition holds:
where \( \Vert \varphi (t,\cdot ) \Vert _{{H}^3({\mathbb {R}}^3)} \rightarrow 0 \) as \(t \rightarrow T^-\).
Remark 1.2
As 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 \(d \geqq 4\). We therefore restrict ourselves to the lowest dimension, and the physically most relevant case, \(d=3\).
Remark 1.3
Due to the embedding \(H^3({\mathbb {R}}^3) \hookrightarrow L^\infty ({\mathbb {R}}^3)\), the conclusion of the theorem implies that the evolution of the perturbation (1.6), when dynamically self-similarly rescaled, converges back to U in \(L^\infty ({\mathbb {R}}^3)\). In other words,
uniformly on \({\mathbb {R}}^3\) as \(t \rightarrow T^-\).
1.2 Related Results for \(d \geqq 3\)
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 \(L^{\infty }({\mathbb {R}}^d)\) as well as in other function spaces, see e.g. [3, 25]. 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 \(L^{d/2}({\mathbb {R}}^d)\)-norm lead to global (weak) solutions [15]. This result was later extended by Calvez, Corrias, and Ebde [10] 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, 4, 40]. Concerning the existence of finite time blowup, the aforementioned works [10, 15] 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 [42]. For other, more recent results see [3, 5, 6, 43, 46, 52]. We point out, however, that in contrast to the \(d=2\) case, for \(d \geqq 3\) 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 straight– forward to conclude that blowup solutions of (1.1) satisfy \(\liminf _{t \rightarrow T^{-}} \,(T-t) \Vert u(t,\cdot ) \Vert _{L^{\infty }({\mathbb {R}}^d)} >0\), see e.g. [44]. Accordingly, singular solutions are classified as type I if
and as type II otherwise. The first formal construction of type II blowup was performed by Herrero, Medina and Velázquez for \(d=3\) in [32]; the singularity they construct consists of a smoothed-out shock wave concentrated in a ring that collapses into a Dirac mass the origin. This blowup mechanism was later observed numerically in [9] for higher dimensions as well, and is furthermore conjectured to be radially stable. A rigorous construction of this solution for all \(d \geqq 3\) came only recently in the work of Collot, Ghoul, Masmoudi and Nguyen [14], who also prove its radial stability. In contrast to these results, if the initial profile of a blowup solution is radially non-increasing 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 [52]. This is in particular the case for self-similar solutions, for which \(\lim _{t \rightarrow T^-} u(t,x) = C|x|^{-2}\) at the blowup time. To add to the importance of self-similar solutions for understanding the structure of singularities, Giga, Mizoguchi and Senba [25] showed that any radial, non-negative type I blowup solution of (1.1) is asymptotically self-similar. For \(3 \leqq d \leqq 9\), it is known that there are infinitely many similarity profiles, while for \(d \geqq 10\) there is at least one, see [51]. 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. [45]. In the mass subcritical case \(d=1\) blowup has been excluded in [12].
1.3 Outline of the Proof of the Main Result
Since we assume radial symmetry, i.e. \(u(t,x) = {{\tilde{u}}}(t,|x|)\), we first reformulate (1.1) in terms of the reduced mass
The advantage of this change of variable lies in the fact that it reduces (1.2) to a local semilinear heat equation on \({\mathbb {R}}^{5}\) for \(w(t,x):= {{\tilde{w}}}(t,|x|)\)
where \(\Lambda f(x):=x \cdot \nabla f(x)\), and the initial datum is radial \(w_0={\tilde{w}}(0,|\cdot |)\). Furthermore, similar to (1.2), Equation (1.7) obeys the scaling law
which leaves the \({\dot{H}}^{\frac{1}{2}}({\mathbb {R}}^5)\) norm invariant. 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 Sect. 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 [22,23,24]. Note that the strip \([0,T)\times {\mathbb {R}}^5\) is mapped via \((t,x) \mapsto (\tau ,\xi )\) into the half-space \([0,\infty ) \times {\mathbb {R}}^5\). Furthermore, by rescaling the dependent variable \((T-t)w(t,x)=:\Psi (\tau ,\xi )\), the solution \(w_T\) becomes \(\tau \)-independent, \(\xi \mapsto \phi (\xi )\). 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 \(\phi \). To study evolutions near \(\phi \), we consider the pertubation ansatz \(\Psi (\tau ,\cdot ) = \phi + \psi (\tau )\), which yields the central evolution equation of this paper:
Here, the linear operator L and the remaining nonlinearity N are explicitly given as
The next step is to establish a well-posedness theory for the Cauchy problem (1.8). To that end, in Sect. 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 Sect. 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 \({\mathcal {L}}\) generates a strongly continuous semigroup \(S(\tau )\) 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 \({\mathcal {L}}f=g\). Thereby we establish existence of the linear flow near \(\phi \). This flow exhibits growth in general, due to the existence of the unstable eigenvalue \(\lambda =1\) of the generator \({\mathcal {L}}\). 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 \({\mathcal {L}}\) in a suitably weighted \(L^2\)-space, we prove the existence of a (non-orthogonal) rank-one projection \({\mathcal {P}}: X^k \rightarrow X^k\) relative to \(\lambda =1\), such that \(\ker {\mathcal {P}}\) in an invariant subspace for \({\mathcal {L}}\), and the linear evolution in \(X^k\) decays exponentially on \(\ker {\mathcal {P}}\). This is expressed formally in the central result of the linear theory, Theorem 4.1.
The existence of \(S(\tau )\) 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 \(S(\tau )\), we show that the operator \(f \mapsto S(\tau )N(f)\) 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 \(S(\tau )\) and those of the accompanying nonlinear Lipschitz estimates comprise the main content of Sect. 5.
With these technical results at hand, in Sect. 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 \({\text {rg}}{\mathcal {P}}\) in the initial data, we employ a Lyapunov-Perron type argument, by means of which we also extract the blowup time T. In Sect. 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 \(n \geqq 6\). This involves carrying out the analogous well-posedness theory for (1.8) in the high-dimensional counterpart of the space \(X^k\)
along with using the same techniques to treat the underlying spectral problem.
1.4 Notation and Conventions
We write \({\mathbb {N}}\) for the natural numbers \(\{1,2,3, \dots \}\), \({\mathbb {N}}_0:= \{0\} \cup {\mathbb {N}}\). Furthermore, \({\mathbb {R}}^+:= \{x \in {\mathbb {R}}: x >0\}\). By \(C_{c}^\infty ({\mathbb {R}}^d)\) we denote the space of smooth functions with compact support. In addition, we define \( C_{\textrm{rad }}^{\infty }({\mathbb {R}}^d):= \{ u \in C^{\infty }({\mathbb {R}}^d): u \text { is radial}\}\) and analogously \(C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^d)\). Also, \({\mathcal {S}}_{\text {rad}}({\mathbb {R}}^d)\) stands for the space of radial Schwartz functions. By \(L^p(\Omega )\) for \(\Omega \subseteq {\mathbb {R}}^d\), we denote the standard Lebesgue space. For the Fourier transform we use the following convention:
For a closed linear operator \((L, {\mathcal {D}}(L))\), we denote by \(\rho (L)\) the resolvent set, and for \(\lambda \in \rho (L)\) we use the following convention for the resolvent operator \(R_{L}(\lambda ):=(\lambda - L)^{-1}\). The spectrum is defined as \(\sigma (L):= {\mathbb {C}}\setminus \rho (L)\). The notation \(a\lesssim b\) means \(a\leqq Cb\) for some \(C>0\), and we write \(a\simeq b\) if \(a\lesssim b\) and \(b \lesssim a\). We use the common notation \(\langle x \rangle := \sqrt{1+|x|^2}\) also known as the Japanese bracket.
2 Equation in Similarity Variables
To restrict to radial solutions of (1.1) we assume that \(u(t,\cdot )={\tilde{u}}(t,|\cdot |)\) and \(u_0={\tilde{u}}_0(|\cdot |)\). Then, we define the so-called reduced mass
and let \(n:=d+2\). The utility of the reduced mass is reflected in the fact that by means of the function \(w(t,x):={\tilde{w}}(t,|x|)\), the initial value problem for the system (1.1) can be rewritten in the form of a single local semilinear heat equation in w
Here \((t,x) \in [0,T) \times {\mathbb {R}}^{n}\),
and
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 \(S_T:=[0,T) \times {\mathbb {R}}^n\) into the upper half-space \(H_+:=[0,+\infty ) \times {\mathbb {R}}^n\). Also, we define the rescaled dependent variable
Consequently, the evolution of w inside \(S_T\) corresponds to the evolution of \(\Psi \) inside \(H_+\). Furthermore, since
we get that the nonlinear heat equation (2.1) transforms into
with the initial datum
For convenience, we denote
Now, we have that for all \(n \geqq 5\) the function
is a static solution to (2.6). To analyze stability properties of \(\phi _n\), we study evolutions of initial data near \(\phi _n\), and for that we consider the perturbation ansatz
This leads to the central evolution equation of the paper,
where
Furthermore, we write the initial datum as
where, for convenience, we denote
Now we fix \(n=5\), and study the Cauchy problem (2.8)–(2.10). For this we need a convenient functional setup.
