The analysis of singularities is a vital part of the modern theory of partial differential equations. Often it concerns finding the fundamental solution of a boundary value problem by taking the delta distribution as the source term or studying the singularities of solutions to partial differential equations.

In this research project, we focus on singularities in the coefficients of the equation. Such types of singularities naturally arise in many physical models, e.g. by jumps in material densities or in other physical quantities. These quantities may be further differentiated or multiplied and so complicating the type of singularity even further. The bottleneck is that, by the Schwartz impossibility result, the equation cannot be interpreted in a distributional sense. With aid of the promising notion of a very weak solution developed by Garetto and Ruzhanksy (Hyperbolic second order equations with non-regular time dependent coefficients. Arch. Rat. Mech. Anal., 217(1), 113–154, 2025) , such equations can be rigorously analysed even if the distributional interpretation of equations does not make sense.

The main aim of this challenging and ambitious project is to further explore and deepen the powerful machinery for dealing with PDEs with (strong) singularities. The goal of the project is to advance the development of a general well-posedness theory for linear and nonlinear initial-boundary value problems with strong singularities and to analyse the quantitative behaviour of very weak solutions. As applications, we target coupled problems, contact problems, and inverse problems.