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Irfan Glogić

Publications and source records attributed to Irfan Glogić.

At least 19 recordsLinked to original sources
arXiv2026

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

We study the focusing semilinear heat equation with an additional defocusing Hénon-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.

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arXiv2026

Three self-similar solutions of Yang-Mills equations in high odd dimensions

We consider spherically symmetric Yang-Mills equations with gauge group $SO(d)$ in $d+1$ dimensional Minkowski spacetime. For any given odd $d\geq 11$, we establish existence and uniqueness (modulo reflection symmetry) of exactly $N$ smooth self-similar solutions, where $N$ is the number of zeros of an explicit polynomial $P_m(z)$ of degree $m=(d-5)/2$ in the interval $0<z<1$. The number $N$ can be determined algorithmically by an explicit computation. Our extensive computations for large odd dimensions suggest that $N=3$ for all odd $d\geq 11$. Two of these self-similar solutions admit closed-form expressions: one has been known previously, while the other appears to be new. Our result points toward a relatively simple landscape of possible blowup scenarios for high-dimensional Yang-Mills equations. Beyond its purely mathematical interest, this rigidity of self-similar blowup may also be relevant from a physical perspective, as it constrains the possible ultraviolet dynamics of non-abelian gauge fields in higher-dimensional Yang-Mills theories arising in string-inspired extra-dimensional setups and in holographic models.

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arXiv2025

Existence and stability of discretely self-similar blowup for a wave maps type equation

We study finite-time blowup for a nonlinear wave equation for maps from the Minkowski space $\mathbb{R}^{1+d}$ into the 1-sphere $\mathbb{S}^1$, whose nonlinearity exhibits a null-form structure. We construct, for every dimension $d \geq 1$, a countable family of discretely self-similar blowup solutions, which are even for $d=1$ and radial for $d \geq 2$. The main contribution of the paper is a detailed nonlinear stability analysis of this family of solutions. For $d \geq 2$, we consider radial data, while in $d=1$ we allow for general perturbations. After linearizing around the self-similar profiles in similarity variables, we construct resolvents of the resulting highly non-self-adjoint operators through Liouville-Green transformations and precise Volterra-type asymptotics. The construction itself, which occupies most of the paper, is technically challenging, as it is performed in arbitrary dimensions and for a countable family of operators in each. Combined with a detailed spectral analysis of the linearized operators, this yields sharp semigroup bounds and allows us to establish nonlinear stability of all discretely self-similar profiles in all dimensions, with precise co-dimension determined by the unstable spectrum. To our knowledge, this is the first result on the existence and stability of discretely self-similar blowup for a geometric wave equation.

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arXiv2025

Self-similar blowup from arbitrary data for supercritical wave maps with additive noise

We consider stochastically perturbed wave maps from $\mathbb{R}^{1+d}$ into $\mathbb{S}^d$, in all energy-supercritical dimensions $d \geq 3$. We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.

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arXiv2025

Non-uniqueness of mild solutions to supercritical heat equations

We consider the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -Δu = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$. It is well-known that if $p$ is large enough then \eqref{Eq:Heat_abstract} is unconditionally locally well-posed in $L^q(\mathbb{R}^d)$ for $q \geq d(p-1)/2$. We prove that this result is optimal in the sense that uniqueness of local solutions fails when $q < d(p-1)/2$ as long as $p < p_{JL}$, where $p_{JL}$ stands for the Joseph-Lundgren exponent. Our proof is based on the method that Jia-Šverák proposed in \cite{JiaSve15} to show non-uniqueness of Leray solutions to incompressible 3d Navier-Stokes equations. In particular, we rigorously verify for \eqref{Eq:Heat_abstract} the (analogue of the) spectral assumption made in \cite{JiaSve15}. To our knowledge, this is the first rigorous implementation of the Jia-Šverák method to a nonlinear parabolic equation without forcing.

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arXiv2023

Globally stable blowup profile for supercritical wave maps in all dimensions

We consider wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere. It is known from the work of Bizoń and Biernat \cite{BizBie15} that in the energy-supercritical case, i.e., for $d \geq 3$, this model admits a closed-form corotational self-similar blowup solution. We show that this blowup profile is globally nonlinearly stable for all $d \geq 3$, thereby verifying a perturbative version of the conjecture posed in \cite{BizBie15} about the generic large data blowup behavior for this model. To accomplish this, we develop a novel stability analysis approach based on similarity variables posed on the whole space $\mathbb{R}^d$. As a result, we draw a general road map for studying spatially global stability of self-similar blowup profiles for nonlinear wave equations in the radial case for arbitrary dimension $d \geq 3$.