3 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 \(X^k\)
Let \(k \in {\mathbb {N}}\) and \(f \in C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5)\). As usual, we define the Sobolev norm \(\Vert f \Vert _{\dot{H}^{k}({\mathbb {R}}^5)}:= \Vert |\cdot |^{k} {\mathcal {F}} f \Vert _{L^2({\mathbb {R}}^5)}\) via the Fourier transform. Since we are concerned with radial functions only, i.e., \(f = {{\tilde{f}}}(|\cdot |)\), we straightforwardly get that
with
where \( \Delta _{\textrm{rad}}:= r^{-4} \partial _r ( r^{4} \partial _r)\) denotes the radial Laplace operator on \({\mathbb {R}}^5\). We also define an inner product on \(C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5)\) by
with corresponding norm \(\Vert f \Vert _{X^k}:= \sqrt{\langle f,f\rangle _{X^k}}\). The central space of our analysis is the completion of \((C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5), \Vert \cdot \Vert _{X^k})\), 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:
Lemma 3.1
Let \(k \in {\mathbb {N}}\). Then
for all \(f \in C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5)\), and all \(\alpha \in {\mathbb {N}}_0^5\) with \(1 \leqq |\alpha |\leqq k.\)
Also, it is straightforward to see that the following result holds:
Lemma 3.2
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). Then
In particular, elements of \(X^k\) can be identified with functions in \(C_\textrm{rad }^{k-3}({\mathbb {R}}^5)\). Furthermore, \(X^{k}\) is a Banach algebra, i.e.,
for all \(f,g \in X^{k}\). In addition, \(X^{k_2}\) embeds continuously into \(X^{k_1}\) for \( k_1 \leqq k_2\); shortly
The next result follows from an elementary approximation argument.
Lemma 3.3
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). Define
Then \({\mathcal {C}} \subset X^{k} \).
Throughout the paper, we frequently use the commutator relation
for \(\alpha \in {\mathbb {N}}_0\), \(|\alpha | = k\), as well as its radial analogue
3.2 Weighted \(L^2-\)Spaces
For \(x \in {\mathbb {R}}^5\), we set
where
If for a non-negative measurable function \(\omega \) on \({\mathbb {R}}^5\) we denote
then we define the following Hilbert spaces of radial functions
with the corresponding inner products
We note that both \({\mathcal {H}}_0\) and \({\mathcal {H}}\) have \(C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\) as a dense subset. An immediate consequence of the exponential decay of the weight functions is the following result:
Lemma 3.4
Let \(k \in {\mathbb {N}}, k \geqq 3\). Then
for every \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\). 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, \(\phi \in X^{k}\) for any \(k \in {\mathbb {N}}, k \geqq 3\), and the same holds for \(\Lambda \phi \).
4 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 \(k \geqq 3\). To accomplish this, we use 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, \(\lambda =1\), with an explicit eigenfunction
According to Lemma 3.3, the function \(\nu \) belongs to \(X^k\), and this allows us to define a projection operator \({\mathcal {P}}: X^k \rightarrow X^k\) by
This whole section is devoted to proving the next theorem, which, in short, states that the linear flow of (2.8) decays exponentially in time on the kernel of \({\mathcal {P}}\).
Theorem 4.1
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). Then the operator \(L: C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5) \subset X^k \rightarrow X^k\) is closable, and its closure generates a strongly continuous semigroup \((S(\tau ))_{\tau \geqq 0}\) of bounded operators on \(X^k\). Furthermore, there exists \(\omega _k \in (0,\frac{1}{4})\) such that
for all \(f \in X^k\) and all \(\tau \geqq 0\).
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 \({\mathcal {H}}\).
4.1 The Linearized Evolution on \({\mathcal {H}}\)
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 [28], Lemma 3.1, for a detailed proof:
Lemma 4.2
The operator \(L_0: {\mathcal {D}}(L_0) \subset {\mathcal {H}}_0 \rightarrow {\mathcal {H}}_0\) is closable, and its closure \(({\mathcal {L}}_0,{\mathcal {D}}({\mathcal {L}}_0))\) generates a strongly continuous semigroup \((S_0(\tau ))_{\tau \geqq 0}\) of bounded operators on \({\mathcal {H}}_0\). Explicitly,
where \(G_\tau (x)=[4\pi \alpha (\tau )]^{-\frac{5}{2}}e^{-|x|^2/4\alpha (\tau )}\) and \(\alpha (\tau )=1-e^{-\tau }\).
The operator L, on the other hand, has a self-adjoint realization in the space \({\mathcal {H}}\); we have the following fundamental result.
Proposition 4.3
The operator \( L: {\mathcal {D}}(L) \subset {\mathcal {H}} \rightarrow {\mathcal {H}}\) is closable, and its closure \(({\mathcal {L}},{\mathcal {D}}({\mathcal {L}}))\) generates a strongly continuous semigroup \((S(\tau ))_{\tau \geqq 0}\) of bounded operators on \({\mathcal {H}}\). The spectrum of \({\mathcal {L}}\) consists of a discrete set of eigenvalues, and moreover,
where \(\lambda =1\) is a simple eigenvalue with the normalized eigenfunction g from (4.2). Furthermore, for the orthogonal projection \({\mathcal {P}}: {\mathcal {H}} \rightarrow {\mathcal {H}}\) defined in (4.2) there exists \(\omega _0 >0\) such that
for all \(f \in {\mathcal {H}}\) and all \(\tau \geqq 0\).
Proof
We define the unitary map
and note that \(-L = U A U^{-1}\) with
where
and \({\mathcal {D}}(A) = U^{-1} {\mathcal {D}}(L)\). Let us denote by \(A_c\) the restriction of A to \(C^{\infty }_c({\mathbb {R}}^+)\). Using standard results, we infer that \(A_c\) is limit-point at both endpoints of the interval \((0,\infty )\) (see, e.g., [54], 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 \({\mathcal {A}}: {\mathcal {D}}({\mathcal {A}}) \subset L^2({\mathbb {R}}^+) \rightarrow L^2({\mathbb {R}}^+)\), where
and \({\mathcal {A}} u = A u\) for \(u \in {\mathcal {D}}({\mathcal {A}})\). The inclusion \(A_c \subset A \subset {\mathcal {A}} = \overline{A_c}\) implies \({\overline{A}} ={\mathcal {A}}\). Now, since q is bounded from below, the same is true for the operator \({\mathcal {A}}\), i.e, there is a constant \(\mu > 0\) such that \(\textrm{Re}\langle {\mathcal {A}} u, u \rangle _{L^2({\mathbb {R}}^+)} \geqq -\mu \Vert u \Vert _{L^2({\mathbb {R}}^+)}\) for all \(u \in {\mathcal {D}}({\mathcal {A}})\). Since \(q(r) \rightarrow \infty \) when \(r \rightarrow \infty \), \({\mathcal {A}}\) 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, \( L: {\mathcal {D}}(L) \subset {\mathcal {H}} \rightarrow {\mathcal {H}}\) is essentially self-adjoint and its closure is given by \({\mathcal {L}} = - U {\mathcal {A}} U^{-1}\) with \({\mathcal {D}}({\mathcal {L}}) = U {\mathcal {D}}({\mathcal {A}})\). Moreover, we have that \(\textrm{Re}\langle {\mathcal {L}} f, f \rangle _{{\mathcal {H}}} \leqq \mu \Vert f \Vert _{{\mathcal {H}}}\) for all \(f \in {\mathcal {D}}({\mathcal {L}})\). This implies that \({\mathcal {L}}\), being self-adjoint, generates a strongly continuous semigroup on \({\mathcal {H}}\).
