Bonn-Cologne Analysis & PDE Workshop 2026

Titles & Abstracts

Speaker: Jean-Marc Delort (Université de Paris XIII (Paris-Nord))

Title: Blowing up solutions for one dimensional Klein-Gordon equations

Abstract: Consider  a quasi-linear (or semi-linear) cubic Klein-Gordon equation in one space dimension with small, smooth and decaying initial data. It is known that when the nonlinearity satisfies a convenient ``null condition'', the solution is global. It is conjectured that, if this null condition is not satisfied, generic initial data should give rise to solutions blowing up at a time of magnitude similar to $\pm e^{S_*/\epsilon^2}$, where $S_*>0$ and $0<\epsilon\ll 1$ is the size of the initial data. 

We consider in this talk equation

\[(\partial_t^2-\partial_x^2 +1)u = (\partial_t u)^3,\]

where the cubic semi-linear nonlinearity does not satisfy the null condition. We construct a blowing up solution at time $T_*(\epsilon) = e^{S_*/\epsilon^2}$, which at time $T_0(\epsilon) = \epsilon^{-2}T_*(\epsilon)^{1-b}$ ($b>0$ small), satisfies smallness and decay conditions, compatible with those allowing one to prove global existence for the similar problem when the nonlinearity satisfies the null condition. Moreover, we have an asymptotic description of $u$ close to the unique blowing up point $(T_*(\epsilon),0)$.

Speaker: Robert Schippa (University of Bonn)

Title: Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows

Abstract: We obtain sharp bilinear estimates for solutions to dispersive equations with C^{1,1}-coefficients. These extend the results of Wolff (2001) and Tao (2003) to the case of coefficients with minimal pointwise regularity. If time permits, we point out the application to local smoothing estimates for rough wave equations. Based on joint works with Daniel Tataru (UC Berkeley) and Jan Rozendaal (IMPAN).

Speaker: Érik De Amorim (University of Cologne)

Title: Exact general-relativistic solutions with finite self-energy for discrete point charges under Bopp-Podolsky electromagnetism

Abstract: We present --- in a self-contained and elementary way --- the existence of a family of static, spherically symmetric spacetimes that solve the Einstein equations of General Relativity coupled to the electric field of a point charge under the generalized laws of electromagnetism of Bopp-Landé-Thomas-Podolsky (BLTP for short). This is mathematically interesting, albeit not physically realistic, because the point charge is not plagued by the problem of the "infinite self-force" that occurs with the physical solution (the so-called Reissner-Nordström spacetime). This means that further rigorous studies of the motion of point charges in GR can in theory be carried out using BLTP theory as a toy model. Time permitting, we will also discuss the difficulties that arise in the same type of study for two interacting particles. All of the needed prerequisites from Physics, like Electromagnetism and Relativity, and Math, like hypergeometric functions and differential equations as perturbation problems, will be covered during the talk.