2 00 2 On the existence of the Møller wave operator for wave equations with small dissipative terms

The aim of this short note is to reconsider and to extend a former result of K. Mochizuki [Moc76], [MN96] on the existence of the scattering operator for wave equations with small dissipative terms. Contrary to the approach used by Mochizuki we construct the wave operator explicitly in terms of the parametrix construction obtained by a (simplified) diagonalization procedure, cf. [Yag97]. The method is based on ODE techniques. These considerations are part of a larger project and the idea is taken from [Wir] and generalized to x-dependent coefficients. AMS subject classification: 35L15; 35P25 keywords: Cauchy problem, wave equation, scattering operator, dissipative term We consider the Cauchy problem u+ b(t, x)ut = 0, u(0, ·) = u1, Dtu(0, ·) = u2 (1) with b ∈ L1(R, L∞(Rn)) ∩ L∞(R). We restrict our calculations to spaces of dimension n ≥ 2. The modification to obtain results also for n = 1 are obvious. We denote by E = Ḣ × L2 the energy space. We prove that in the energy space (u,Dtu) converges to the local energy of a solution ũ of the free wave equation ũ = 0. 1 As usual we use the representation Ḣ = |D|L2, |D|−1 beeing the Riesz potential operator of order 1, inverse to the operator |D| = √ −∆. In our calculations we use this isomorphism Ḣ ≃ L2 to restrict ourselves to calculations in L2-space. Theorem 1. Assume b ∈ L1(R, L∞(Rn)) ∩ L∞(R). There exist isomorphisms W± : E → E of the energy space such that for u = u(t, x) the solution of (1) to data (u1, u2) ∈ E and for (ũ1, ũ2) = W±(u1, u2) and ũ the solution of the free wave equation ũ = 0 to data ũ(0, ·) = ũ1, Dtũ(0, ·) = ũ2 the asymptotic relation ||(u,Dtu)− (ũ,Dtũ)||E → 0 as t → ±∞ holds. We subdivide the proof in some steps and construct these wave operators explicitly in terms of the solution representation. Let U = (|D|û,Dtû) . Then U satisfies the equation DtU = ( |D| |D| )