Next, we describe the spectral properties of \({\mathcal {L}}\). Obviously, \(g \in {\mathcal {D}}({\mathcal {L}})\) and \( L g = g\) by explicit calculation. Hence, \({\tilde{g}}:= U^{-1} g \) satisfies \((1 + A) {\tilde{g}} = 0\). Moreover, \({\tilde{g}}\) is strictly positive on \((0, \infty )\). Hence, we have the factorization \(A = A^{-} A^{+}-1\), where
We define \( A_{{\mathcal {S}}}:= A^{+} A^{-} - 1\). Explicitly, we have
with
This gives rise to the maximally defined self-adjoint operator \({\mathcal {A}}_{{\mathcal {S}}}: {\mathcal {D}}({\mathcal {A}}_{{\mathcal {S}}}) \subset L^2({\mathbb {R}}^+) \rightarrow L^2({\mathbb {R}}^+)\) which is, by construction, isospectral to \({\mathcal {A}}\) except for \(\lambda = -1\). For a detailed discussion on the above process of “removing” an eigenvalue, see [29], Section B.1. Now, to prove (4.5), it is enough to show that \({\mathcal {A}}_S\) does not have any eigenvalues in \((-\infty ,0]\). For this, we use a so-called GGMT criterion, see e.g. Appendix A in [16], that we adapted to our problem in [28], Theorem A.1. In particular, we show that for \(p=2\),
where \(\alpha = 6\) and \(Q_S^-(r) = \min \{ Q_S(r), 0 \}\). By calculating explicitly the root of \(Q_S\) one can easily check that \(Q_S(r) > 0\) for \(r \geqq 5\), hence
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 Equation (4.7) for \(p=2\) and \(\alpha =6\). Theorem A.1 in [28] now implies that \(\sigma ({\mathcal {A}}_S) \subset (0,+\infty )\), and (4.5) follows. Furthermore, a simple ODE analysis yields that the geometric eigenspace of \(\lambda =1\) is equal to \(\langle g \rangle \). Consequently, \({\mathcal {P}}\) is the orthogonal projection onto \({\text {rg}}{\mathcal {P}}\) in \({\mathcal {H}}\), and we readily get (4.6). \(\square \)
The next lemma shows that the growth bounds are preserved when the linear evolution is measured in graph norms associated to fractional powers of the operator \(1- {\mathcal {L}}\). This result will be crucial in Sect. 4.2, in particular, in conjunction with Lemma 4.6 below.
Lemma 4.4
There is a unique self-adjoint, positive operator \((1 - {\mathcal {L}})^{\frac{1}{2}}\) with \(C^\infty _{c,\textrm{rad}}({\mathbb {R}}^5)\) as a core, such that \(\big ((1 - {\mathcal {L}})^{\frac{1}{2}}\big )^2 = 1 - {\mathcal {L}}\). For \(k \in {\mathbb {N}}_0\) and \(f \in {\mathcal {D}}((1- {\mathcal {L}})^{k/2})\) we define the graph norm
and infer that
for every \(k \in {\mathbb {N}}_0\), \(f \in {\mathcal {D}}((1- {\mathcal {L}})^{k/2})\) and all \(\tau \geqq 0\).
Proof
By standard results (see, e.g., [38], p. 281, Theorem 3.35), the square root \((1 - {\mathcal {L}})^{\frac{1}{2}}\) exists and commutes with any bounded operator that commutes with \({\mathcal {L}}\). We show that \(C^\infty _{c,\textrm{rad}}({\mathbb {R}}^5)\) is a core of \((1-{\mathcal {L}})^\frac{1}{2}\). Let \(\varepsilon > 0\). Since \({\mathcal {D}}({\mathcal {L}})\) is core for \((1 - {\mathcal {L}})^{\frac{1}{2}}\) and \(C^\infty _{c,\textrm{rad}}({\mathbb {R}}^5)\) is a core for \(1 - {\mathcal {L}}\) there is \({{\tilde{f}}} \in C^\infty _{c,\textrm{rad}}({\mathbb {R}}^5)\) such that \(\Vert f - {{\tilde{f}}} \Vert _{{\mathcal {H}}} + \Vert (1- {\mathcal {L}})^{\frac{1}{2}} (f - {{\tilde{f}}}) \Vert _{{\mathcal {H}}} < \varepsilon \), by using that \(\Vert (1- {\mathcal {L}})^{\frac{1}{2}} f\Vert _{{\mathcal {H}}} \lesssim \Vert (1- {\mathcal {L}})f\Vert _{{\mathcal {H}}} + \Vert f \Vert _{{\mathcal {H}}}\). The growth bounds for the semigroup can be proved by induction using the fact that \((1-{\mathcal {L}})^{\frac{1}{2}}\) commutes with the projection and the semigroup. \(\square \)
We conclude this section by proving two technical results that will be crucial in the sequel.
Lemma 4.5
Let \(k \in {\mathbb {N}}_0\) and \(R >0\). Then
for all \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\) and all \(\alpha \in {\mathbb {N}}_0^5\) with \(|\alpha | = k\).
Proof
First, for \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\) and \(\mu (x):= |x|^4 e^{-|x|^2/4 }\) 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,
for \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\). 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 \(k \geqq 1\). Then we have that for all \(1 \leqq j \leqq k\)
Lemma A.1 then implies the claim for \(|\alpha |=k+1\). \(\square \)
Finally, we show that graph norms can be controlled by \(X^k\)-norms.
Lemma 4.6
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). Then
for all \(f \in X^k\) and all \(\kappa \in \{0, \dots , k \}\).
Proof
We prove the statement for \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\). The claim then follows by density and the closedness of \((1-{\mathcal {L}})^{\kappa /2}\). First, it is easy to see that
for smooth, radial and polynomially bounded functions \(p_\alpha \). Using the exponential decay of the weight function \(\sigma \) along with interpolation, see Lemma 3.1, and Hardy’s inequality, we get
\(\square \)
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 \(X^k\). Since \(C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\) is dense in \(X^{k}\), we consider L as defined in (4.4).
Proposition 4.7
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). The operator \(L: {\mathcal {D}}(L) \subset X^k \rightarrow X^k\) is closable and with \(({\mathcal {L}}_k, {\mathcal {D}}({\mathcal {L}}_k))\) denoting the closure we have that \({\mathcal {C}} \subset {\mathcal {D}}({\mathcal {L}}_k)\). Furthermore, the operator \({\mathcal {L}}_k\) generates a strongly continuous semigroup \((S_{k}(\tau ))_{\tau \geqq 0}\) of bounded operators on \(X^k\), which coincides with the restriction of \(S(\tau )\) to \(X^k\), i.e,
for all \(\tau \geqq 0\).
Proof
We prove the first part of the statement by an application of the Lumer-Phillips theorem. For this, we show that
for some \({\bar{\omega }}_k >0\) and all \(f \in {\mathcal {D}}(L)\). 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 \(R \geqq 1\), there are constants \(C_k, C_{R,k} > 0\) such that for all \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\),
An application of Lemma 4.6 to Equation (4.9) immediately implies Equation (4.8). Equation (4.9) will be proved in several steps. First, recall that \(L = L_0 + L'\). By partial integration,
Based on the identity
which easily follows from integration by parts and the commutator relation Equation (3.3), we get
Consequently,
and thus
To obtain Equation (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
with
Note that \(V \in C^{\infty }_{\textrm{rad}}({\mathbb {R}}^5)\) and the properties of \(\phi \) imply that
for \(\alpha \in {\mathbb {N}}_0^5\). First, we prove that
for suitable constants \(C_{R}, C >0\). For this, we use the fact that
for all \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\). Hence, we can apply the standard Leibniz rule to estimate products. More precisely, for \(\alpha ,\beta ,\gamma \in {\mathbb {N}}_0^5\) for which \(|\alpha |=|\beta +\gamma | = k\), we estimate
By Equation (4.14) and Lemmas 4.5 and 3.1 the last term can be bounded by
Similarly, we have that
For \(\gamma = 0\), we estimate the last term by Hardy’s inequality,
For \(\gamma \ne 0\), by interpolation and equivalence of norms, see Lemma (3.1), we obtain
which implies Equation (4.15). To estimate the second term in Equation (4.13), we use the relation
with certain smooth functions \(|\varphi _{\beta }(x)| \lesssim \langle x \rangle ^{-2}\). Note that, by partial integration, we have the 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., [20], p. 82, Proposition 3.14-(iv)), and that the closure \({\mathcal {L}}_k: {\mathcal {D}}({\mathcal {L}}_k) \subset X^k \rightarrow X^k\), satisfies
for all \(f \in {\mathcal {D}}({\mathcal {L}}_k)\) and some suitable \({\bar{\omega }}_k > 0\).
Next, we prove that \({\mathcal {C}} \subset {\mathcal {D}}({\mathcal {L}}_k)\) by showing that for \(f \in {\mathcal {C}}\) there is a sequence \((f_n) \subset C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\), such that \(f_n \rightarrow f\) and \(L f_n \rightarrow v\) in \(X^k\). Then, by definition of the closure, \(f \in {\mathcal {D}}({\mathcal {L}}_k)\) and \({\mathcal {L}}_k f = v\). We set \(f_n = \chi (\cdot /n) f\), where \(\chi \) a smooth, radial cut-off function equal to one for \(|x| \leqq 1\) and zero for \(|x| \geqq 2\). It is easy to see that \((f_n) \subset C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\) converges to f in \(X^k\). More precisely, by exploiting the decay of f,
we have \(f_n \rightarrow f\) in \(L^{\infty }({\mathbb {R}}^5)\). Furthermore, for \(m \leqq n\), \(f_n - f_m = 0\) on \({\mathbb {B}}^5_m\) and thus
We have
and similarly \(\Vert D^{k}(\chi _n f) \Vert _{L^2({\mathbb {R}}^5{\setminus } {\mathbb {B}}^5_m)} \lesssim n^{-\frac{1}{2}}\). Thus, \(\Vert f_n - f_m \Vert _{X^k} \lesssim n^{-\frac{1}{2}} + m^{-\frac{1}{2}}\) and \((f_n)\) converges in \(X^k\) to some limiting function which must be equal to f by the \(L^{\infty }-\)embedding and the uniqueness of limits.
