EXISTENCE OF SCATTERING STATES FOR WAVE EQUATIONS

WITH DISSIPATIVE TERMS IN STRATIFIED MEDIA

Mitsuteru Kadowaki

1. Introduction.

Let n ≧ 1 and x ∈ R^{n}, y ∈ R. In this paper we study the following wave equations with dissipative terms :

(1.1)  ∂_{t}^{2}u(x, y, t) + b(x, y)∂_{t}u(x, y, t) − c_{0}(y)^{2}△u(x, y, t) = 0,

where (x, y, t) ∈ R^{n} × R × [0, ∞), △ is Laplacian for (x, y),

c_{0}(y) = \{ \begin{matrix}c_{+} (y ≧ h) \\ c_{h} (0 < y < h) \\ c_{−} (y ≦ 0), \end{matrix}

and c_{+}, c_{−}, c_{h} and h are positive constants. Our aim is to show the existence of the scattering states to (1.1) for t → +∞. We deal with the following assumption for b(x, y) : There exist C > 0 and θ > 1 such that

0 ≦ b(x, y) ≦ C(1 + |x|^{2} + y^{2})^{− \frac{θ}{2}} ,

for any (x, y) ∈ R^{n} × R.

Considering the case c_{h} < min(c_{+}, c_{−}) we find the guided waves (cf. Wilcox [8] or Weder [7]). In brief we deal with c_{h} < c_{+} ≦ c_{−} only.

L^{2}(R^{n+1}) is the usual L^{2} space defined on R^{n+1}. For s ∈ R, let be L_{s}^{2}(R^{n+1}) the weighted L^{2} space defined by

L_{s}^{2}(R^{n+1}) = \{u(x) : X_{−s}u(x) ∈ L^{2}(R^{n+1})\}, X_{−s} = (1 + |x|^{2} + y^{2})^{\frac{s}{2}} .

Let E and F be Banach spaces. Then we denote by ∥ · ∥_{E} the norm of E. If T is considered as an operator from E into F, we denote by ∥T ∥_{E→F} the operator norm. If in particular T is considered as an operator from L^{2}(R^{n+1}) into itself, then its norm is denoted by the notation ∥T ∥.

We define the symmetric operator L_{0} as :

L_{0} = −c_{0}(y)^{2}△,

Then L_{0} admits a unique self-adjoint realizations in L^{2}(R^{n+1}; c_{0}^{−2}(y)dxdy) . Then L_{0} is a non-negative operator ( zero is not an eigenvalue ) and the domain of L_{0}, D(L_{0}), is given by H^{2}(R^{n+1}), H^{s}(R^{n+1}) being Sobolev space of order s over R^{n+1}. We also denoted by R(ζ; L_{0}) the resolvent (L_{0} − ζ)^{−1} of L_{0} for Imζ ≠ 0.

We remark that Weder [7] has showed the absence of eigenvalues and the limiting absorption principle for L_{0}.

To state our main results we introduce the free wave equations :

(1.2)  ∂_{t}^{2}u(x, y, t) + L_{0}u(x, y, t) = 0.

We put f = (u(t, x, y), ∂_{t}u(t, x, y)). Then (1.1) and (1.2) can be written as ∂_{t}f = −iAf and ∂_{t}f = −iA_{0}f respectively, where

A = i ( \begin{matrix}0 & 1 \\ −L_{0} & −b(x, y) \end{matrix} ) , A_{0} = i ( \begin{matrix}0 & 1 \\ −L_{0} & 0 \end{matrix} ) .

Let H be Hilbert spaces with inner products

< f, g >_{H}= ∫_{R_{n+1}}(∇f_{1}(x, y)\overline{∇g_{1}(x, y)} + f_{2}(x, y)\overline{g_{2}(x, y)}c_{0}^{−2}(y))dxdy,

and the corresponding norms ∥ · ∥_{H} where ∇ = (∂/∂x_{1}, ∂/∂x_{2}, ⋯, ∂/∂x_{n}, ∂/∂y) and f =^{t} (f_{1}, f_{2}), g =^{t} (g_{1}, g_{2}).

H is the energy space for (1.1) and (1.2). Then A and A_{0} generate a contraction semi-group {V (t)}_{t≧0} and a unitary group {U_{0}(t)}_{t∈R} on H respectively. Their domains are

D(A_{0}) = {f ∈ H; △f_{1} ∈ L^{2}(R^{n+1}), f_{2} ∈ H^{1}(R^{n+1})}

and D(A) = D(A_{0}). Main results is as follows:

Theorem 1.1. Assume that c_{+} = c_{−}. Then the wave oprators

W = s − lim_{t→+∞} U_{0}(−t)V (t)

exists. Moreover W is not identically zero in H.

