On the wave equation with multiplicities and space-dependent irregular coefficients

Abstract

In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in Garetto and Ruzhansky [Arch. Ration. Mech. Anal. 217 (2015), pp. 113–154], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples.

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

“…with initial conditions u(0, π) k,l = u 0 (π ) k,l and u t (0, π) k,l = u 1 (π ) k,l , for all π ∈ G and for any k, l ∈ N, where now f (t, π) k,l can be regarded as the source term of the second order differential equation as in (12). Now, let us decouple the matrix equation in (12) by fixing π ∈ G, and treat each of the equations represented in (12) individually. If we denote by…”

“…The wave type equations with time-dependent coefficients on graded Lie groups were analysed in [7] for Hölder coefficients, and in [8] for distributional time-dependent coefficients, using the notion of very weak solutions. All these works deal with the time-dependent equations and in the recent papers [9][10][11][12], the authors start to develop the notion of very weak solutions for equations with (irregular) space-depending coefficients.…”

Heat equation with singular thermal conductivity

“…In these papers, the authors dealt with equations with time-dependent coefficients. In [Gar21,ARST21] the authors started using the concept of very weak solutions for equations with coefficients depending on the spatial variable. Recently, the authors in [CRT22a,CRT22b] used the approach of very weak solutions for the heat equation and the fractional Schrödinger equation for general hypoelliptic operators in the setting of graded Lie groups.…”

On the wave equation with space dependent coefficients: singularities and lower order terms

“…For this reason, as in [G21,GR14] we look for solutions of the Cauchy problem (1) in the very weak sense. In other words we replace the equation under consideration with a family of regularised equations obtained via convolution with a net of mollifiers.…”