Now, \(\Lambda f \in {\mathcal {C}}\) for \(f \in {\mathcal {C}}\). Since \(\Lambda f_n = f \Lambda \chi _n + \chi _n \Lambda f\) and
we have \(\Lambda f_n \in C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5) \rightarrow \Lambda f\) in \(L^{\infty }({\mathbb {R}}^5)\). By similar considerations as above one finds that \((\Lambda f_n)\) is Cauchy in \(X^k\) and thus \(\Lambda f_n \rightarrow \Lambda f\) in \(X^k\) for \(n \rightarrow \infty \). Now,
Arguments as above and the Banach algebra property of \(X^k\) imply that \((L f_n)\) converges in \(X^k\) to some limiting function \(v \in X^k\). By Sobolev embedding, \(L f_n \rightarrow v\) in \(L^{\infty }({\mathbb {R}}^5)\) and by convergence of the individual terms, \(v = \Delta f - \frac{1}{2} \Lambda f - f + 2 \Lambda (\phi f) + 12 \phi f.\) This shows in particular, that \({\mathcal {L}}_k\) acts as a classical differential operator on \({\mathcal {C}}\).
For the invocation of the Lumer–Phillips theorem, it is left to prove the density of the range of \(\lambda _k - {\mathcal {L}}_k\) for some \(\lambda _k > {\bar{\omega }}_k\). 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 \(f \in C^{\infty }_{c,\textrm{rad }}({\mathbb {R}}^5)\) such that \(f = {{\tilde{f}}} (|\cdot |)\). By Lemma C.1, there exists \(\lambda > {\bar{\omega }}_k\) such that the ODE
with \(\phi = {{\tilde{\phi }}}(|\cdot |)\), has a solution \({{\tilde{u}}} \in C^1[0,\infty ) \cap C^{\infty }(0,\infty )\) satisfying \({{\tilde{u}}}'(0) = 0\) as well as \({{\tilde{u}}}^{(j)}(\rho ) = {\mathcal {O}}(\rho ^{-3-j})\) for \(j \in {\mathbb {N}}_0\) as \(\rho \rightarrow \infty \). By setting \(u:= {{\tilde{u}}}(|\cdot |)\), we obtain a classical solution to the equation
on \({\mathbb {R}}^5\setminus \{0\}\). Since u belongs to \(H^1({\mathbb {R}}^5)\), it solves (4.20) weakly on \({\mathbb {R}}^5\), and by elliptic regularity we infer that \(u \in C^{\infty }_{\textrm{rad}}({\mathbb {R}}^5)\). The decay of \({{\tilde{u}}}\) at infinity implies that \(u \in {\mathcal {C}}\). Hence, \(u \in {\mathcal {D}}( {\mathcal {L}}_k)\) which implies the claim.
An application of the Lumer–Phillips Theorem now proves that \(({\mathcal {L}}_k, {\mathcal {D}}({\mathcal {L}}_k))\) generates a strongly continuous semigroup \((S_k(\tau ))_{\tau \geqq 0}\) on \(X^k\). In view of the embedding \(X^k \hookrightarrow {\mathcal {H}}\) and the fact that \(C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\) is a core for \({\mathcal {L}}\) and \({\mathcal {L}}_{k}\) for any k, an application of Lemma C.1 in [27] proves the claimed restriction properties. \(\square \)
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 Equation (4.12) and setting \(L'=0\) in the subsequent arguments.
Lemma 4.8
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). The operator \(L_0: {\mathcal {D}}(L_0) \subset X^k \rightarrow X^k\) is closable and its closure \(({\mathcal {L}}_{0,k}, {\mathcal {D}}({\mathcal {L}}_{0,k}))\) generates a strongly continuous semigroup on \(X^k\), which coincides with the restriction of the \(S_0(\tau )\) to \(X^k\) for \(\tau \geqq 0\). Furthermore,
for all \(f \in X^k\) and \(\tau \geqq 0\).
4.3 Proof of Theorem 4.1
First, we show that the operator \({\mathcal {P}}\) from (4.2) induces a non-orthogonal rank-one projection on \(X^k\). To indicate the dependence on k, we write
for \(f \in X^k\). By its decay properties, the function g is an element of \({\mathcal {C}} \subset X^k\) for any \(k \in {\mathbb {N}}\). In view of the embedding \(X^k \hookrightarrow {\mathcal {H}}\), the inner product makes sense for \(f \in X^k\) and by definition \({\mathcal {P}}_{X^{k}}^2 = {\mathcal {P}}_{X^{k}}\). The fact that the projection commutes with \(S_{k}(\tau )\) follows from the respective properties on \({\mathcal {H}}\).
Let \(f \in C^\infty _{c,\textrm{rad}}({\mathbb {R}}^5)\). By Proposition 4.7, \({\tilde{f}}:=(1-{\mathcal {P}}_{X^{k}})f \in {\mathcal {C}} \subset {\mathcal {D}}({\mathcal {L}}_k)\). Using Equation (4.9), Lemmas 4.4 and 4.6, we infer that for \(R \geqq 1\) sufficiently large,
for \(c = \frac{1}{2} \min \{\omega _0, \frac{1}{8} \}\). Hence,
and integration yields
for some suitably chosen \(\omega _k > 0\) and all \(\tau \geqq 0\). For \(f \in X^k\), the same bound follows by density. Again, for simplicity, we write \({\mathcal {P}} f= {\mathcal {P}}_{X^k} f\) for \(f \in X^k\).
5 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 \(k \geqq 3\) the space \(X^k\) embeds into \(C^{k-3}_{\text {rad}}({\mathbb {R}}^5)\). 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:
Lemma 5.1
Let \(k \in {\mathbb {N}}, k \geqq 4\). Given \(f,g \in X^k\) we have that \(\Lambda (fg) \in X^{k-1}\). Furthermore,
for all \(f,g \in X^k\).
Proof
In this proof we crucially rely on a recently established inequality for the weighted \(L^\infty \)-norms of derivatives of radial Sobolev functions; see [27], Proposition B.1. For convenience, we copy here the version of this result in five dimensions. Namely, given \(s \in (\frac{1}{2},\frac{5}{2})\) and \(\alpha \in {\mathbb {N}}_0^5\), we have that
for all \(u \in C^\infty _{c,\text {rad}}({\mathbb {R}}^5)\). Now we turn to proving (5.1). Due to the \(W^{1,\infty }\)-embedding of \(X^k\) for \(k \geqq 4\) and the fact that \((f,g) \mapsto \Lambda (fg)\) is bilinear, it is enough to show (5.1) for \(f,g \in C^\infty _{c,\text {rad}}({\mathbb {R}}^5)\). To estimate the \({\dot{H}}^{k-1}\) part, we do the following. If k is odd, then
where the last estimate follows from a combination of the \(L^\infty \)-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 \(|\alpha | \leqq |\beta |\). We then separately treat the integrals corresponding to the unit ball and its complement. For the unit ball, we first assume that \(\alpha =0\). Then
by the \(L^\infty \)-embedding. If \(\alpha \ne 0\), we have that
by (5.2) and Hardy’s inequality. For the complement of the unit ball, we have
by (5.2) 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 \({\dot{H}}^1\) part of the norm is treated in the same fashion. \(\square \)
Now we establish the crucial smoothing properties of \(S_0(\tau )\).
Proposition 5.2
Let \(k \in {\mathbb {N}}, k \geqq 3\) and \(l \in {\mathbb {N}}_0\). Then, given \(f \in X^k\) the function
maps \((0,\infty )\) continuously into \(X^{k+l}\). Furthermore, denoting \(\beta (\tau ):=\alpha (\tau )^{-\frac{1}{2}}\), where \(\alpha \) is defined in Lemma 4.2, we have that
for all \(\tau >0\) and all \(f \in X^k\).
