arXiv subjects

Jacek Jendrej

Publications and source records attributed to Jacek Jendrej.

At least 19 recordsLinked to original sources

arXiv2026

Construction of two-bubble solutions for the energy-critical NLS in dimension 6

We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N = 6$. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $e^{-|t|}$. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension $N \geq 7$. The six-dimension case presents specific difficulties, as the ground state does not belong to $\dot H^{-1}$. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.

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arXiv2026

Nonexistence of blow-up solutions with smooth radiation for energy-critical equivariant wave maps

We study $k$-equivariant energy critical wave maps $\mathbb{R}^{1+2} \to \mathbb{S}^2$, for any equivariance degree $k\ge 2$. We prove that the radiation associated with any finite-energy blow-up solution cannot satisfy a certain regularity condition; in particular, it cannot be smooth. The assumption $k \geq 2$ is necessary, since for $k = 1$ such solutions are known to exist. The starting point of our analysis is the soliton resolution theorem. The key ingredient is a novel application of the modulation method, in which we compare the effects of the radiation and inner bubbles to study the dynamic behavior of the widest bubble.

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arXiv2026

Construction of multi-bubble solutions for the energy-critical wave equation in dimension four

For any $N\geq 2$, we construct a global solution of the energy-critical focusing wave equation in dimension four which blows up in infinite time at $N$ prescribed points $z_1,\ldots,z_N\in \mathbb R^4$, provided that the points form one orbit under a finite group of orthogonal symmetries. We denote by $c:=2\sum_{j\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \[ \log\frac{1}{λ(t)} = \left(\frac{9c}{4}\right)^{1/3}t^{2/3}+O(t^{1/3}) \qquad \text{as } t\to+\infty . \] This concentration rate comes from a genuinely four-dimensional effect: the borderline decay of the ground state makes the interaction between different bubbles enter the leading order parameter dynamics.

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arXiv2026

Rigidity of the multi-bubble solutions to the energy critical wave equation in dimension five

We study the asymptotic dynamics of multi-bubble solutions to the focusing energy-critical wave equation in five dimensions. Assuming that the solution asymptotically decomposes into a finite superposition of spatially separated bubbles with comparable scales, we prove a rigidity result that describes the precise long-time behavior of these scales. More precisely, we show that all scaling parameters are necessarily of order $t^{-2}$, and that the corresponding renormalized modulation vector converges to a connected component of a finite-dimensional algebraic set determined by the limiting spatial configuration of the bubbles. This algebraic system encodes the strong interactions between the polynomial tails of the bubbles and governs the effective asymptotic dynamics of the multi-bubble regime.

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arXiv2026

High-order long-time asymptotics for small solutions to the one-dimensional nonlinear Schrödinger equation

We investigate the global well-posedness and modified scattering for the one-dimensional Schrödinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we establish global existence of solution together with persistence of the localization of the associated profile. We further provide a rigorous derivation of the asymptotic expansion at arbitrary order of such solutions, taking into account long-range effects induced by the cubic component of the nonlinearity. Our analysis relies on the space-time resonance method.

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arXiv2026

Construction of two-bubble solutions for the energy-critical Hartree equation

We construct a pure two-bubble solution for the focusing, energy-critical Hartree equation in space dimension $N \geq 7$. The constructed solution is spherically symmetric, global in (at least) the negative time direction and asymptotically behaves as a superposition of two ground states (or bubbles) both centered at the origin, with the ratio of their length scales converging to $0$ and the phases of the two bubbles form the right angle. The main arguments are the modulation analysis, the bootstrap argument and the topological argument. The main novelty with respect to existing constructions of pure two-bubble solutions is the nonlocal interaction, which is more complex to analyze.

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arXiv2026

Scalar behavior for a complex multi-soliton arising in blow-up for a semilinear wave equation

This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution behaves exactly as in the real-valued case. Namely, up to a rotation in the complex plane, the solution decomposes into a sum of a finite number of decoupled solitons with alternate signs. The main novelty of our proof is a resolution of a complex-valued first order Toda system governing the evolution of the positions and the phases of the solitons.

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arXiv2025

Rate of convergence to equilibrium for the heated string

In the present manuscript, we calculate the exponential rate of convergence of the heated string system (a mixed-type hyperbolic-parabolic system of PDEs) towards the equilibrium, independently of the initial data. As a by-product of our analysis, we obtain an enhanced time decay of the solution. The main tool of our reasoning consists of asymptotic analysis at the Fourier side of the linearized problem in the spirit of Kato. The latter method fits very well to the estimates obtained earlier for the system. The matching between the linear problem and the nonlinear original system requires delicate estimates of the nonlinear terms at the Fourier side as well as careful spectral analys

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arXiv2025

Concentric bubbles concentrating in finite time for the energy critical wave maps equation

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0, t_0]\times \mathbb{R}^2$, $t_0>0$, and concentrating two concentric bubble profiles at the frequency scales $λ_1(t) = e^{α(t)},\,α(t)\sim \big|\log t\big|^{β+1}$, as well as $λ_2(t) = t^{-1}\cdot \big|\log t\big|^β$. The parameter $β>\frac32$ can be chosen arbitrarily. This shows that soliton resolution scenarios with finite time blow up and $N = 2$ collapsing profiles, i. e. bubble trees, do occur for this equation.

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arXiv2025

Multisoliton solutions for equivariant wave maps on a $2+1$ dimensional wormhole

We study equivariant wave maps from the $2+1$ dimensional wormhole to the 2-sphere. This model has explicit harmonic map solutions which, in suitable coordinates, have the form of the sine-Gordon kinks/anti-kinks. We conjecture that there exist asymptotically static chains of $N\geq 2$ alternating kinks and anti-kinks whose subsequent rates of expansion increase in geometric progression as $t\rightarrow \infty$. Our argument employs the method of collective coordinates to derive effective finite-dimensional ODE models for the asymptotic dynamics of $N$-chains. For $N=2,3$ the predictions of these effective models are verified by direct PDE computations which demonstrate that the $N$-chains lie at the threshold of kink-anti-kink annihilation.

