arXiv subjects

Sarah Kistner

Publications and source records attributed to Sarah Kistner.

3 recordsLinked to original sources

arXiv2026

Existence of a stable shrinker for the corotational harmonic map heat flow in higher space dimensions

We study singularity formation for the heat flow of harmonic maps from $\R^d$ into $\mathbb{S}^d$ in supercritical dimensions $d \in \{3,4,5,6\}$. It is well known that in each of these dimensions there exist infinitely many self-similar solutions that provide examples of loss of regularity in finite time. In this paper, we extend the results of \cite{BieDon18}, \cite{BieDonSch17} for $d=3$ to higher space dimensions $d \in \{4,5,6\}$ and prove the existence of a monotonically increasing self-similar profile $f_0$, which is asymptotically stable under small corotational perturbations. To construct the solution and resolve the spectral problem, we use rigorous computer assistance. As a byproduct of our stability analysis, we also obtain finite-codimension stability of arbitrary self-similar profiles within the corotational class.

math.AP↗

arXiv2026

Stable blowup profile for a semilinear heat equation with spatially inhomogeneous nonlinearity

We study the focusing semilinear heat equation with an additional defocusing H\'enon-type nonlinearity, the coupling of which is measured by a constant $c >0$. For $c \in (0,c^*)$, the model admits a closed-form self-similar blowup solution in every space dimension $d \geq 1$. Restricting ourselves to the three-dimensional case, we study the stability of this solution under small non-radial perturbations. By working in intersection Sobolev spaces with additional angular regularity, we prove finite co-dimension stability for all admissible values of $c$. Furthermore, we analyze the spectrum of the underlying linearized operator and we prove stable blowup for the cubic-quintic case and $c$ sufficiently close to $c^*$. Finally, we discuss the situation for small values of $c$ and use a modified version of the classical GGMT criterion to give an upper bound on the number of unstable eigenvalues.

math.AP↗

arXiv2023

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

We study singularity formation for the heat flow of harmonic maps from $\R^d$. For each $d \geq 4$, we construct a compact, $d$-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly \emph{shrinker}) that represents a stable blowup mechanism for the corresponding Cauchy problem.

math.AP↗