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arXiv2023

Global-in-space stability of singularity formation for Yang-Mills fields in higher dimensions

We continue our work \cite{Glo22a} on the analysis of spatially global stability of self-similar blowup profiles for semilinear wave equations in the radial case. In this paper we study the Yang-Mills equations in $(1+d)$-dimensional Minkowski space. For $d \geq 5$, which is the energy supercritical case, we consider an explicitly known equivariant self-similar blowup solution and establish its nonlinear global-in-space asymptotic stability under small equivariant perturbations. The size of the initial data is measured in terms of, in a certain sense, optimal Sobolev norm above scaling. This result complements the existing stability results in odd dimensions, while for even dimensions it is new.

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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.

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arXiv2023

On blowup for the supercritical quadratic wave equation

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for all $d \geq 7$. For $d=9$, we study the stability of $u^*$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$. In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d=7$ and $d=9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

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arXiv2022

Stable singularity formation for the Keller-Segel system in three dimensions

We consider the parabolic-elliptic Keller-Segel system in dimensions $d \geq 3$, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example has been found more than two decades ago by Brenner et al. \cite{BCKSV99}, and is conjectured to be nonlinearly radially stable. We prove this conjecture for $d=3$. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we crucially rely on a technique we recently developed in \cite{GloSch20}. To our knowledge, this provides the first result on stable self-similar blowup for the Keller-Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

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arXiv2022

Co-dimension one stable blowup for the quadratic wave equation beyond the light cone

We study the stability of an explicitly known, non-trivial self-similar blowup solution of the quadratic wave equation in the lowest energy supercritical dimension $d = 7$. This solution blows up at a single point and extends naturally away from the singularity. By using hyperboloidal similarity coordinates, we prove the conditional nonlinear asymptotic stability of this solution under small, compactly supported radial perturbations in a region of spacetime which can be made arbitrarily close to the Cauchy horizon of the singularity. To achieve this, we rigorously solve the underlying spectral problem and show that the solution has exactly one genuine instability. The unstable nature of the solution requires a careful construction of suitably adjusted initial data at $t = 0$, which, when propagated to a family of spacelike hypersurfaces of constant hyperboloidal time, takes the required form to guarantee convergence. By this, we introduce a new canonical method to investigate unstable self-similar solutions for nonlinear wave equations within the framework of hyperboloidal similarity coordinates.

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arXiv2022

Stable blowup for the supercritical hyperbolic Yang-Mills equations

We consider the Yang-Mills equations in $(1+d)$-dimensional Minkowski spacetime. It is known that in the supercritical case, i.e., for $d \geq 5$, these equations admit closed form equivariant self-similar blowup solutions \cite{BieBiz15}. These solutions are furthermore conjectured to be the universal attractors for generic large equivariant data evolutions. In this paper we partially prove this conjecture. Namely, we show that for all odd $d \geq 5$ the blowup mechanism exhibited by these solutions is stable.

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arXiv2021

Co-dimension one stable blowup for the supercritical cubic wave equation

For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution $u^*_T$, which is defined on the whole space and exists in all supercritical dimensions $d \geq 5$. For $d=7$, we analyze its stability properties without any symmetry assumptions and prove the existence of a set of perturbations which lead to blowup via $u^*_T$ in a backward light cone. Moreover, this set corresponds to a co-dimension one Lipschitz manifold modulo translation symmetries in similarity coordinates.

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arXiv2020

Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case

We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger for $d=5$ to higher dimensions. At the same time, we provide a general framework for proving stability of self-similar blowup solutions to a large class of semilinear heat equations in arbitrary space dimension $d \geq 3$, including a robust and simple method for solving the underlying spectral problems.

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arXiv2020

Threshold for blowup for the supercritical cubic wave equation

We consider the focusing cubic wave equation in the energy supercritical case, i.e., in dimensions $d \geq 5$. For this model an explicit nontrivial self-similar blowup solution was recently found by the first and third author in \cite{GlogicSchoerkhuber}. Furthermore, the solution is proven to be co-dimension one stable in $d=7$. In this paper, we study the equation from a numerical point of view. For $d=5$ and $d=7$ in the radial case, we provide evidence that this solution is at the threshold between generic ODE blowup and dispersion. In addition, we investigate the spectral problem that underlies the stability analysis and compute the spectrum in general supercritical dimensions.

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arXiv2019

Strichartz estimates for the one-dimensional wave equation

We study the hyperboloidal initial value problem for the one-dimensional wave equation perturbed by a smooth potential. We show that the evolution decomposes into a finite-dimensional spectral part and an infinite-dimensional radiation part. For the radiation part we prove a set of Strichartz estimates. As an application we study the long-time asymptotics of Yang-Mills fields on a wormhole spacetime.

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arXiv2017

On blowup of co-rotational wave maps in odd space dimensions

We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $d\geq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on a method developed by the second author and Schörkhuber, we prove the asymptotic nonlinear stability of the "ground-state" self-similar solution.

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