Work

Asymptotics and scattering for critically weakly hyperbolic and singular systems

Year: 2026

Type: article

Abstract: We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain lower-order coefficients that blow up at a single time $t = 0$. In "critical" weakly hyperbolic settings, it is well-known that solutions lose a finite amount of regularity at the degenerate time $t = 0$. In this paper, we both improve upon the results in the weakly hyperbolic setting, and we extend this analysis to systems containing critically singular coefficients, which may also exhibit significantly modified asymptotics at $t = 0$. In particular, we give precise quantifications for (1) the asymptotics of solutions as $t$ approaches $0$; (2) the scattering problem of solving the system with asymptotic data at $t = 0$; and (3) the loss of regularity due to the degeneracies at $t = 0$. Finally, we discuss a variety of applications for these results, including to weakly hyperbolic and singular wave equations, equations of higher order, and equations arising from relativity and cosmology, e.g. at big bang singularities.

Source: Advances in Mathematics

DOI: 10.1016/j.aim.2026.111212

Authors: Bolys Sabitbek, Arick Shao

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,790

Funders: Ministry of Education and Science of the Republic of Kazakhstan, Engineering and Physical Sciences Research Council

Awards: Rigidity Problems in Holography and Relativity, Hyperbolic problems with discontinuous coefficients