Proof
The proof follows from a straightforward computation on the Fourier side. Explicitly, by definition of \(S_0(\tau )\), see Lemma 4.2, and scaling of Sobolev norms we get
for all \(s \geqq 0\) and all \(f \in C^{\infty }_{c,\textrm{rad}}({\mathbb {R}}^5)\). We infer that for all \(f \in X^k\),
and
where we used that \(\Vert |\cdot |^l \alpha (\tau )^{\frac{l}{2}} \hat{G}_{\tau }\Vert _{L^{\infty }({\mathbb {R}}^5)} \lesssim 1\) for all \(\tau \geqq 0\). This implies (5.4). Continuity of the map \(\tau \mapsto S_0(\tau )f: (0,\infty ) \rightarrow X^{k+l}\) follows from the continuity of the kernel maps \(\hat{G}_{\tau }, |\cdot | \alpha (\tau )^{\frac{1}{2}} \hat{G}_{\tau }: (0,\infty ) \rightarrow L^{\infty }({\mathbb {R}}^5)\). \(\square \)
Now we propagate the smoothing estimates of \(S_0(\tau )\) to \(S(\tau )\). We take a perturbative approach, and for that we need the following result:
Lemma 5.3
Let \(f \in C^\infty _{c,rad }({\mathbb {R}}^5)\), \(k \geqq 3\), and \(\tau \geqq 0\). Then the following relations hold in \(X^k\):
Proof
Define \(\xi _f:[0,\tau ] \rightarrow X^k\) by \(s \mapsto S(\tau -s)S_0(s)f\). We prove that \(\xi _f\) is continuously differentiable. More precisely, we show that
which is a continuous function from \([0,\tau ]\) into \(X^k\). To show this, we first write
and then by letting \(h \rightarrow 0\) we get (5.7). For the first term above, this follows from the fact that \(S_0(s)f \in {\mathcal {C}} \subset {\mathcal {D}}({\mathcal {L}}_k)\) and that \({\mathcal {L}}_kf=Lf\) for \(f \in {\mathcal {C}}.\) The conclusion for the second term follows by similar reasoning for \(S_0(\tau )\), together with the strong continuity of \(S(\tau )\) in \(X^k\). Now, continuity of \(\xi '_f\) follows from the continuity of the map
and the strong continuity of \(S(\tau )\) in \(X^k\). We note that, according to the definition of \(L'\), the continuity of (5.8) follows from the strong continuity of \(S_0(\tau )\) on \(X^{k+1}\) and the estimate (5.1). Finally, by integrating (5.7), we get (5.5). To prove (5.6), 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 \({\mathcal {D}}({\mathcal {L}}_{k+1}) \subset {\mathcal {D}}({\mathcal {L}}_{0,k})\), and therefore \(S(s)f \in {\mathcal {D}}({\mathcal {L}}_{0,k})\) for every \(k \geqq 3\). Continuity of \(\eta '_f\), similarly to above, follows from the continuity of \(s \mapsto L' S(s)f:[0,\tau ] \rightarrow X^k\) and the strong continuity of \(S_0(\tau )\) in \(X^k\). \(\square \)
Recall the operator N from (3.4). According to Lemma 5.1 we have that \(N: X^{k} \rightarrow X^{k-1}\) for \(k \geqq 4\). Also, recall the projection operator \({\mathcal {P}}={\mathcal {P}}_{X^k}\) from (4.21). Now we prove the central result of this section.
Proposition 5.4
Let \(k \geqq 3\). If \(f \in X^k\) then
is a continuous map. Furthermore, there exists \(\tilde{\omega }_k >0\) such that
for all \(\tau > 0\) and all \(f \in X^k\), with \(\omega _{k+1} > 0\) from Theorem 4.1.
Proof
The proof we give here is based exclusively on semigroup methods. In the Appendix 7 we provide a more standard proof by energy methods. We first prove (5.10). To this end, we use (5.6), (5.4), and (5.1) to obtain
for all \(\tau >0\) and \(f \in C^\infty _{c,\text {rad}}({\mathbb {R}}^5)\). Now, a generalization of Gronwall’s lemma to weakly singular kernels (see, e.g., Henry [31], p. 188, Theorem 7.1.1) yields (5.10) for all \(\tau \in (0,2)\). To treat higher values of \(\tau \) we first note that, according to (5.6), for \(\tau \geqq 2\) we have that
From here, by the previous step (for small \(\tau \)), Theorem 4.1, Lemma 5.1 and smoothing of \(S_0(\tau )\), we get
wherefrom follows the estimate (5.10) for all \(\tau \geqq 2\).
Now we use (5.10), (5.5) and the decay of \(S(\tau )\) on the orthogonal space to establish (5.11). We separately treat small and large values of \(\tau \). First note that from (5.10) it follows that (5.11) holds for all \(\tau \in (0,2)\). Now assume \(\tau \geqq 2\). Then, we can write
Now, denote \(\tilde{f}:=(1-{\mathcal {P}})f\). Then, according to Theorem 4.1, estimate (5.10), and smoothing of \(S_0(\tau )\) we have that
for all \(\tau \geqq 2\). From here, estimate (5.11) follows. The continuity of the map \(S(\tau )f: (0,\infty ) \rightarrow X^{k+1}\) for \(f \in X^{k}\) follows from (5.10) and the strong continuity of \(S(\tau )\) in \(X^k\). \(\square \)
As the last result of this section, we prove the local Lipschitz continuity in \(X^4\) of the composition of \({\mathcal {P}}\) and N.
Lemma 5.5
We have that
for all \(f,g \in X^4\).
Proof
By definition, for \(u,v \in X^4\) we have
Therefore, by Cauchy-Schwarz, the embedding \(X^3 \hookrightarrow {\mathcal {H}}\), and Lemma 5.1, we get that
for all \(u,v \in X^4\). The estimate (5.12) then follows by letting \(u=f+g\) and \(v=f-g\). \(\square \)
6 Construction of Strong Solutions
For simplicity, from now on we will drop the subscript in \(\Vert \cdot \Vert _{X^4}\), 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 \({\mathcal {L}}\) generates the semigroup \(S(\tau )\), to rewrite (6.1) into integral form
Then, as \(S(\tau )\) 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 \(\lambda =1\). 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 \(\omega _4\) is from Proposition 5.4. Then, we denote
Now, we define a correction function \(C: X^4 \times {\mathcal {X}} \rightarrow X^4\) by
and a map \( K_{u}: {\mathcal {X}} \rightarrow C([0,\infty ),X^4)\) by
The fact that \(K_u(\psi )(\tau )\) is a well-defined element of \(X^4\) for every \(\tau \geqq 0\), follows from Theorem 4.1, Proposition 5.4 and the integrability of \(\beta \). Similarly, the continuity of \(K_u(\psi ):[0,\infty ) \rightarrow X^4\) follows.
Proposition 6.1
For all sufficiently small \(\delta >0\) and all sufficiently large \(C > 0\) the following holds. If \(u \in {\mathcal {B}}_{\delta /C}\) then there exits a unique \(\psi = \psi ({u}) \in {\mathcal {X}}_\delta \) for which
Furthermore, the map \({u} \mapsto \psi ({u}): {\mathcal {B}}_{\delta /C} \rightarrow {\mathcal {X}}\) is Lipschitz continuous.
Proof
To utilize the decay of \( S(\tau )\) on the stable subspace, we write \( K_{u}\) in the following way:
Then, according to Proposition 5.4 we get that if \(\psi (s) \in {\mathcal {B}}_\delta \) for all \(s \geqq 0\) then
Furthermore, since \(\beta \) is integrable, if \( u \in {\mathcal {B}}_{\delta /C}\) and \(\psi \in {\mathcal {X}}_{\delta }\) then the above estimate implies the bound
Also, we similarly get that
for all \(\psi ,\varphi \in {\mathcal {X}}_\delta \). Now, the last two displayed equations imply that for all small enough \(\delta \) and for all large enough C, given \( u \in {\mathcal {B}}_{\delta /C}\) the operator \( K_{ u}\) is contractive on \({\mathcal {X}}_\delta \), with the contraction constant \(\frac{1}{2}\). Consequently, the existence and uniqueness of solutions to (6.4) follows from the Banach fixed point theorem. To show continuity of the map \( u \mapsto \psi ( u) \) we utilize the contractivity of \( K_{ u}\). Namely, we have the estimate
wherefrom the Lipschitz continuity follows. \(\square \)
Lemma 6.2
Recall the initial data operator U from (2.10). For \(\delta \in (0,\frac{1}{2}]\) and \({ v} \in X^4 \) the map
is continuous. In addition, we have that
for all \({ v} \in X^4 \) and all \(T \in [\frac{1}{2},\frac{3}{2}]\).