Assuming c_{+} = c_{−} we show the existence of scattering states to (1.1) as an immediate consequence of Theorem 1.1.

Considering solutions to (1.1) in any low energy space we find scattering states for the case c_{+} ≠ c_{−} too. This results is as follows :

Theorem 1.2. Assume that c_{+} ≠ c_{−}. Then the local wave operators

W_{l} = s − lim_{t→+∞} E_{0}((−l, l))U_{0}(−t)V (t)

exists for any l > 0, where E_{0}((−l, l)) is the spectral projection of A_{0}onto (−l, l). Moreover W_{l} is not identically zero in H.

It seems that there are no works dealing with scattering problem for dissipative wave equations in stratified media. Our proof of Theorem 1.1 and 1.2 are due to Mochizuki [4] (see section 4). He has proved existence of scattering states for wave equations with disspative terms in the case c_{h} = c_{+} = c_{−} = 1. His idea is baesd on the combination Kato’s smooth pertabation theory with a resolvent estimates for Laplacian in R^{n}(n ≠ 2). To consider scattering problem for stratified media, we also need Mochizuki’s estimates for L_{0}. That statement is as follows : There exist η > 0 and C_{η} > 0 such that for any ζ ∈ C\R satisfing |Imζ| ≦ η

(1.3)  sup_{|Imζ|≦η} ∥X_{\frac{θ}{2}} R(ζ^{2}; L_{0})X_{\frac{θ}{2}} ∥ ≦ C_{η}|ζ|^{−1},

where X_{\frac{θ}{2}} = (1 + |x|^{2} + y^{2})^{−θ/4}. (1.3) follows from low and high resolvent estimates. In section 2 and section 3 the idea of the proof for low and high estimates is given respectively.

2. Low energy resolvent estimates.

The idea is due to Kuroda [3] using the satandard trace operator. We deal with s, only 1/2 < s < 1. This condition is sufficient to prove Theorem 1.1 and 1.2.

We state Weder’s result [14]. That is needed to prove our results. a and b are used as follows :

c_{+} = c_{−} ⟹ 0 ≦ a < b < \frac{c_{+}^{2}}{h^{2}(\frac{c_{+}^{2}}{c_{h}^{2}} − 1)}π^{2}

or

c_{+} ≠ c_{−} ⟹ 0 ≦ a < b < \frac{c_{+}^{2}}{h^{2}(\frac{c_{+}^{2}}{c_{h}^{2}} − 1)} arctan^{2} \begin{matrix}√ & \overline{1 − \frac{c_{+}^{2}}{c_{−}^{2}}} \\ √ & \overline{\frac{c_{+}^{2}}{c_{h}^{2}} − 1} \end{matrix} .

We write S_{r}^{n} as the sphere of center zero and radius in R^{n}. Then We define

S_{c} = S_{+} ∪ S_{0} ∪ S_{−},

where

S_{+} = \{ω ∈ Sn_{\frac{1}{c_{+}}} : ω_{0} > (\frac{c_{−}^{2}}{c_{+}^{2}} − 1)^{1/2}|\overline{ω}|\}

S_{0} = \{ω ∈ Sn_{\frac{1}{c_{+}}} : 0 < ω_{0} < (\frac{c_{−}^{2}}{c_{+}^{2}} − 1)^{1/2}|\overline{ω}|\}

S_{−} = \{ω ∈ Sn_{\frac{1}{c_{−}}} : ω_{0} < 0\},

ω = (\overline{ω}, ω_{0}) and |\overline{ω}|^{2} = ω_{1}^{2} + ω_{2}^{2} + ⋯ + ω_{n}^{2}.

According to Lemma 2.3 in Weder [7] there exist trace operators, T_{0}(μ) and T_{1}(μ), from L_{s}^{2}(R^{n+1}) into L^{2}(S_{c}) and L^{2}(S_{1}^{n−1}) respectively such that

\begin{matrix}X_{s}E_{0}((a, b))R(ζ^{2}; L_{0})X_{s} \\ = \{ \begin{matrix}∫_{a}^{b} \frac{1}{μ − ζ^{2}} X_{s}T_{0}^{∗}(μ)T_{0}(μ)X_{s}dμ + ∫_{a}^{b} \frac{1}{μ − ζ^{2}} X_{s}T_{1}^{∗}(μ)T_{1}(μ)X_{s}dμ, (c_{+} = c_{−}) \\ ∫_{a}^{b} \frac{1}{μ − ζ^{2}} X_{s}T_{0}^{∗}(μ)T_{0}(μ)X_{s}dμ, (c_{+} ≠ c_{−}), \end{matrix} \end{matrix}

where ζ^{2} ∈ C\R, L^{2}(S_{c}) and L^{2}(S_{1}^{n−1}) are usual L^{2} spaces defined on S_{c} and S_{1}^{n−1} respectively.

