arXiv subjects
Publications and source records attributed to Birgit Schörkhuber.
We consider Ornstein-Uhlenbeck operators perturbed by a radial potential. Under weak assumptions we prove a spectral mapping theorem for the generated semigroup. The proof relies on a perturbative construction of the resolvent, based on angular separation, and the Gearhart-Prüß Theorem.
We study the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $t=T>0$ given by \[ ψ^T(t,x)=c_p (T-t)^{-\frac{2}{p-1}} \] where $c_p$ is a suitable constant. We prove that the blow up described by $ψ^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that lead to a solution which converges to $ψ^T$ as $t\to T-$ in the backward lightcone of the blow up point $(t,r)=(T,0)$.
We consider the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $1 0$ and $κ_p$ is a $p$-dependent constant. We prove that the blow up described by $ψ^T$ is stable against small perturbations in the energy topology. This complements previous results by Merle and Zaag. The method of proof is quite robust and can be applied to other self-similar blow up problems as well, even in the energy supercritical case.