Search

Global behavior of solutions is studied for the nonlinear Klein-Gordon equation with a focusing power nonlinearity and a damping term in the energy space on the Euclidean space. We give a complete classification of solutions into 5 types of global behavior for all initial data in a small neighborhood of each superposition of two ground states (2-solitons) with the opposite signs and sufficient spatial distance. The neighborhood contains, for each sign of the ground state, the manifold with codimension one in the energy space, consisting of solutions that converge to the ground state at time infinity. The two manifolds are joined at their boundary by the manifold with codimension two of solutions that are asymptotic to 2-solitons moving away from each other. The connected union of these three manifolds separates the rest of the neighborhood into the open set of global decaying solutions and that of blow-up. The main ingredient in the proof is a difference estimate on two solutions starting near 2-solitons and asymptotic to 1-solitons. The main difficulty is in controlling the direction of the two unstable modes attached to 2-solitons, while the soliton interactions are not uniformly integrable in time. It is resolved by showing that the non-scalar part of the interaction between the unstable modes is uniformly integrable due to the symmetry of the equation and the 2-solitons., Comment: 62 pages

The Cauchy problem for the Zakharov system in the energy-critical dimension $d=4$ is considered. We prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a Strichartz estimate for the Schr\"odinger equation with a potential. More precisely, a Strichartz estimate is proved to hold uniformly for any potential solving the free wave equation with mass below the ground state constraint. The key new ingredient is a bilinear (adjoint) Fourier restriction estimate for solutions of the inhomogeneous Schr\"odinger equation with forcing in dual endpoint Strichartz spaces., Comment: Partly, the notation in Section 2 is adopted from arXiv:1912.05820. v2: references fixed, further minor corrections. v3: Minor change of Theorem 4.1 and its proof, which includes low frequeny cases now. v4: fixed references to [7]

The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence on the initial data. In addition, under a condition on the data for the Schr\"odinger equation at the lowest admissible regularity, global well-posedness and scattering is proved. The results cover energy-critical and energy-supercritical dimensions $d \geqslant 4$., Comment: v2: typos fixed, new Section 5 on a simplified approach to LWP and small data GWP and scattering results in non-endpoint cases. v3: minor corrections in the proofs of Thm 7.6 and 7.7. v4: slight improvement of Prop. 6.1 and 6.2 to fix the proof of the uniqueness claim

We consider nonlinear Schr\"odinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities., Comment: 12 pages

We construct a singular solution of a stationary nonlinear Schr\"{o}dinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.

We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Ap\'ery's constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$., Comment: 41 pages

Observational study of galactic magnetic fields is hampered by the fact that the observables only probe various projections of the magnetic fields. Comparison with numerical simulations is helpful to understand the real structures, and observational visualization of numerical data is an important task. In this paper, we investigate 8~GHz radio synchrotron emission from spiral galaxies, using the data of global three-dimensional magneto-hydrodynamic simulations. We assume a frequency independent depolarization in our observational visualization. We find that the appearance of the global magnetic field depends on the viewing angle: a face-on view seemingly has hybrid magnetic field types combining axisymmetric modes with higer order modes; at a viewing angle of $\sim 70\degr$, the galaxy seems to contain a ring-like magnetic field structure; while in edge-on view, only field structure parallel to the disk can be seen. The magnetic vector seen at 8~GHz traces the global magnetic field inside the disk. These results indicate that the topology of global magnetic field obtained from the relation between azimuthal angle and Faraday depth strongly depends on the viewing anglue of the galaxy. As one of the examples, we compare our results at a viewing angle of $25\degr$ with the results of IC342. The relation between azimuthal angle and Faraday depth of the numerical result shows a tendency similar to IC342, such as the peak numbers of the Faraday depth., Comment: 10 pages, 7 figures accepted in MNRAS