Analyzing T_{0}(μ) and T_{1}(μ) we obtain the following theorem. Theorem 2.1. As |ζ| → 0, one has

∥X_{s}R(ζ^{2}; L_{0})X_{s}∥ = \{ \begin{matrix}O(|ζ|^{2s−2}) (if n ≧ 2) \\ O(|ζ|^{γ−1}) (if n = 1), \end{matrix}

for any γ < s − 1/2.

3. High energy resolvent estimates.

By a technical reason we deal with only the case c_{+} = c_{−} except for Lemma 3.4 and its introduction. Theorem 3.1. Let s > 1/2. Assume that c_{+} = c_{−}. Then one has

∥X_{s}R(λ ± iκ; L_{0})X_{s}∥ = O(λ^{− \frac{1}{2}} ) (λ → ∞),

uniformly in κ > 0, where X_{s} = (1 + |x|^{2} + y^{2})^{−s/2}.

It follows from Theorem 3.1 that Corollary 3.2. Assume that c_{+} = c_{−}. Then there exist λ_{1} >> 1, η_{1} > 0 and C_{λ_{1},η_{1}} > 0 such that for any ζ ∈ C\R satisfing |Reζ| ≧ λ_{1} and |Imζ| ≦ η_{1}

sup_{|Reζ|≧λ_{1},|Imζ|≦η_{1}} ∥X_{\frac{θ}{2}} R(ζ^{2}; L_{0})X_{\frac{θ}{2}} ∥ ≦ C_{λ_{1},η_{1}}|ζ|^{−1}.

To prove Theorem 3.1 we define the self-adjoint operator L_{0}(λ) on L^{2}(R^{n+1}),

\{ \begin{matrix}L_{0}(λ) = −△ − λ(c_{0}^{−2}(y) − c_{+}^{−2}) \\ D(L_{0}(λ)) = H^{2}(R^{n+1}). \end{matrix}

This operator has been introduced by Weder [7]. Theorem 3.1 is obtained as an immediate consequence of the following proposition Proposition 3.3. Assume that c_{+} = c_{−}. Then one has

∥X_{s}G_{κ}^{±}(0; λ)X_{s}∥ = O(λ^{− \frac{1}{2}} ) (λ → ∞), uniformly in κ > 0, where

G_{κ}^{±}(0; λ) = (L_{0}(λ) − λc_{+}^{−2} ∓ iκc_{0}^{−2}(y))^{−1}

for κ > 0.

By using Mourre’s method (cf. Mourre’s [5]) Proposition 3.3 can be proven. Concering with L_{0}(λ) there exist F_{0}(λ), F_{1}(λ) and G_{j}(λ)(j = 1, 2, 3 ⋯ Q(λ)) which are partially isometric operators from L^{2}(R^{n+1}) onto L^{2}(R_{k}^{n}), L^{2}(Ω_{0}) and L^{2}(Rn ; \overline{k}) respectively, where k = (\overline{k}, k_{0}) ∈ R^{n} × R, Ω_{0} = \{k ∈ R^{n+1}; 0 < k_{0} < \sqrt{q_{−}(λ)} =

\sqrt{λ(c_{+}^{−2} − c_{−}^{−2})}\} and Q(λ) is finite number in N. Using F_{0}(λ), F_{1}(λ) and G_{j}(λ) we construct the conjugate operators and show Mourre’s estimates (3.1) (cf. Mourre [5]). First we define conjugate operator, D(λ), as follows (cf. Kadowaki[1]):

F_{0}(λ)^{∗}(−D_{n+1})F_{0}(λ) + F_{1}(λ)^{∗}(−D_{n})F_{1}(λ) + ∑_{j=1}^{Q(λ)} G_{j}(λ)^{∗}(−D_{n})G_{j}(λ),

where

D_{n+1} = \frac{1}{2i}(k · ∇_{k} + ∇_{k} · k), D_{n} = \frac{1}{2i}(\overline{k} · ∇\overline{k} + ∇\overline{k} · \overline{k}).

We consider the commutator i[L_{0}(λ), D(λ)] as a form on H^{2}(R^{n+1}) ∩ D(D(λ)). Moreover we can extented the commutator as a bounded operators,i[L_{0}(λ), D(λ)]^{0}, from H^{1}(R^{n+1}) to H^{−1}(R^{n+1}) and obtain Mourre’s estimates as follows

Lemma 3.4. Let λ > 1, take f_{λ}(r) ∈ C_{0}^{∞}(R), 0 ≦ f_{λ} ≦ 1 such that f_{λ} has support in ((c_{+}^{−2} − c_{−}^{−2}/2)λ, 2c_{+}^{−2}λ) and f_{λ} = 1 on [(c_{+}^{−2} − c_{−}^{−2}/4)λ, 3c_{+}^{−2}λ/2]. Then there exists a positive constant C which is independent of λ such that

(3.1)  f_{λ}(L_{0}(λ))i[L_{0}(λ), D(λ)]^{0}f_{λ}(L_{0}(λ)) ≧ Cλf_{λ}(L_{0}(λ))^{2}

in the form sense.

