×

Wave equations with mass and dissipation. (English) Zbl 1329.35180

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.

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
PDFBibTeX XMLCite
Full Text: arXiv Euclid