L1 Estimates for Oscillating Integrals Related to Structural Damped Wave Models
The goal of this paper is to derive L p − L q estimates away from the conjugate line for structural damped wave models. The damping term interpolates between exterior damping and viscoelastic damping. The crucial point is to derive at first L 1 − L 1 esti
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L1 Estimates for Oscillating Integrals Related to Structural Damped Wave Models Takashi Narazaki and Michael Reissig
Abstract The goal of this paper is to derive L p − Lq estimates away from the conjugate line for structural damped wave models. The damping term interpolates between exterior damping and viscoelastic damping. The crucial point is to derive at first L1 − L1 estimates. Depending on the behavior of the characteristic roots of the operator, one has to take into consideration oscillations in one part of the extended phase space. The radial symmetric behavior of the roots allows to apply the theory of modified Bessel functions. Oscillations may produce unbounded time-dependent constants (either for small times close to 0 or for large times close to infinity) in the L1 − L1 estimates. Some interpolation techniques imply the desired L p − Lq estimates away from the conjugate line. Key words: Modified Bessel functions, L1 –L1 estimates, Structural damping, Wave models 2010 Mathematics Subject Classification: 35L99, 35B40, 35A23.
11.1 Introduction Let us consider the Cauchy problem for the structural damped wave equation utt − Δ u + μ (−Δ )σ ut = 0, u(0, x) = u0 (x), ut (0, x) = u1 (x),
(11.1)
T. Narazaki Department of Mathematical Sciences, Tokai University, Kitakaname, Kanagawa, 259-1292 Japan e-mail: [email protected] M. Reissig () Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Prüferstr. 9, D-09596 Freiberg, Germany e-mail: [email protected] M. Cicognani et al. (eds.), Studies in Phase Space Analysis with Applications to PDEs, 215 Progress in Nonlinear Differential Equations and Their Applications 84, DOI 10.1007/978-1-4614-6348-1__11, © Springer Science+Business Media New York 2013
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T. Narazaki and M. Reissig
where μ is a positive constant and σ ∈ (0, 1]. Here we have two limit cases. The case σ = 0 describes waves with external damping, and σ = 1 describes waves with viscoelastic damping. In both cases L p − Lq estimates for the solutions are well understood; see [4–7, 10] and [11] for the case σ = 0 (linear or semi-linear models) and [8] for the case σ = 1. Applying partial Fourier transformation to (11.1), we have the following representation of solution (let us assume λ1 = λ2 ): λ eλ 2 t − λ eλ 1 t 1 2 v0 (ξ ) λ1 − λ2 e λ 1 t − e λ 2t v1 (ξ ) , + F −1 λ1 − λ2
u(t, x) = F −1 (v(t, ξ )) = F −1
where
λ1,2 =
− μ | ξ |2 σ ±
μ 2 |ξ |4σ − 4|ξ |2 , v0 (ξ ) = F(u0 )(ξ ), v1 (ξ ) = F(u1 )(ξ ). 2 (11.2)
We introduce the notations λ eλ 2 t − λ eλ 1 t 1 2 v0 (ξ ) , λ1 − λ2 e λ 1 t − e λ 2t v1 (ξ ) . J1 (t, x)(u1 ) := F −1 λ1 − λ2 J0 (t, x)(u0 ) := F −1
Let χ = χ (|ξ |) be a smooth function which localizes to small frequencies, then we shall estimate the Fourier multipliers λ eλ 2 t − λ eλ 1 t 1 2 χ (|ξ |)v0 (ξ ) , λ1 − λ2 λ eλ 2 t − λ eλ 1 t 1 2 (1 − χ (|ξ |))v0(ξ ) , J02 (t, x)(u0 ) := F −1 λ1 − λ2 eλ 1 t − eλ 2 t J11 (t, x)(u1 ) := F −1 χ (|ξ |)v1 (ξ ) , λ1 − λ2 eλ 1 t − eλ 2 t (1 − χ (|ξ |))v1(ξ ) . J12 (t, x)(u1 ) := F
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