It follows from (4.3) that f_{λ}(L_{0}(λ))i[L_{0}(λ), D(λ)]^{0}f_{λ}(L_{0}(λ)) is non-negative and hence we define an operator, G_{κ}^{±}(ϵ; λ), on L^{2}(R^{n+1}) by

G_{κ}(ϵ; λ) = (L_{0}(λ) − λc_{+}^{−2} ∓ iκc_{0}^{−2}(y) ∓ iϵM_{0}(λ))^{−1}

for κ > 0 and ϵ > 0, where M_{0}(λ) = f_{λ}(L_{0}(λ))i[L_{0}(λ), D(λ)]^{0}f_{λ}(L_{0}(λ)).

We write

F_{κ}^{±}(ϵ; λ) = λ^{\frac{1}{2}} Z_{s}(ϵ, λ)G_{κ}^{±}(λ^{− \frac{1}{2}} ϵ; λ)Z_{s}(ϵ, λ),

where Z_{s}(ϵ, λ) = (λ^{\frac{1}{2}} + |D(λ)|)^{−s}(λ^{\frac{1}{2}} + ϵ|D(λ)|)^{s−1}.

This is due to Yafaev [9]. But we do not use the scaling argument for λ (see the definition of G_{κ}^{±}(0; λ) and D(λ)). Using (3.1) and differeting F_{κ}^{±}(ϵ; λ) for ϵ > 0 we obtain Proposition 3.3. (cf. Kikuchi-Tamura[2]).

4. Proof of Theorem 1.1 and 1.2.

The proof of Theorem 1.1 and 1.2 are same as in §3 in Mochizuki[4]. Here we give only a sketch of the proof of Theorem 1.1.

proof of Theorem 1.1. Noting that the local boundedness of R(ζ′; L_{0}) for ζ′ ∈ C_{±}( see Weder[14]) we obtain (1.3) by Theorem 2.1 and Corollary 3.2. It follows from (1.3) that

sup_{|Imζ|≦η} ∥Λ_{\frac{θ}{2}} (A_{0} − ζ)^{−1}Λ_{\frac{θ}{2}} ∥_{H→H} ≦ C_{η},

where C_{η} is a positive constant which depend on η only and

Λ_{\frac{θ}{2}} = ( \begin{matrix}0 & 0 \\ 0 & (1 + |x|^{2} + y^{2})^{− \frac{θ}{4}} \end{matrix} ) .

This means that Λ_{\frac{θ}{2}} is A_{0}-smooth (cf. Reed-Simon [6]). Therefore we have that {U_{0}(−t)V (t)f}_{t≧0} is Chauchy sequence as t → ∞ in H for f ∈ H.

To prove W ≢ 0 we assume that

lim_{t→∞} ∥V (t)f∥ = 0,

for any f ∈ H. Then we can show a contradiction. □

References

[1] M. Kadowaki, Low energy resolvent estimstes for acoustic propagators in perturbed stratified media, J. Differential Equations. 141 (1997), 25-53.

[2] K. Kikuchi - H. Tamura, The limiting amplilude principle for acoustic propagators in perturbed stratified fluids, J. Differential Equations. 93 (1991), 260-282.

[3] S.T. Kuroda, An Introduction to Scattering Theory, Lecture Note Series N^{o} 51, Matematisk Institut. Aarhus University, 1980.

[4] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. RIMS, Kyoto Univ 12 (1976), 383-390.

[5] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys 78 (1981), 391-408.

[6] M. Reed - B. Simon, Methods of Modern Mathematical Physics, Vol.3, Academic Press, New York, San Francisco, London, 1979.

[7] R. Weder, Spectral and scattering theory for wave propagation in perturbed stratified media, Applied Mathematical Sciences 87, Springer-Verlag, New york, Berlin, Heidelberg, 1991.

[8] C. Wilcox, Sound propagation in stratified fluids, Applied Mathematical Sciences 50, Springer-Verlag, New york, Berlin, Heidelberg, 1984.

[9] D.R. Yafaev, Eikonal approximation and the aymptotics of the total scattering cross-section for the Schrödinger equation, Ann. Inst. Henri Poincaré, Phys. théor 44 (1986), 397-425.

Tokyo Metoropolitan College of Aeronautical Eingineering, 8-52-1, Minamisenju, Arakawaku, Tokyo 116-0003 Japan

E-mail address: kadowaki@kouku-k.ac.jp