On Global Attraction to Stationary States for Wave Equations with Concentrated Nonlinearities

The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \pm \infty $$\end{document}t→±∞ to stationary states. The attraction is caused by nonlinear energy radiation.


Introduction
The paper concerns a nonlinear interaction of the real wave field with a point oscillator.The system is governed by the following equations where G is the Green's function of operator − in R 3 , i.e.
All derivatives here and below are understood in the sense of distributions. The nonlinearity admits a potential F(ζ ) = U (ζ ), ζ ∈ R, U ∈ C 2 (R). (1.2) We assume that U (ζ ) → ∞, ζ → ±∞. Furthermore, we assume that the set Q = {q ∈ R : F(q) = 0} is nonempty. Then the system (1.1) admits stationary solutions qG(x), where q ∈ Q. We suppose that the set Q satisfies the following condition [a, b] ⊂ Q for any a < b. (1.4) LetH 1 (R 3 ) be the completion of the space C ∞ 0 (R 3 ) in the norm ∇ψ(x) L 2 (R 3 ) . Equivalently, using Sobolev's embedding theorem,H 1 (R 3 We consider Cauchy problem for system (1.1) with initial data (x, 0) = (ψ(x, 0),ψ(x, 0)) which can be represented as the sum of regular component fromH 2 (R 3 ) ⊕H 1 (R 3 ) and singular component proportional to G(x) (see Definition 2.1). Our main goal is the global attraction of the solution (x, t) = (ψ(x, t),ψ(x, t)) to stationary states: where the asymptotics hold in local L 2 ⊕ L 2 -seminorms. Similar global attraction was established for the first time (i) in [6][7][8] for 1D wave and Klein-Gordon equations coupled to nonlinear oscillators, (ii) in [9,10] for nD Klein-Gordon and Dirac equations with mean field interaction, and (iii) in [5] for discrete in space and time nD Klein-Gordon equation equations interacting with a nonlinear oscillator.
In the context of the Schrödinger and wave equations the point interaction of type (1.1) was introduced in [1,2,4,11,12], where the well-posedness of the Cauchy problem and the blow up solutions were studied. The orbital and asymptotic stability of soliton solutions for the Schrödinger equation with the point interaction has been established in [3]. The global attraction for 3D equations with the point interaction was not studied up to now. In the present paper we prove for the first time the global attraction in the case of 3D wave equation.
Let us comment on our approach. First, similarly to [8][9][10], we represent the solution as the sum of dispersive and singular components. The dispersive component is a solution of the free wave equation with the same initial data (x, 0). The singular component is a solution of a coupled system of wave equation with zero initial data and a point source, and of a nonlinear ODE.
We prove the long-time decay of the dispersive component in local H 2 ⊕ H 1 -seminorms. To establish the decay for regular part of the dispersive component, corresponding to regular initial data from H 2 ⊕ H 1 , we apply the strong Huygens principle and the energy conservation for the free wave equation. For the remaining singular part we apply the strong Huygens principle. The dispersive decay is caused by the energy radiation to infinity.
Finally, we study the nonlinear ODE with a source. We prove that the source decays and then the attractor of the ODE coincides with the set of zeros of the nonlinear function F, i.e. with the set Q. This allows us to prove the convergence of the singular component of the solution to one of the stationary solution in local L 2 ⊕ L 2 -seminorms.

Model
We fix a nonlinear function F : R → R and define the domain Let H F be a nonlinear operator on the domain D F defined by Let us introduce the phase space for Eq. (2.3). Denote the spacė Obviously, D F ⊂Ḋ.

