Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case

Marianna Chatzakou,Michael Ruzhansky,Niyaz Tokmagambetov

doi:10.1080/17476933.2021.1950146

Abstract

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part [Altybay et al. Fractional Klein-Gordon equation with singular mass. Chaos Solitons Fractals. 2021;143:Article ID 110579] which was devoted to the classical Euclidean Klein-Gordon equation.

Highlights

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution
Such operators are Rockland operators on Rd, as we explain
Let us briefly recall some basic concepts, terminology and notation on graded Lie groups that will be useful for the ideas we develop throughout this paper

Summary

Preliminaries

Let us briefly recall some basic concepts, terminology and notation on graded Lie groups that will be useful for the ideas we develop throughout this paper. Graded Lie groups are naturally homogeneous Lie groups; that is g is equipped with a one-parameter family {Dr}r>0 of automorphisms of g of the form Dr = exp(A logr), with A being a diagonalisable linear operator on g with positive eigenvalues A remarkable class among left-invariant operators, that generalises the notion of the sub-Laplacian on the bigger class of graded groups, is that of Rockland operators, which are usually denoted by R The latter is a class of operators that are hypoelliptic on G [HN79], and homogeneous of a certain positive degree. For s > 0, p > 1, and R a positive homogeneous Rockland operator of degree ν, we define the R-Sobolev spaces as the space of tempered distributions.

Estimates for the classical solution

Existence and uniqueness of the very weak solution

Consistency of the very weak solution with the classical one