Proof
Fix \(\delta \in (0,\frac{1}{2}]\) and \( v \in X^4\). Then for \(T,S \in [1-\delta ,1+\delta ]\) we have that
Let \(\varepsilon >0\). By density, we know that there exists \({\tilde{w}}_0 \in C^\infty _{c,\text {rad}}({\mathbb {R}}^5)\) for which \(\Vert w_0 - {\tilde{w}}_0 \Vert < \varepsilon \). Now, by writing
and using the fact that \(\lim _{S \rightarrow T} \Vert {\tilde{w}}_0(\sqrt{T}\cdot ) - {\tilde{w}}_0(\sqrt{S}\cdot ) \Vert =0, \) from (6.6) we see that
Then, continuity follows by letting \(\varepsilon \rightarrow 0\). For the second part of the lemma, we write U(v, T) in the following way
From here, the estimate (6.5) follows. \(\square \)
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 \(\psi \in C([0,\infty ),X^4)\) to (6.2).
Theorem 6.3
There exist \(\delta ,N>0\) such that the following holds. If
then there exist \(T \in [1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\) and \( \psi \in {\mathcal {X}}_{\delta }\) such that (6.2) holds for all \(\tau \geqq 0.\)
Proof
Lemma 6.2 and Proposition 6.1 imply that for all small enough \(\delta \) and all large enough N we have that if v satisfies (6.8) and \(T \in [1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\) then there is a unique \(\psi =\psi ({ v},T) \in {\mathcal {X}}_\delta \) that solves
We remark that \(\psi (\tau )\) is real-valued for all \(\tau \geqq 0\), since the set of real-valued functions in \(X^4\) is invariant under the action of both \( S(\tau )\) and \({\mathcal {P}}\). Now, to construct solutions to (6.2), we prove that there is a choice of \(\delta \) and N such that for any v that satisfies (6.8) there is \(T=T( v) \in [1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\) for which the correction term in (6.9) vanishes. As C takes values in \({\text {rg}}{\mathcal {P}} = \langle g \rangle \), it is enough to show existence of T for which
We therefore consider the real function \(T \mapsto \langle C( U( v,T), \psi ({ v},T)), g \rangle _{X^4} \) and employ the intermediate value theorem to prove that it vanishes on \([1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\). 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 \(R_1( v,T) \lesssim \delta /N^2\).
Furthermore, based on the definition of the correction function C, we similarly conclude that
where \(T \mapsto R_2( v,T)\) is a continuous, real-valued function on \([1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\), for which \(R_2( v,T) \lesssim \delta /N^2 + \delta ^2.\) Therefore, there is a choice of sufficiently large N and sufficiently small \(\delta \) such that \(|R_2( v,T)| \leqq c \Vert g \Vert ^2 \frac{\delta }{N}\). Based on this, we get that (6.10) is equivalent to
for some function F which maps the interval \([1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\) continuously into itself. Consequently, by the intermediate value theorem we infer the existence of \(T \in [1-\frac{\delta }{N}, 1+\frac{\delta }{N}]\) for which (6.11), and therefore (6.10), holds. The claim of the theorem follows. \(\square \)
7 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 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.
Proposition 7.1
If v from Theorem 6.3 belongs to the radial Schwartz class \({\mathcal {S}}_{rad }({\mathbb {R}}^5)\), then the function \(\Psi (\tau ,\xi ):=\psi (\tau )(\xi )\) belongs to \(C^\infty ([0,\infty )\times {\mathbb {R}}^5)\) and satisfies
in the classical sense.
Proof
Instead of \(\psi (\tau )\) solving (6.2), we analyze the corresponding strong solution to (2.6). More precisely, by means of relation (5.6), we get that \(\tau \mapsto \varphi (\tau ) := \phi + \psi (\tau )\) belongs to \(C([0,\infty ),X^4)\) and satisfies
where \(U_0(T)=\phi +U(v,T)\). First, we show that \(\varphi (\tau )\) belongs to \(C^{\infty }({\mathbb {R}}^5)\) for all \(\tau \geqq 0\). To this end, we use the following two estimates
for \(k \geqq 6\), and
for \(k > 2 + l\) and \(l \geqq 0\). To remove any possible ambiguity in notation, by \(X^s\) for a non-integer s, we denote, analogous to the integer exponent case, the closure of the test space \(C^\infty _{\text {rad}}({\mathbb {R}}^5)\) under the norm \(\Vert \cdot \Vert ^2_{X^s} = \Vert \cdot \Vert ^2_{\dot{H}^1({\mathbb {R}}^5)} + \Vert \cdot \Vert ^2_{\dot{H}^s({\mathbb {R}}^5)}\). The estimates (7.3) and (7.4) are proved by using standard arguments from interpolation theory (see, e.g., Bergh-Löfström [2], Section 6.4, as well as Theorem 4.4.1, p. 96) together with (5.1) and Proposition 5.2. Now, from (7.2) and the above estimates, we have
which implies that \(\varphi (\tau ) \in X^{\frac{9}{2}}\) for all \(\tau \geqq 0\). Then, inductively we get that \(\varphi (\tau ) \in X^\frac{k}{2}\) for all \(k \geqq 9\) and \(\tau \geqq 0\). Then, by the embedding \(X^k \hookrightarrow C^{k-3}_{\text {rad}}({\mathbb {R}}^5)\) we conclude that \(\psi (\tau ) \in C^\infty ({\mathbb {R}}^5)\) for all \(\tau \geqq 0\). To establish regularity in \(\tau \), we do the following. First, since \(\varphi \) is a locally bounded curve in \(X^5\) we use Gronwall’s inequality (see, e.g., Cazenave-Haraux [11], p. 55, Lemma 4.2.1) to show from (7.2) that \(\varphi :[0,{\mathcal {T}}] \rightarrow X^4\) is Lipschitz continuous for every \({\mathcal {T}}>0\). Consequently, according to (5.1) we have that \(\tau \mapsto N(\psi (\tau )):[0,{\mathcal {T}}] \rightarrow X^3\) is Lipschitz continuous for every \({\mathcal {T}} >0\). This, together with the fact that \(U_0(T) \in {\mathcal {D}}( {\mathcal {L}}_0)\) implies that \(\varphi \in C^1([0,\infty ),X^3)\), and therefore \(\psi (\tau )=\varphi (\tau ) - \phi \) satisfies (6.1) in \(X^3\) in the operator sense (see, e.g., [11], p. 51, Proposition 4.1.6, (ii)). Furthermore, as \(X^3\) is continuously embedded in \(L^\infty ({\mathbb {R}}^5)\) the \(\tau \)-derivative holds pointwise. Consequently, by (a strong version of) the Schwarz theorem (see, e.g., Rudin [49], p. 235, Theorem 9.41), we conclude that mixed derivatives of all orders in \(\tau \) and \(\xi \) exist, and we thereby infer smoothness of \((\tau ,\xi ) \mapsto \psi (\tau )(\xi )\) and the fact that it satisfies (7.1) classically. \(\square \)
Proof of Theorem 1.1
Due to Lemma B.1 we can choose \(\varepsilon >0\) small enough such that
for \(\delta ,N\) from Theorem 6.3. Then, according to Theorem 6.3 there exists a solution \(\psi \in C([0,\infty ),X^4)\) to (6.2), for which
Now, since, by assumption, v belongs to \({\mathcal {S}}_{\text {rad}}({\mathbb {R}}^5)\), Proposition 7.1 implies that \(\Psi (\tau ,\xi ) = \phi (\xi ) + \psi (\tau )(\xi )\) is smooth and solves (2.6) classically. Therefore,
belongs to \(C^\infty ([0,T)\times {\mathbb {R}}^5)\) and solves the system (2.1) on \([0,T)\times {\mathbb {R}}^5\) classically. This then yields a smooth solution to (1.1)
where, according to (B.1) and (7.5) we have
as \(t \rightarrow T^-\). \(\square \)
Data Availability
No datasets were generated or analyzed during the current study.
Change history
05 June 2024
A Correction to this paper has been published: https://doi.org/10.1007/s00205-024-02004-9
References
Ascasibar, Y., Granero-Belinchón, R., Moreno, J.M.: An approximate treatment of gravitational collapse. Phys. D: Nonlinear Phenom. 262, 71–82, 2013
Bergh, J., Löfström J.: Interpolation spaces. An introduction. In: Grundlehren der Mathematischen Wissenschaften, No. 223, pp. x+207. Springer-Verlag, Berlin-New York (1976)
Biler, P: Singularities of solutions to chemotaxis systems, volume 6 of De Gruyter Series in Mathematics and Life Sciences. De Gruyter, Berlin, 2020.