3) has a unique strong solution ψ(t) such that (ii) The energy is conserved:
(iii) The following a priori bound holds This result is proved in [12,Theorem 3.1]. For the convenience of readers, we sketch main steps of the proof in Appendix in the case t ≥ 0 clarifying some details of [12]. As the result the solution ψ(x, t) to (2.3) with initial data ψ(0) = ψ 0 ∈ D F ,ψ(0) = π 0 ∈Ḋ can be represented as the sum is a unique solution of the Cauchy problem for the free wave equation Here is a unique solution to the Cauchy problem for the following firstorder nonlinear ODE 1 4πζ Next lemma implies that limit (2.9) is well defined, and there exists λ(0+) = lim t→0+ λ(t).
with zero initial data. The unique solution to (2.10) is the spherical wave where θ is the Heaviside function. This is well-known formula [14, Section 175] for the retarded potential of the point particle. Hence, (ii) We have lim (iii) Due to (2.12) it remains to show thatψ f,reg (0, t) ∈ L 2 loc ([0, ∞)). This follows immediately from [12,Lemma 3.4].
Our main result is the following theorem.

4) hold and let ψ(x, t) be a solution to eq. (2.3) with initial data
where the convergence hold in L 2 loc (R 3 ) ⊕ L 2 loc (R 3 ). It suffices to prove Theorem 2.5 for t → +∞.

Dispersion Component
We will only consider the solution ψ(x, t) restricted to t ≥ 0. In this section we extract regular and singular parts from the dispersion component ψ f (x, t) and establish their local decay. First, we represent the initial data (ψ(0),ψ(0)) = (ψ 0 , π 0 ) ∈ D F as (ψ 0 , π 0 ) = (ψ 0,reg , π 0,reg ) where a cut-of function χ ∈ C ∞ 0 (R 3 ) satisfies Indeed, On the other hand, where ϕ and ψ G are defined as solutions to the following Cauchy problems: , t), (3.5) and study the decay properties of ψ G and ϕ.
The following lemma states a local decay of solutions to the free wave equation with regular initial data from

Lemma 3.2 Let ϕ(t) be a solution to (3.4) with initial data
where B R is the ball of radius R.
Proof For any r ≥ 1 denote χ r = χ(x/r ), where χ(x) is a cut-off function defined in (3.1). Let u r (t) and v r (t) be the solutions to the free wave equations with the initial data χ r φ 0 and (1 − χ r )φ 0 , respectively, so that u(t) = u r (t) + v r (t). By the strong Huygens principle u r (x, t) = 0 for t ≥ |x| + 2r.
To conclude (3.7), it remains to note that due to the energy conservation for the free wave equation. We also use the embedding H 1 (R 3 ) ⊂ L 6 (R 3 ). The right-hand side of (3.8) could be made arbitrarily small if r ≥ 1 is sufficiently large.

Singular Component
Due to (3.9) to prove Theorem 2.5 it suffices to deduce the convergence to stationary states for the singular component ψ S (x, t) of the solution.
Proposition 4.1 Let assumptions of Theorem 2.5 hold, and let ψ S (t) be a solution to (2.7). Then where the convergence holds in L 2 Proof The unique solution to (2.7) is the spherical wave First, we obtain a convergence of ζ(t).

Lemma 4.2 There exists the limit
where q + ∈ Q.
Proof From (2.4) it follows that ζ(t) has the upper and lower limits: Suppose that a < b. Then the trajectory ζ(t) oscillates between a and b. Assumption (1.4) implies that F(ζ 0 ) = 0 for some ζ 0 ∈ (a, b). For the concreteness, let us assume that F(ζ 0 ) > 0. The convergence (3.9) implies that Hence, for sufficiently large T we have Then for t ≥ T the transition of the trajectory from left to right through the point ζ 0 is impossible by (2.8). Therefore, a = b = q + . Finally F(q + ) = 0 by (2.8). Further, uniformly in |x| ≤ R. Then (4.1) and (4.2) imply that where the convergence holds in L 2 loc (R 3 ). It remains to deduce the convergence ofψ S (t). We haveψ From (4.2), (2.8) and (4.3) it follows thatζ (t) → 0 as t → ∞. Theṅ by (4.4). This completes the proof of Proposition 4.1 and Theorem 2.5.
The solution (t) = (ψ(t),ψ(t)) ∈ D constructed in Proposition 5.2 exists for 0 ≤ t ≤ τ , where the time span τ in Lemma 5.1 depends only on ( 0 ). Hence, the bound (5.10) at t = τ allows us to extend the solution to the time interval [τ, 2τ ]. We proceed by induction to obtain the solution for all t ≥ 0.