Microlocal analysis for waves propagating in Einstein \& de Sitter spacetime
This paper deals with regularity properties of solutions of the equation \[ \psi_{tt}-t^{-4/3}\Delta\psi+{2\over t}\psi_t=f(x,t), \] with \(t>0\), \(x\in{\mathbb R}^n\) and \(f\) given. It is a sequel to a previous paper by the authors and \textit{T. Kinoshita} [J. Math. Phys. 51, No. 5, 052501, 18 p. (2010; Zbl 1310.58021)]. Letting \(\psi=u/t\), the authors reduce the problem to \[ u_{tt}-t^{-4/3}\Delta u=g(x,t), \] where \(g=tf\), and give an integral representation formula for solutions in terms of hypergeometric functions (that take an elementary form in this case), and solve a singular initial-value problem, pointing out the similarity with the Euler-Poisson-Darboux (EPD) equation to ``give a motivation for the choice of initial conditions.'' Uniqueness and regularity of the solution are also discussed. It is not clear whether the authors are aware that the equation for \(u\) is actually reducible to an inhomogeneous EPD equation (by introducing a new time variable \(T=t^{1/3}\)). Indeed, on page 3 or the previous paper, we read ``[t]o us, this approach seems to be more immediate than the one that uses the Euler-Poisson-Darboux equation.''
Euler-Poisson-Darboux equation, Sobolev estimates
scientific article; zbMATH DE number 6370127
scientific article