We consider the wave equation with an energy supercritical focusing nonlinearity in general odd dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space is global and scatters to a linear solution.

We initiate the study of the soliton resolution conjecture for the energy critical nonlinear wave equation in the nonradial case, by considering compactly supported data in dimensions N = 3, 4, 5. Our starting point is the sequential decomposition established in [6]. Under the assumption that all the solitons in the sequential decomposition are ground state traveling waves with distinct velocities, i.e. of the form +/- W & ell;i where W is the ground state, and W & ell;i is its Lorentz transform with velocity i pound, and where i= pound/j pound for i =/ j, we show a dichotomy. Either the soliton resolution holds for all times, or the resolution holds up to a collision on infinitely many time intervals. We then rule out this exotic scenario in the case where the number of solitons is smaller than or equal to two.

We study the initial value problem (IVP) associated to the semi-linear fractional Schr & ouml;dinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modeling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

This note concerns an exterior energy bound (channel of energy property) for the linear radial wave equations, which is crucial in the proof of the soliton resolution for the energy-critical wave equations. We give a short and synthetic proof that this property in general odd space dimension follows from the case of space dimension 3, which is elementary. This gives a simple proof of the channel of energy property in all odd space dimensions. We also show by the same proof that the analogous bound in even space dimensions follows from the case of space dimension 4.

Let u u be a harmonic function in a C 1 C^1 domain D ⊂ R d D\subset {\mathbb {R}}^d , which vanishes on an open subset of the boundary. In this note we study its critical set C ( u ) := { x ∈ D ¯ : ∇ u ( x ) = 0 } \mathcal {C}(u): = \{x \in \overline {D}: \nabla u(x) = 0 \} . When D D is a C 1 , α C^{1,\alpha } domain for some α ∈ ( 0 , 1 ] \alpha \in (0,1] , we give an upper bound on the ( d − 2 ) (d-2) -dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all C 1 C^1 -Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set S ( u ) := { x ∈ D ¯ : u ( x ) = 0 = | ∇ u ( x ) | } \mathcal {S}(u): = \{x \in \overline {D}: u(x) = 0 = |\nabla u(x)| \} (see [Arch. Ration. Mech. Anal. 245 (2022), pp. 1–88] and [Adv. Nonlinear Stud. 23 (2023)]).

We survey recent results on soliton resolution for the energy critical nonlinear wave equation.

We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported on Sobolev spaces with increasing regularity. The fact that we work with complex valued functions makes the analysis of the invariance much harder compared to the real valued case, that can be handled instead following the ideas used by Zhidkov [73].

We consider the quadratic semilinear wave equation in six dimensions. This energy critical problem admits a ground state solution, which is the unique (up to scaling) positive stationary solution. We prove that any spherically symmetric solution, that remains bounded in the energy norm, evolves asymptotically to a sum of decoupled modulated ground states, plus a radiation term. As a by-product of the approach we prove the non-existence of multisoliton solutions that do not emit any radiation. The proof follows the method initiated for large odd dimensions by the last three authors, reducing the problem to ruling out the existence of such non-radiative multisolitons, by deriving a contradiction from a finite dimensional system of ordinary differential equations governing their modulation parameters. In comparison, the difficulty in six dimensions is the failure of certain channel of energy estimates and the related existence of a linear resonance. We use the obtention of new channel of energy estimates, from our previous article (Int. Math. Res. Not., 2022), as well as the classification of non-radiative solutions with small energy, from our work (arXiv:2211.16085, 2022).

Channels of energy estimates control the energy of an initial data from that which it radiates outside a light cone. For the linearised energy critical wave equation they have been obtained in the radial case in odd dimensions, first in $3$ dimensions by Duyckaerts, Kenig and Merle (Camb. J. Math., 2013), then for general odd dimensions by the same authors (Comm. Math. Phys., 2020). We consider even dimensions, for which such estimates are known to fail (C\^ote, Kenig and Schlag, Math. Ann., 2014). We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us to prove the soliton resolution conjecture in six dimensions (Collot, Duyckaerts, Kenig and Merle, arXiv preprint 2201.01848, 2022 versions 1 and 2).

Consider the energy-critical focusing wave equation in odd space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. The proof essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in our previous work, Cambridge Journal of Mathematics 2013, arXiv:1204.0031) by reducing the study of a multisoliton solution to a finite dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multisolitons.

In this short note, we discuss a boundary value problem for a matrix valued partial derivative equation.

Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures.

Non-radiative solutions of energy critical wave equations are such that their energy in an exterior region $|x|>R+|t|$ vanishes asymptotically in both time directions. This notion, introduced by Duyckaerts, Kenig and Merle (J. Eur. Math. Soc., 2011), has been key in solving the soliton resolution conjecture for these equations in the radial case. In the present paper, we first classify their asymptotic behaviour at infinity, showing that they correspond to a $k$-parameters family of solutions where $k$ depends on the dimension. This generalises the previous results (Duyckaerts, Kenig and Merle, Camb. J. Math., 2013 and Duyckaerts, Kenig, Martel and Merle, Comm. Math. Phys., 2022) in three and four dimensions. We then establish a unique maximal extension of these solutions.

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form Delta u + W . del u = 0 in R-2 , where W = W-1 + iW(2) with each W-j being real-valued. Under the assumptions that W-j is an element of L-qj for some q(1) is an element of [2,infinity] , q2 is an element of (2,infinity] and that W-2 exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.

Let u be a harmonic function in a C^1 -Dini domain D⊂ℝ^d such that u vanishes on a boundary surface ball ∂ D∩ B_5R(0) . We consider an effective version of its singular set (up to boundary) 𝒮(u):={X∈D: u(X) = |∇ u(X)| = 0} and give an estimate of its (d-2) -dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of u centered at 0. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space for a wave-type equation. We also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.