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A lower bound for the life span of solutions to the Kirchhoff equation with Gevrey data

We provide a new lower bound for the life span of solutions to the Kirchhoff equation for which the initial data belongs to the Gevrey space. This lower bound strictly improves the classical one in the case when the frequency spectrum of the initial data is concentrated at the origin.


Introduction
In this article, we concern ourselves with Kirchhoff-type equations of the form where we always assume that ϕ(ρ) is a locally Lipschitz function on [0, ∞) for which there exists a real ν 0 > 0 such that (1.2) ϕ(ρ) ≥ ν 0 for all ρ ≥ 0.
In 1876, Kirchhoff [8] proposed the special case of n = 1, ϕ(ρ) = ν 0 + aρ (ν 0 , a > 0), for the equation (1.1) to describe the transversal motions of the elastic string. When looking at the general case, several authors have investigated the global existence for the Kirchoff-type equations when the initial data is real analytic. In 1940, Bernstein [3] first studied the global existence for analytic data in one space dimension. After him, in 1975, Pohozaev [13] extended Bernstein's result to several space dimensions. Later, the global solvability in the real analytic class was studied by D'Ancona and Spagnolo [5] (see also [2]) under the additional assumption that ϕ is continuous on [0, ∞), ϕ(ρ) ≥ 0, for all ρ ≥ 0.
Kajitani and Yamaguti [7] obtained the same result under a more general principal term. It is of course natural to ask whether the Cauchy problem (1.1) admits a unique global solution with initial data in larger function spaces, such as e.g. the quasianalytic class or Sobolev spaces. The global solvability for quasi-analytic data was studied by Nishihara [12] and Ghisi and Gobbino [6]. Manfrin [9] discovered spectral gap data which assure global solvability of the Kirchhoff equation. It should be noted that the space in [6,12] is included in the Gevrey spaces.
It has been a long-standing open problem whether or not, one can prove the existence of time global solutions in the Sobolev spaces without smallness condition on the initial data. In fact, the existence of local solutions in low regular Sobolev spaces, say, H σ × H σ−1 , σ ∈ [1, 3/2), is still not known. The main idea of the proof of the global existence of high regular solutions is to obtain boundedness of the local solutions in the H 3/2 -norm at the life span. On the one hand, the main difficulty lies in controlling an intensive oscillation of the coefficient ϕ( ∇u(t) 2 L 2 ). On the other hand, when the data is very small, one can overcome such an oscillation problem to get global solutions (see [10] and the references therein). However, if one does not impose extra conditions, no results have been obtained as of yet.
As an intermediate step before considering the global solvability of the Kirchhoff equation, it is interesting to look at the existence of a life span with respect to certain initial data. In [1] it is shown that for any nontrivial (u 0 , . Note that T m = +∞ corresponds to (1.1) being globally solvable for the initial data (u 0 , u 1 ). Now, if we put ϕ(ρ), and L := sup then the following classical lower bound for T m was found in [1, Equation (2.13)]: where E 3/2 (u; t) is the energy of order 3/2 of the solution (see (2.2)).
In this paper, we will consider the case where the initial data is contained in the Gevrey spaces, which lie in between the real analytic class and the Sobolev spaces. For s ≥ 1, we denote by γ s L 2 (R n ) the Roumieu-Gevrey space of order s on R n , endowed with its natural (LB)-space topology, where f belongs to γ s η, here (F f )(ξ) stands for the Fourier transform of f (x). If f, g ∈ γ s η,L 2 , we also consider the norm η,L 2 . Note that in the particular case s = 1, γ 1 L 2 (R n ) is exactly the real analytic class, and its global existence was proved by Bernstein [3] for n = 1 and by Pohozaev [13] for n ≥ 2. For s > 1, the well-posedness of the Kirchhoff equation with initial data in γ s L 2 (R n ) was first considered in [11]. Here, we will provide an explicit lower bound for T m in function of the Gevrey norm of the initial data. In fact, we have the following result.
Theorem 1.1. Suppose that ϕ(ρ) is a locally Lipschitz function on [0, ∞) satisfying the non-degeneracy condition (1.2). Let s > 1 and suppose Depending on the data, the lower bound given in (1.6) will be strictly larger than the one classically given by (1.5) (see Remark 3.5). This seems to be especially the case when the frequency spectrum of the initial data is concentrated at the origin. Moreover, we also mention that our proof could be adapted, similarly as in [11], to find an analogous result for the initial-boundary value problems of the Kirchhoff equation with initial data in the Gevrey class. We have organized the paper as follows: We first state some known results on local existence theorems in Section 2, after which we prove our main result in Section 3.