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arXiv2024

Classification of kink clusters for scalar fields in dimension 1+1

We consider a real scalar field equation in dimension 1+1 with an even, positive self-interaction potential having two non-degenerate zeros (vacua) 1 and -1. Such a model admits non-trivial static solutions called kinks and antikinks. We define a kink n-cluster to be a solution approaching, for large positive times, a superposition of n alternating kinks and antikinks whose velocities converge to $0$. They can be equivalently characterized as the solutions of minimal possible energy containing n transitions between the vacua, or as the solutions whose kinetic energy decays to 0 in large time. Our first main result is a determination of the main-order asymptotic behavior of any kink n-cluster. The proof relies on a reduction,using appropriately chosen modulation parameters, to an n-body problem with attractive exponential interactions. We then construct a kink n-cluster for any prescribed initial positions of the kinks and antikinks, provided that their mutual distances are sufficiently large. Next, we prove that the set of all the kink n-clusters is an n-dimensional topological manifold, and we show how it can be parametrized by the positions of the kinks in the configuration. The proof relies on energy estimates and the contraction mapping principle, using the Lyapunov-Schmidt reduction technique. Finally, we show that kink clusters are universal profiles for the formation/collapse of multikink configurations. In this sense, they can be interpreted as forming the stable/unstable manifold of the multikink state given by a superposition of n infinitely separated alternating kinks and antikinks.

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arXiv2024

Wave maps in dimension $1+1$ with an external forcing

This paper aims to establish the local and global well-posedness theory in $L^1$, inspired by the approach of Keel and Tao [Internat. Math. Res. Notices, 1998], for the forced wave map equation in the ``external'' formalism. In this context, the target manifold is treated as a submanifold of a Euclidean space. As a corollary, we reprove Zhou's [Math. Z., 1999] uniqueness result, leading to the uniqueness of weak solutions with locally finite energy. Additionally, we achieve the scattering of such solutions through a conformal compactification argument.

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arXiv2024

Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations

We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t, x) = Q_v(x - vt - x_0)$, for $v$ in some range $(0, v_*)$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to\infty}\|u(t) - (Q_v(\cdot - x_1(t)) \pm Q_v(\cdot - x_2(t)))\|_{H^1} = 0, \qquad \text{with}\quad\lim_{t \to \infty}x_2(t) - x_1(t) = \infty. \end{equation} Existence of such solutions is known for $f(u) = |u|^{p-1}u$ with $p \in \mathbb{Z} \setminus \{5\}$ and $p > 2$. We describe the~dynamical behavior of any solution satisfying \eqref{eq:abstract} under the assumption that $Q_v$ is linearly unstable (which corresponds to $p > 5$ for power nonlinearities). We prove that in this case the sign in \eqref{eq:abstract} is necessarily "$+$", which corresponds to an attractive interaction. We also prove that the~distance $x_2(t) - x_1(t)$ between the solitons equals $\frac{2}{\sqrt v}\log(κt) + o(1)$ for some $κ= κ(v) > 0$.

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arXiv2023

Continuous in time bubble decomposition for the harmonic map heat flow

We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.

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arXiv2023

Dynamics of kink clusters for scalar fields in dimension 1+1

We consider a real scalar field equation in dimension 1+1 with an even positive self-interaction potential having two non-degenerate zeros (vacua) 1 and -1. It is known that such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0. They can be equivalently characterised as the solutions of minimal possible energy containing a given number of transitions between the vacua, or as the solutions whose kinetic energy decays to 0 for large time. Our main result is a determination of the main-order asymptotic behaviour of any kink cluster. Moreover, we construct a kink cluster for any prescribed initial positions of the kinks and antikinks, provided that their mutual distances are sufficiently large. Finally, we show that kink clusters are universal profiles for the formation/collapse of multi-kink configurations. The proofs rely on a reduction, using appropriately chosen modulation parameters, to an n-body problem with attractive exponential interactions.

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arXiv2023

Asymptotic stability and classification of multi-solitons for Klein-Gordon equations

Focusing on multi-solitons for the Klein-Gordon equations, in first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to multi-solitons in the energy space as $t\rightarrow\infty$. Using Strichartz estimates developed in our earlier work \cite{CJ2} and the modulation techniques, we show that if a solution stays close to the multi-soliton family, then it scatters to the multi-soliton family in the sense that the solution will converge in large time to a superposition of Lorentz-transformed solitons (with slightly modified velocities), and a radiation term which is at main order a free wave. Moreover, we construct a finite-codimension centre-stable manifold around the well-separated multi-soliton family. Finally, given different Lorentz parameters and arbitrary centers, we show that all the corresponding pure multi-solitons form a finite-dimension manifold.

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arXiv2023

Strichartz estimates for Klein-Gordon equations with moving potentials

We study linear Klein-Gordon equations with moving potentials motivated by the stability analysis of traveling waves and multi-solitons. In this paper, Strichartz estimates, local energy decay and the scattering theory for these models are established. The results and estimates obtained in this paper will be used to study the interaction of solitons and the stability of multisolitons of nonlinear Klein-Gordon equations.

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arXiv2022

Bubble decomposition for the harmonic map heat flow in the equivariant case

We consider the harmonic map heat flow for maps from the plane taking values in the sphere, under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times -- the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval motivated by the authors' recent work on the soliton resolution problem for equivariant wave maps.

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