Singular convergence for semilinear wave equations with steep potential well

In this paper we consider the semilinear wave equation (1.1) u tt + γu t + V β (x)u -∆u = f β (x, u), (t, x) ∈ [0, +∞[×R 3 in the energy space H 1 (R 3 ) × L 2 (R 3 ). The damping coefficient γ is a non-negative (possibly zero) constant and the function V β (x) has the form V β (x) = 1 + βV (x), where β is a (large) positive parameter and V (x) is a "well potential". This means that there exists a bounded open domain Ω on which V (x) ≡ 0, while V (x) is strictly positive on R 3 ∖ Ω and lim |x|→∞ V (x) = 1. The functions f β (x, u) are C 1 with respect to u and satisfy a growth condition of type |∂ u f β (x, u)| ≤ C(a(x) + |u| 2 ), so they define locally Lipschitz continuous mappings from H 1 (R 3 ) to L 2 (R 3 ); moreover, as β → +∞, the family f β converges in some sense to a function f Ω with analogous properties. It is well known that the Cauchy problem for (1.1) is well posed in H 1 (R 3 ) × L 2 (R 3 ). Our aim is to investigate the behaviour of the solutions when the parameter β tends to infinity. There are several results concerning the stationary solutions of (1.1) (see e.g. [2,3,10,11,22]) and the eigenvales of the linear operator -∆ + V β (x) (see e.g. [2,13,20]). Both physical and mathematical evidence confirm that as β → +∞ the solutions of the nonlinear elliptic equation

tend to remain confined within the region Ω, and become closer and closer to solutions of the elliptic Dirichlet problem

Our aim is to prove that the same is true for non stationary solutions, and to provide a framework to study the (singular) convergence of the dynamics generated by (1.1) in the space H 1 (R 3 ) × L 2 (R 3 ) to the one generated by the “limit” equation (1.4) u tt + γu t + u -∆u = f Ω (x, u), (t, x) ∈ [0, +∞[×Ω u = 0, (t, x) ∈ [0, +∞[×∂Ω in the space H 1 0 (Ω) × L 2 (Ω). In the spirit of works like [1,4,5,15,16] and others, we have in mind the persistence of attractors or more general invariant sets, the robustness of exponential dichotomies, and other dynamical aspects of equation (1.1).

The paper is organized as follows:

In Section 2 we introduce the functional analytic setting for the linear elliptic problem and we prove the (singular) convergence of the resolvent of the operator -∆+V β (x) in R 3 to the resolvent of the operator -∆+I in Ω (with Dirichlet boundary condition) when β → +∞.

In Section 3 we study the behaviour of the spectrum of -∆ + V β (x) as β → +∞. We show that as β increases, the bottom of the essential spectrum moves towards +∞ and more and more eigenvalues appear. The eigenvalues converge to the ones of the limit operator -∆ + I in Ω (with Dirichlet boundary condition), and so do the corresponding eigenfunctions. This spectral convergence can be used e.g. in studying exponential dichotomies for the linearization of (1.1) near an equilibrium.

In Section 4 we introduce the functional analytic setting for the linear wave equation (1.5) u tt + γu t + V β (x)u -∆u = 0, (t, x) ∈ [0, +∞[×R 3 and we prove that such equation generates a strongly continuous semigroup T β (t) in the space H 1 (R N ) × L 2 (R 3 ), with bounds independent of β.

In Section 5 we prove a singular Trotter-Kato theorem for the semigroups T β (t) as β → +∞, showing that they converge to the semigroup T Ω (t) generated by the linear wave equation (1.6) u tt + γu t + u -∆u = 0, (t, x) ∈ [0, +∞[×Ω u = 0, (t, x) ∈ [0, +∞[×∂Ω In Section 6 we finally prove that, as β → +∞, the solutions of the semilinear equation (1.1) converge to the solutions of (1.4) uniformly on compact time intervals.

In this paper we denote by | ⋅ | the euclidean norm in R N . Let Ω ⊂ R N be an open bounded set with C 2 -boundary and let V ∶ R N → R be a bounded measurable function. Throughout the paper we make the following assumption: Hypothesis 2.1. The function V satisfies the following properties:

(1) 0 ≤ V (x) ≤ 1 almost everywhere in R N ;

(2) V (x) = 0 almost everywhere in Ω;

(3) ess inf{V (x) | x ∈ B r (x 0 )} > 0 for every closed ball B r (x 0 ) ⊂ R N ∖ Ω;

(4) ess lim |x|→∞ V (x) = 1.

For every β ≥ 0 we define (2.1)

For β ≥ 0, in H 1 (R N ) we define the following bilinear form:

It turns out that a β (⋅, ⋅) is a scalar product in H 1 (R N ), equivalent to the standard one. We shall denote this scalar product by ⟨⋅, ⋅⟩ 1,β . Setting ∥u∥ 1,β ∶= ⟨u, u⟩ 1/2 1,β , we have:

The scalar product ⟨⋅, ⋅⟩ 1,β induces a bounded linear map

One can easily check that

, which returns exacly the norm ∥ ⋅ ∥ -1,β and makes Λ β an isometry. The part of Λ β in L 2 (R N ) defines a selfadjoint operator in L 2 (R N ), which we denote by A β . The domain of

Concretely, the operator A β acts according to the rule

By regularity theory for elliptic equations (see e.g. [8]), it turns out that D(A β ) = H 2 (R N ), and

(2.7)

We introduce the following closed subspace of H 1 (R N ):

(2.8)

), and we observe that (2.9)

We denote by H -1 Ω (R N ) the dual space of H 1 Ω (R N ) and we observe that H -1 Ω (R N ) can be naturally identified with H -1 (Ω).

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