Local existence theorems
In the context of the Sobolev spaces, the Kirchhoff equation has a first integral.
is the solution to (1.1). If we define the energy then, we have Proof. The proof is straightforward: Multiplying (1.1) by ∂ t u and integrating over R n gives d dt H(u; t) = 0, as desired.
For σ ∈ R, we denote the homogeneous counterpart of the fractional Sobolev spaces byḢ . We now define the energy of order 3/2 for any u ∈ 1 j=0 C j ([0, T ];Ḣ σ−j (R n )) as follows: The following result was shown by Arosio and Garavaldi.
and one of the following statements is true: We remark here that the life span T m is to be understood as follows: It should be noted that, however big the regularity of the data is, T m depends only on the norm of the data inḢ 3/2 (R n ) ×Ḣ 1/2 (R n ). This means that when one would show the global existence of solutions to (1.1), it suffices to obtain that the norm of The local existence theorem for Gevrey spaces is now immediately obtained as a consequence of Theorem 2.2, and the life span depends only on the constants H(u; 0) and E 3/2 (u; 0). More precisely, we have the following: there exists a life span T m = T m (u 0 , u 1 ) > 0 depending only on H(u; 0) and E 3/2 (u; 0) such that the Cauchy problem (1.1) admits a unique solution u(t, x) in the class and one of the following statements is true: (i) T m = +∞; (ii) T m < +∞ and lim sup Proof. We may see the initial data in (−∆) − 3 4 γ s η,L 2 (R n )×(−∆) − 1 4 γ s η,L 2 (R n ) as elements of the phase spaceḢ 3/2 (R n ) ×Ḣ 1/2 (R n ). Let T ∈ (0, T m ) be arbitrarily fixed. By Theorem 2.2, with σ = 3/2, we know that the Cauchy problem (1.1) admits a unique solution u such that . It follows by the theory of linear partial differential equations that the Cauchy problem Then we conclude that v = u, i.e., the Cauchy problem (1.1) admits a unique solution u such that From here the result follows.
We end this section with a remark on the constants in (1.3) and (1.4). We now focus on proving Theorem 1.1. To do this, we will consider linear Cauchy problems of the form

Remark 2.4. Depending on the initial data (u
In the case where the derivative of c has a pole at T , we find the following result when u 0 and u 1 belong to γ s L 2 (R n ) (see also [4]).
For the remainder of this section, we fix 0 . Moreover, we will assume that T m = T m (u 0 , u 1 ) < +∞, as otherwise Theorem 1.1 is trivial. Our proof is based on a contradiction argument, that is, we will from now on suppose that (1.6) is false and from there show that the life span is then strictly larger than T m . For this, we will consider the following class of functions. We define the class K as all those functions c on [0, We endow K with the topology induced by the Fréchet space L ∞ loc ([0, T m )). Note that if (1.6) doesn't hold, then this implies exactly that (3.4) holds with K as in (3.10) and q = (s + 1)/s. Consequently, by Proposition 3.1, for any c ∈ K the Cauchy problem (3.1) with initial data (u 0 , u 1 ) has a unique solution v(t, x) ∈ C 1 [0, T m ]; γ s L 2 (R n ) . We now consider the function Then ϕ * ∈ Lip loc ([0, ∞)), and note that by Remark 2.4, in case of the initial data (u 0 , u 1 ), we may exchange ϕ with ϕ * in (1.1) and obtain the same solution u(·, x) on [0, T m ). Given a c ∈ K, we define the function Theorem 1.1 will then follow from the following two crucial results.
Lemma 3.3. The mapping is well-defined and continuous.
Lemma 3.4. K is a convex and compact Fréchet space.
Before showing these lemmas, let us first demonstrate how they entail the proof of Theorem 1.1.
Proof of Theorem 1.1. By Lemmas 3.3 and 3.4, it follows from the Schauder-Tychonoff theorem that the mapping Θ in (3.11) has a fixed point c 0 in K. Consequently, the solution v(t, x) to the Cauchy problem (3.1) with c = c 0 and initial data (u 0 , u 1 ) is also a solution u = u(t, x) to the non-linear Cauchy problem (1.1) with initial data (u 0 , u 1 ) on [0, T m ]. Hence u exists at the endpoint T m , so that E 3/2 (u; T m ) < +∞, contradicting Proposition 2.3. Therefore, we may conclude that (1.6) holds.
We now move on to prove the lemmas.
Proof of Lemma 3.3. We first show that Θ is well-defined, i.e. that for every c ∈ K also c v ∈ K. It is clear that c v ∈ Lip loc ([0, T m )), and by the definition of ϕ * we have that For the derivative, take (3.12) η ′ = η − (Ks + 2M)ν −1 0 > 0, then, by Proposition 3.1, we deduce that almost everywhere Combining these two estimates together, we find that almost everywhere Consequently, c v ∈ K, so that Θ is well-defined. Next, we show that Θ is continuous. To do this, let us take a sequence (c k (t)) k∈N in K such that and let v k (t, x) and v(t, x) be the corresponding solutions to the linear Cauchy problem (3.1) with the coefficients c k (t) and c(t), respectively. Then it is sufficient to prove that the imagesc k (t) := Θ(c k (t)) andc(t) := Θ(c(t)) satisfy (3.13)c k (t) →c(t) in L ∞ loc ([0, T m )), k → ∞.
We define the energies where c ′ (t 0 ) exists. For h > 0 small enough, we then have (T m − t 0 + h) 1/s , so that by taking the limit h → 0 + , we find We may conclude that c ∈ K, which completes the proof.
We end this section with the following remark.