do Nascimento, Wanderley Nunes; Wirth, Jens Wave equations with mass and dissipation. (English) Zbl 1329.35180 Adv. Differ. Equ. 20, No. 7-8, 661-696 (2015). The authors are interested in the following Cauchy problem for wave equations with time-dependent mass and dissipation terms: \[ u_{tt}-\Delta u+ b(t)u_t+ m(t)u= 0,\quad u(0, x)= \varphi(x),\quad u_t(0,x) = \psi(x). \] The mass and dissipation terms are chosen in such a way that both are non-effective, that is, qualitative properties of solutions are closely related to those for free waves. A difference to earlier papers is that the authors use the technique of asymptotic integration in the pseudo-differential zone to derive estimates for the fundamental solution. This helps to understand critical constants \(b_0\) and \(m_0\) appearing in corresponding non-effective scale-invariant models with \(b(t)={b_0\over 1+t}\) and \(m(t)= {m_0\over (1+ t)^2}\).In the hyperbolic zone, a standard approach basing on diagonalization techniques is used to get explicit representations of solutions in terms of Fourier multipliers. Gluing the results which are obtained in both zones allows to prove \(L^2\)-\(L^2\) estimates with and without additional regularity for the data and \(L^p\)-\(L^q\) estimates on the conjugate line. The results are sharp. The authors obtain the sharpness after proving a modified scattering result. Reviewer: Michael Reissig (Freiberg) Cited in 8 Documents MSC: 35L15 Initial value problems for second-order hyperbolic equations 35L05 Wave equation 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs Keywords:damped Klein-Gordon equation; energy decay; \(L^p\)-\(L^q\) estimates; scattering theory; WKB representation; pseudo-differential zone; hyperbolic zone PDFBibTeX XMLCite \textit{W. N. do Nascimento} and \textit{J. Wirth}, Adv. Differ. Equ. 20, No. 7--8, 661--696 (2015; Zbl 1329.35180) Full Text: arXiv Euclid