On Global attraction to solitary waves for Klein-Gordon equation with concentrated nonlinearity

The global attraction is proved for the nonlinear 3D Klein-Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each"finite energy solution"to the manifold of all solitary waves as $t\to\pm\infty$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. Then the application of the Titchmarsh Convolution Theorem reduces the spectrum of each omega-limit trajectory to a single frequency $\omega\in[-m,m]$.


Introduction
The paper concerns a nonlinear interaction of the Klein-Gordon field with a point oscillator. The point interaction are widely used in physical works. One of the well-known application in dimension one is the Kronig-Penney model [25]. I n 3D case a rigorous mathematical definition of point interactions was given by Berezin and Faddeev [6]. For the numerous literature concerning the models with a point interactions we refer to [5].

Model
We fix a nonlinear function F : C → C and define the domain which generally is not a linear space. Let H F be a nonlinear operator on the domain D F defined by Let us introduce the phase space for equation (2.3). Denote the spacė Obviously, D F ⊂Ḋ.
where B R is the ball of radius R.
Remark 2.3. The spaces L 2 loc are metrisable. The metrics can be defined by (2.8)

Global well-posedness
For the global well-posedness, we assume that . The next theorem is proved in [23]. (1.4) and (2.9) hold. Then (i) For every initial data Ψ(0) = Ψ 0 = (ψ 0 , π 0 ) ∈ D the Cauchy problem for (2.3) has a unique solution ψ(t) such that (ii) The energy is conserved: The following a priori bound holds The identity (1.6) implies that the set S is invariant under multiplication by e iθ , θ ∈ R. Let us note that since F(0) = 0 by (1.5), then for any ω ∈ R there is a zero solitary wave with ψ ω (x) ≡ 0.
where q ω is the solution to m − m 2 − ω 2 = 4πb(|q ω | 2 ). (2.14) Proof. We can split ψ( Evidently, ψ reg (·,t) ∈ H 2 (R 3 ), ζ (·) ∈ C b (R). Finally, the second equation of (1.1) together with (1.5) give At last, we assume that the nonlinearity is polynomial. This assumption is crucial in our argument since it will allow us to apply the Titchmarsh convolution theorem. Now all our assumptions on F can be summarized as follows.
In particular, this assumption guarantees that the nonlinearity F satisfies the bound (2.9) from Theorem 2.4. Our main result is the following theorem.

Theorem 2.7 (Main Theorem). Let Assumption (2.15) be satisfied. Then for any
It suffices to prove Theorem 2.7 for t → +∞. We will only consider the solution ψ(x,t) restricted to t ≥ 0.

Dispersive component
Let J 1 be the Bessel function of order 1, and θ be the Heaviside function. In [23] we proved that the solution ψ(x,t) to (2.3) with initial data ψ 0 = ψ 0,reg + ζ 0 G ∈ D F , π 0 = π 0,reg +ζ 0 G ∈ D is given by ) is a unique solution to the Cauchy problem for the free Klein-Gordon equation.
and ζ (t) ∈ C 1 ([0, ∞)) is a unique solution to the Cauchy problem for the following first-order nonlinear integro-differential equation with delayζ Note that the limit is well defined, λ (t) is continuous for t > 0, and it admits a limit as t → +0 (see [23]). The integral in (3.3) is bounded for all t ≥ 0 due to well known properties of the Bessel function J 1 : J 1 (r) ∼ r −1/2 for r → ∞, and J 1 (r) ∼ r as r → 0 (see for example [29]). Now we study the decay properties of the dispersive component ψ f (x,t) for t → ∞.
where ψ f ,reg and ψ f ,G are defined as solutions to the following Cauchy problems: Since (ψ 0,reg , π 0,reg ) ∈ X , then evidently, The following lemma states well known decay in local seminorms for the free Klein-Gordon equation.
where U (t) is the dynamical group of the free Klein-Gordon equation.
Therefore, the first dispersive component ψ f ,reg (x,t) decays in X loc seminorms. That is, ∀R > 0 Now we consider the second dispersive component ψ f ,G .
Proof. Let η(x) be a smooth function with a support in B 1 , such that η(x) = 1 for x ∈ B 1/2 . We split G as Hence it suffices to prove that where (u(t),u(t)) := U (t) ζ 0 ηG,ζ 0 ηG . The matrix kernel U (x − y,t) of the dynamical group U (t) can be written as Well known asymptotics of the Bessel function imply that (3.14) Hence, for |x| ≤ R and t > 2(R + 1), we obtain In conclusion, let us show that Indeed, the energy conservation for the free Klein-Gordon equation implies that .
where C + := {z ∈ C : Im z > 0}. Note thatψ S (·, ω) is an L 2 -valued analytic function of ω ∈ C + due to (4.3). Equation whereζ (ω) is the Fourier-Laplace transform of ζ (t): Applying the Fourier transform to (4.5), we getψ Then κ(ω) is the analytic function on C + , andψ S (x, ω) is given bỹ We then have, formally, for any ε > 0: Traces on the real line It is the boundary value of the analytic function (4.4) in the following sense: where the convergence holds in S ′ (R, L 2 (R 3 )). Indeed, ). Therefore, (4.11) holds by the continuity of the Fourier trans- Similarly to (4.11), the distributionζ (ω), ω ∈ R, is the boundary values of the analytic in C + functionζ (ω), ω ∈ C + : since the function θ (t)ζ (t) is bounded. The convergence holds in the space of tempered distributions S ′ (R). Let us justify that the representation (4.9) forψ S (x, ω) is also valid when ω ∈ R \ {−m; m}, if the multiplication in (4.9) is understood in the sense of distribution. Namely, holds in the sense of distributions.

Compactness
We are going to prove compactness of the set of translations of {ψ S (x,t + s) : s ≥ 0}. We start from the following lemma Lemma 5.1. For any sequence s j → ∞ there exists an infinite subsequence (which we also denote by s j ) such that for some η ∈ C b (R). The convergence is uniform on [−T, T ] for any T > 0. Moreover, η(t) is the solution to Proof. Theorem 2.4-iv), Corollary 3.4 and equation (3.3) imply that ζ ∈ C 1 b (R). Then (5.1) follows from the Arzelá-Ascoli theorem. Further, for any t ∈ R we get which imply (5.7) by (5.6). 6 Nonlinear spectral analysis We call an omega-limit trajectory any function β S (x,t) that can appear as a limit in (5.4). Proposition 3.1 demonstrates that the long-time asymptotics of the solution ψ(x,t) in L 2 loc depends only on the singular component ψ S (x,t). Namely, the convergences (5.4), and system (1.1) together with (3.1), (3.4) and (3.15) imply that any β S (x,t) is a solution to (1.1) with η(t) instead ζ (t): In this section we prove the following proposition.
Using (4.13) and taking into account that V (x, ω) is smooth for ω = ±m and x = 0, we obtain the following relation, which holds in the sense of distributions: Since V (x, ω) = 0 for ω ∈ R it follows from Lemma 6.2 that

Spectral inclusion and the Titchmarsh theorem
We will derive (6.1) from the following identity which will be proven in three steps. We start with an investigation of suppη.