We prove explicit doubling inequalities and obtain uniform upper bounds (under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubli
ng inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain analyticity, and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,,u_2$ are two suitable solutions of the equation defined in $mathbb R^ntimes[0,T]$ such that for some
non-empty open set $Omegasubset mathbb R^ntimes[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) in Omega$, then $u_1(x,t)=u_2(x,t)$ for any $(x,t)inmathbb R^ntimes[0,T]$. The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann in cite{GhSaUh}, and some extensions obtained here.
Consider the energy-critical focusing wave equation in odd space dimension $Ngeq 3$. 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 ener
gy 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.
By definition, the exterior asymptotic energy of a solution to a wave equation on $mathbb{R}^{1+N}$ is the sum of the limits as $tto pminfty$ of the energy in the the exterior ${|x|>|t|}$ of the wave cone. In our previous work (JEMS 2012, arXiv:1003.
0625), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension $N$ is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential begin{equation} label{abstractLW} tag{*} partial_t^2u+L_Wu=0,quad L_W:=-Delta -frac{N+2}{N-2}W^{frac{4}{N-2}} end{equation} obtained by linearizing the energy critical wave equation at the ground-state solution $W$, still in odd space dimension. This equation admits nonzero solutions of the form $A+tB$, where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of eqref{abstractLW} is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of $W$, and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions $3$ and $5$.
Consider the energy-critical focusing wave equation in space dimension $Ngeq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. It is conjectured (soliton resolution) 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. A nonradiative solution of the equation is by definition a solution whose energy in the exterior ${|x|>|t|}$ of the wave cone vanishes asymptotically as $tto +infty$ and $tto -infty$. In a previous work (Cambridge Journal of Mathematics 2013, arXiv:1204.0031), we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, $0$ and $pm W$. This was crucial in the proof of soliton resolution in 3 space dimension. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behaviour as $rto infty$. We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag and the second author (Advances in Mathematics, 2015, arXiv:1409.3643) . We also study the propagation of the support of nonzero radial solutions with compactly supported initial data, and prove that these solutions cannot be nonradiative.
We prove that if $u_1,,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)inR times [0,T]$ which agree in an open set $Omegasubset R times [0,T]$, then $u_1equiv u_2$. We extend this uniqueness result to a general class of equations of
Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In cite{IsazaLinaresPonce} it was established that if the initial datun $u_0in H^l((b,infty))$ for some $linmathbb Z^+$ and $bin
mathbb R$, then the corresponding solution $u(cdot,t)$ belongs to $H^l((beta,infty))$ for any $beta in mathbb R$ and any $tin (0,T)$. Our goal here is to extend this result to the case where $,lin mathbb R^+$.
Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded o
n the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W, then the decomposition holds without a restriction to a subsequence.
Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. In a previous paper, we proved that any solution which is bounded in the energy space converges, along a sequence of times and in some weak sense, to a solution with th
e compactness property, that is a solution whose trajectory stays in a compact subset of the energy space up to space translation and scaling. It is conjectured that the only solutions with the compactness property are stationary solutions and solitary waves that are Lorentz transforms of the former. In this note we prove this conjecture with an additional non-degeneracy assumption related to the invariances of the elliptic equation satisfied by stationary solutions. The proof uses a standard monotonicity formula, modulation theory, and a new channel of energy argument which is independent of the space dimension.
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a s
um of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrodinger equation.
