The Shifted Wave Equation on Damek–Ricci Spaces and on Homogeneous Trees

Abstract

We solve explicitly the shifted wave equationDamek-Ricci spaces, usingÁsgeirsson's theorem and the inverse dual Abel transform. As an application, we investigate Huygens' principle. A similar analysis is carried out in the discrete setting of homogeneous trees.

A semi-linear energy critical wave equation with an application

“…In this work, we will prove that similar results as mentioned in the previous subsection still hold for the equation (CP1), at least for those φ's that satisfy some additional condition besides (1).…”

A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on $\mathbb {R}^2$

“…for each t 0 ∈ (−T − , T + ). The combination of this estimate with the fact that (u(•, t 0 ), 2 . By considering the backward Cauchy problem (3) with initial data (u(•, t 0 ), ∂ t u(•, t 0 )), we obtain E(u 0 , u 1 ) ≤ E(u(•, t 0 ), ∂ t u(•, t 0 )) by the same argument above.…”

“…More recently Anker, Pierfelice and Vallarino gave a wide range of Strichartz estimates as well as a brief description on the local well-posedness theory in [3]. Global well-posedness is also considered in [2].…”

Poincaré and Hardy inequalities on homogeneous trees

“…T q+1 has been the object of investigation of many papers either in the field of harmonic analysis or of PDEs, see e.g. to [2,3,7,12,13,14,15,16,19,30]. In particular, the homogeneous tree is in many respects a discrete analogue of the hyperbolic plane; we refer the reader to [7] for a discussion on this point.…”