Biler, P., Karch, G., Pilarczyk, D.: Global radial solutions in classical Keller–Segel model of chemotaxis. J. Differ. Equ. 267(11), 6352–6369, 2019
Biler, P., Karch, G., Zienkiewicz, J.: Optimal criteria for blowup of radial and \(N\)-symmetric solutions of chemotaxis systems. Nonlinearity 28(12), 4369–4387, 2015
Biler, P., Zienkiewicz, J.: Blowing up radial solutions in the minimal Keller–Segel model of chemotaxis. J. Evol. Equ. 19(1), 71–90, 2019
Blanchet, A., Carrillo, J.A., Masmoudi, N.: Infinite time aggregation for the critical Patlak–Keller–Segel model in \(\mathbb{R} ^2\). Comm. Pure Appl. Math. 61(10), 1449–1481, 2008
Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 44, 1–33, 2006
Brenner, M.P., Constantin, P., Kadanoff, L.P., Schenkel, A., Venkataramani, S.C.: Diffusion, attraction and collapse. Nonlinearity 12(4), 1071–1098, 1999
Calvez, V., Corrias, L., Ebde, M.A.: Blow-up, concentration phenomenon and global existence for the Keller–Segel model in high dimension. Comm. Part. Differ. Equ. 37(4), 561–584, 2012
Cazenave, T., Haraux, A.: An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors.
Childress, S., Percus, J.K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56(3–4), 217–237, 1981
Collot, C., Ghoul, T., Masmoudi, N., Nguyen, V.T.: Refined description and stability for singular solutions of the 2D Keller–Segel system. Comm. Pure Appl. Math. 75(7), 1419–1516, 2022
Collot, C., Ghoul, T., Masmoudi, N., Nguyen, VT..: Collapsing-ring blowup solutions for the Keller–Segel system in three dimensions and higher. arXiv e-prints, arXiv:2112.15518, 2021.
Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28, 2004
Creek, M., Donninger, R., Schlag, W., Snelson, S.: Linear stability of the skyrmion. Int. Math. Res. Not. IMRN 8, 2497–2537, 2017
Davila, J., del Pino, M., Dolbeault, J., Musso, M., Wei, J.: Infinite time blow-up in the Patlak–Keller–Segel system: existence and stability. arXiv e-prints, arXiv:1911.12417, 2019.
Donninger, R., Schörkhuber, B.: A spectral mapping theorem for perturbed Ornstein–Uhlenbeck operators on \(L^2(\mathbb{R} ^d)\). J. Funct. Anal. 268(9), 2479–2524, 2015
Donninger, R., Schörkhuber, B.: Stable blowup for the supercritical Yang–Mills heat flow. J. Differ. Geom. 113(1), 55–94, 2019
Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
Ghoul, T.-E., Masmoudi, N.: Minimal mass blowup solutions for the Patlak–Keller–Segel equation. Comm. Pure Appl. Math. 71(10), 1957–2015, 2018
Giga, Y., Kohn, R.V.: Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math. 38(3), 297–319, 1985
Giga, Y., Kohn, R.V.: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36(1), 1–40, 1987
Giga, Y., Kohn, R.V.: Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math. 42(6), 845–884, 1989
Giga, Y., Mizoguchi, N., Senba, T.: Asymptotic behavior of type I blowup solutions to a parabolic–elliptic system of drift-diffusion type. Arch. Ration. Mech. Anal. 201(2), 549–573, 2011
Glogić, I.: Stable blowup for the supercritical hyperbolic Yang-Mills equations. Adv. Math. 408, 108633, 2022
Glogić, I.: Globally stable blowup profile for supercritical wave maps in all dimensions. arXiv e-prints, arXiv:2207.06952, 2022.
Glogić, I., Schörkhuber, B.: Nonlinear stability of homothetically shrinking Yang–Mills solitons in the equivariant case. Comm. Part. Differ. Equ. 45(8), 887–912, 2020
Glogić, I., Schörkhuber, B.: Co-dimension one stable blowup for the supercritical cubic wave equation. Adv. Math. 390, 107930, 2021
Grafakos, L.: Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, second edition, 2008
Henry, D.: Geometric theory of semilinear parabolic equations, vol. 840. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York (1981)
Herrero, M.A., Medina, E., Velázquez, J.J.L.: Finite-time aggregation into a single point in a reaction–diffusion system. Nonlinearity 10(6), 1739–1754, 1997
Herrero, M.A., Medina, E., Velázquez, J.J.L.: Self-similar blow-up for a reaction–diffusion system. J. Comput. Appl. Math. 97(1–2), 99–119, 1998
Herrero, M.A., Velázquez, J.J.L.: Singularity patterns in a chemotaxis model. Math. Ann. 306(3), 583–623, 1996
Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105(3), 103–165, 2003
Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106(2), 51–69, 2004
Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329(2), 819–824, 1992
Kato, T: Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26(3), 399–415, 1970
Lemarié-Rieusset, P.G.: Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller–Segel equations in the whole space. Adv. Differ. Equ. 18(11–12), 1189–1208, 2013
Mizoguchi, N.: Refined asymptotic behavior of blowup solutions to a simplified chemotaxis system. Commun. Pure Appl. Math. 75(8), 1870–1886, 2022
Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5(2), 581–601, 1995
Naito, Y.: Blow-up criteria for the classical Keller–Segel model of chemotaxis in higher dimensions. J. Differ. Equ. 297, 144–174, 2021
Naito, Y., Senba, T.: Blow-up behavior of solutions to a parabolic–elliptic system on higher dimensional domains. Discrete Contin. Dyn. Syst. 32(10), 3691–3713, 2012
Naito, Y., Suzuki, T.: Self-similarity in chemotaxis systems. Colloq. Math. 111(1), 11–34, 2008
Ogawa, T., Wakui, H.: Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions. Anal. Appl. (Singap.) 14(1), 145–183, 2016
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: editors. NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX).
Raphaël, P., Schweyer, R.: On the stability of critical chemotactic aggregation. Math. Ann. 359(1–2), 267–377, 2014
Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill Book Co., New York-Auckland-Düsseldorf (1976)
Schlag, W., Soffer, A., Staubach, W.: Decay for the wave and Schrödinger evolutions on manifolds with conical ends. I. Trans. Amer. Math. Soc. 362(1), 19–52, 2010
Senba, T.: Blowup behavior of radial solutions to Jäger–Luckhaus system in high dimensional domains. Funkcial. Ekvac. 48(2), 247–271, 2005
Souplet, P., Winkler, M.: Blow-up profiles for the parabolic-elliptic Keller–Segel system in dimensions \(n\ge 3\). Comm. Math. Phys. 367(2), 665–681, 2019
Velázquez, J.J.L.: Stability of some mechanisms of chemotactic aggregation. SIAM J. Appl. Math. 62(5), 1581–1633, 2002
Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, vol. 1258. Springer-Verlag, Berlin (1987)
Wolansky, G.: On steady distributions of self-attracting clusters under friction and fluctuations. Arch. Ration. Mech. Anal. 119(4), 355–391, 1992
Funding
Open access funding provided by University of Innsbruck and Medical University of Innsbruck. Irfan Glogić is supported by the Austrian Science Fund FWF, Projects P 30076 and P 34378.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Communicated by Nader Masmoudi
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original online version of this article was revised: several errors within the text
Appendices
Appendix A. Estimates of Local Sobolev Norms
Lemma A.1
Let \(k \in {\mathbb {N}}\) and \(R>0.\) Then
for all \(u \in C^\infty _{rad }({\mathbb {B}}^5_R)\) and all \(\alpha \in {\mathbb {N}}_0^5\) with \(|\alpha |=k\).
Proof
We prove the claim for \(R=1\) as the general case follows by scaling. Let \(\chi : {\mathbb {R}}^5 \rightarrow [0,1]\) be a smooth radial function such that \(\chi (x) =0\) for \(|x|\leqq \frac{5}{4}\) and \(\chi (x) = 0\) for \(|x| \geqq \frac{3}{2}\). We then define for \(u={\tilde{u}}(|\cdot |) \in C_{\text {rad}}^k({\mathbb {B}}^5)\) the extension operator
and then by means of the cut-off \(\chi \) we let
Note that \(E: C^k_{\text {rad}}({\mathbb {B}}^5) \rightarrow C^k_{c,\text {rad}}({\mathbb {R}}^5)\). By denoting with \({\tilde{D}}^iu\) the radial profile of \(D^i u\) we have that
Furthermore, by the fundamental theorem of calculus,
and by Hardy’s inequality (see, e.g., [26], Lemma 2.12) we infer that
Therefore, from (A.3), (A.4) and (A.5), we have that
Now, based on these results, from (A.2) we get that, for \(i \leqq k\),
Therefore, we finally infer that
for all \(u \in C^\infty _{\text {rad}}({\mathbb {B}}^5)\). \(\square \)
Appendix B. Equivalence of Sobolev Norms for the Reduced Mass
Lemma B.1
Let \(d \in {\mathbb {N}}\). For every \(u={\tilde{u}}(|\cdot |) \in C^\infty _{c,rad }({\mathbb {R}}^d)\) define \(w={\tilde{w}}(|\cdot |)\) by
Then given \(k \in {\mathbb {N}}_0\) we have that
for all \(u \in C^\infty _{c,rad }({\mathbb {R}}^d)\).
Proof
The proof relies on the Bessel function representation of the Fourier transform of radial functions. Recall our convention (1.11). Then, for a radial Schwartz function \(f={\tilde{f}}(|\cdot |)\), we have that
(see, e.g., Grafakos [30], p. 429). Now, for \(\rho >0\) by partial integration we have
where we used the recurrence relation for J-Bessel functions
According to (B.3) and (B.2) we have that
\(\square \)
Appendix C. An ODE Result
In this section, we use the following notation
and note \(f \in C_{\textrm{rad }}^{\infty }({\mathbb {R}}^d)\) if and only if \(f = {{\tilde{f}}}(|\cdot |)\) with \({{\tilde{f}}} \in C_{e}^{\infty }[0,\infty )\). For \(\rho \in [0,\infty )\) we set
where \({{\tilde{\phi }}}(\rho ) = \frac{2}{2+\rho ^2}\). Also, we let \({\bar{\omega }}_k\) denote the constant in Equation (4.18).
Lemma C.1
Let \(k \in {\mathbb {N}}\), \(k \geqq 3\). Let f be an element of \(C_{e}^{\infty }[0,\infty )\) with bounded support, and let \(\lambda > \max \{2,{\bar{\omega }}_k \}\). Then there exists a function \(u \in C^1[0,\infty ) \cap C^{\infty }(0,\infty )\) which solves the equation
on the interval \((0,\infty )\), satisfies \(u'(0) = 0\), and given \(j \in {\mathbb {N}}_0\) obeys the estimate
as \(\rho \rightarrow \infty \).
Proof
First, we construct a fundamental system for the homogeneous equation
We note that the origin \(\rho = 0\) is a regular singular point. Hence, by the Frobenius method, there is a fundamental system \(\{ u_0, u_1 \}\) on \((0,\infty )\), where \(u_0\) is analytic at \(\rho =0\) with \(u_0(0) = 1, u_0'(0)=0\), and \(u_1(\rho ) \sim \rho ^{-3}\) near \(\rho =0\). To analyze the behavior of solutions at infinity we write the equation in normal form. With \(\omega (r):= e^{\frac{r^2}{2}} r^{-2} (1+ 2 r^2)^{-1}\) and \(v(r)\omega (r) = u(2r)\), Equation (C.2) transforms into
with \(\mu = 4\lambda - 5 > 0\) and
By transforming the solutions of Equation (C.2) we obtain a fundamental system \(\{v_0,v_1 \}\) for Equation (C.3) with \(v_0(r) \sim r^2\) and \(v_1(r) \sim r^{-1}\) for \(r \rightarrow 0^+\).
For large values of the argument, the situation is more involved. For \(r \geqq 1\) and \(V =0\), a fundamental system can be given in terms of parabolic cylinder functions \(\{ U(\frac{\mu }{2}, \sqrt{2} \cdot ), V(\frac{\mu }{2}, \sqrt{2} \cdot ) \}\), with asymptotic behavior
for \(r \rightarrow \infty \), see for example [47]. Our goal is to construct perturbatively a solution to Equation (C.3), linearly independent of \(v_0\), that behaves like \(U(\tfrac{\mu }{2}, \sqrt{2} \cdot )\) at infinity. We make this fully explicit by considering a slightly different ‘free’ equation first, namely,
with potential
This equation has an explicit fundamental system (see [19], Section 4.1.1),
with \( \xi (r) = \frac{1}{2} \log ( r + \sqrt{1+r^2}) + \frac{1}{2}r \sqrt{1+r^2}\) and Wronskian \(W(v^{-},v^{+}) = 1\). Note that
with \(c_{\mu } \in {\mathbb {R}}\) and \(\varphi _{\mu }(r) = {\mathcal {O}}(r^{-2})\) for \(r \rightarrow \infty \), hence
for \(r \rightarrow \infty \). We add \(Q_{\mu }\) to both sides of Equation (C.3) and put the potential V to the right hand side to obtain
Assuming \(r \geqq 1\), we show by a perturbative argument the existence of a solution \(v_{\infty }\) to (C.5) which behaves like \( v^{-}\) at infinity. For this, we set up a Volterra iteration by reformulating Equation (C.5) as an integral equation using the variation of constants formula. More precisely, we look for a solution \(v_{\infty }\) that satisfies
Noting that \(v^{-}(r) > 0\) for all \(r >0\), we set \(h(r):= \frac{ v_{\infty }(r)}{ v^{-}(r)}\) and write the above equation as
where
Explicitly,
Using the fact that \(\xi \) is monotonically increasing, we obtain the bound
for \(1 \leqq r \leqq s\). Thus,
and we can apply standard results on Volterra equations (see, e.g., [50], Lemma 2.4) which yield the existence of a solution h on \([1,\infty )\) with \(|h(r)| \lesssim 1\) and
By inspection (see also Remark 4.4 in [18]), one finds that
for all \(k \in {\mathbb {N}}\). This yields a smooth solution
to Equation (C.3) on \([1,\infty )\), where the error term behaves like a symbol under differentiation.
Now, by linearity, we have the representation
for some constants \(c_0,c_1 \in {\mathbb {C}}\). Suppose that \(c_1 = 0\), i.e., \(v_{\infty }\) and \(v_0\) are linearly dependent. By transforming back, we would obtain a function \(u \in C_{e}^{\infty }[0,\infty )\) with \(u(\rho ) = {\mathcal {O}}(\rho ^{-2 - 2\lambda })\) as \(\rho \rightarrow \infty \). In particular, \(u(|\cdot |)\) would belong to \({\mathcal {C}}\) and satisfy \((\lambda - {\mathcal {L}}_k)u(|\cdot |) = 0\) for some \(\lambda > {\bar{\omega }}_k\). This, however, contradicts Equation (4.18) stated in the proof of Proposition 4.7. We conclude that \(\{ v_{\infty }, v_0\}\) is a fundamental system for Equation (C.3) on \((0,\infty )\), and we denote by \(W:= W(v_{\infty },v_0)(1)\) its Wronskian.
Now we turn to the inhomogeneous Equation (C.1), which transforms into
By the variation of constants formula we find a particular solution
Obviously, \(v \in C^{\infty }(0,\infty )\). Since f has bounded support, the first integral vanishes for large r and therefore there is a constant c such that \(v(r)= cv_{\infty }(r)\) for all large enough r. For \(r \rightarrow 0\), the first integral converges, hence the behavior of the first term is governed by \(v_0\). The second integral is of order \({\mathcal {O}}(r^5)\) which compensates the singular behavior of \(v_{\infty }\) at the origin. In particular, there is a constant C such that \(r^{-2}v(r) \rightarrow C\) and \(r^{-1}v'(r) \rightarrow 2C\) when \(r \rightarrow 0^{+}\). By transforming back, we obtain a solution \(u \in C^1[0,\infty ) \cap C^{\infty }(0,\infty )\). By inspection, \(u'(0) = 0\) and \(u^{(k)}(\rho ) = {\mathcal {O}}(\rho ^{-2 - 2\lambda - k})\) for \(\rho \rightarrow \infty \) and \(k \in {\mathbb {N}}_0\). \(\square \)
Appendix D. Smoothing of \(S(\tau )\) via energy methods
We give an alternative proof of the key estimate (5.11) of Proposition 5.4 for some \(\omega _{k+1} > 0\). The estimate (5.10) can be proved similarly. As usual, it is enough to assume \(f \in C^\infty _{c,\text {rad}}({\mathbb {R}}^5)\). Denote \(\tilde{f}:=(1-{\mathcal {P}}_{X^{k}})f \in {\mathcal {C}} \subset {\mathcal {D}}({\mathcal {L}}_k)\). First, we refine the proof of Theorem 4.1. Namely, for \(R\geqq 1\) sufficiently large we have
for \(c = \frac{1}{2} \min \{\omega _0, \frac{1}{8} \}\). Hence,
and integration yields
for some suitably chosen \(\omega _k > 0\) and all \(\tau \geqq 0\). Now, by using this estimate we can similarly get
By integration we get
for all \(\tau \geqq 0\). Now, to show (5.11) we treat two cases.
Case 1: \(\tau \in (0,2)\). We get from (C.12) and (C.11) that
for all \(\tau \in (0,2)\).
Case 2: \(\tau \geqq 2\). From the semigroup property, the Case 1, and (C.11) we have
for all \(\tau \geqq 2\).
From these two cases and (C.11), the estimate (5.11) follows.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Glogić, I., Schörkhuber, B. Stable Singularity Formation for the Keller–Segel System in Three Dimensions. Arch Rational Mech Anal 248, 4 (2024). https://doi.org/10.1007/s00205-023-01947-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00205-023-01947-9