We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than the ground state, which is written using the Aubin-Talenti function. The dichotomy is characterized by the critical mass of the wave component of the ground state. The result is similar to that by Kenig and Merle for the energy-critical nonlinear Schrodinger equation (NLS). Unlike NLS, however, the most difficult interaction in the proof stems from the free wave component. In order to control it, the main novel ingredient we develop in this paper is a uniform global Strichartz estimate for the linear Schrodinger equation with a potential of subcritical mass solving a wave equation. This estimate, as well as the proof, may be of independent interest. For the scattering proof, we follow the idea by Dodson and Murphy., Comment: 61 pages

We consider the final-data problem for systems of nonlinear Schr\"odinger equations with $L^2$ subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in $L^2(\mathbb{R}^d)$, extending the result of J. Murphy to powers equal to or lower than the Strauss exponent. In particular, systems with quadratic nonlinearity can be treated in three space dimensions, and by the same argument, the Gross-Pitaevskii equation in the energy space. The extension is by use of the Strichartz estimate with a time weight., Comment: 18 pages

We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schr\"odinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable L\'evy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.

We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by G\'erard \cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering., Comment: 28 pages; Correct some mistakes, the main results remain the same

Consider the focusing nonlinear Schr\"odinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into 9 sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time., Comment: 50 pages

Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with the paper by Schlag and the first author on the subcritical equation, the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference from the subcritical case is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region., Comment: 37 pages

This article resolves some errors in the paper "Scattering threshold for the focusing nonlinear Klein-Gordon equation", Analysis & PDE 4 (2011) no. 3, 405-460. The errors are in the energy-critical cases in two and higher dimensions., Comment: 11 pages

Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints., Comment: 43 pages. A gap has been fixed in Section 8 in the case of two non-scattering profiles, with Lemma 7.1 modified and a more general statement (Theorem 7.2)

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results is restricted to radially symmetric data., Comment: 25 pages, 2 figures

We construct the 'threshold manifold' near the soliton for the mass critical gKdV equation, completing results obtained in arXiv:1204.4625 and arXiv:1204.4624. In a neighborhood of the soliton, this C1 manifold of codimension one separates solutions blowing up in finite time and solutions in the 'exit regime'. On the manifold, solutions are global in time and converge locally to a soliton. In particular, the soliton behavior is strongly unstable by blowup.

The aim of this article is to prove that for the 2+1-dimensional equivariant Faddeev model, which is a quasilinear generalization of the corresponding nonlinear sigma model, small initial data in critical Besov spaces evolve into global solutions which scatter., Comment: 16 pages

We obtain scattering for the 3D Zakharov system with non-radial small data in the energy space with angular regularity of degree one. The main ingredient is a generalized Strichartz estimate for the Schr\"odinger equation in the space of $L^2$ angular integrability., Comment: 20 pages

We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1206.2457

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper. We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states., Comment: 42 pages

In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

For the critical focusing wave equation $\Box u = u^5$ on $\R^{3+1}$ in the radial case, we establish the role of the "center stable" manifold $\Sigma$ constructed in \cite{KS} near the ground state $(W,0)$ as a threshold between type I blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizo\'n, Chmaj, Tabor. The underlying topology is stronger than the energy norm.

We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with radial symmetry. The idea of proof is the same as the Zakharov system studied in \cite{GN}, namely to combine the normal form reduction and the radial-improved Strichartz estimates

We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup., Comment: 28 pages

We prove small energy scattering for the 3D Zakharov system with radial symmetry. The main ingredients are normal form reduction and the radial-improved Strichartz estimates., Comment: 11 pages, corrected minor errors, published online in IMRN. arXiv admin note: text overlap with arXiv:1208.1657

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in the natural energy space with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as time approaches infinity. In our previous paper arxiv:1010.3799 we treated the radial case, and this paper provides the natural nonradial extension of these results. However, the present paper is self-contained and in fact develops a somewhat different formalism in order to handle the more complex nonradial situation., Comment: arXiv admin note: substantial text overlap with arXiv:1010.3799. This is the final version as it appears in Discrete and Continuous Dynamical Systems Volume 33, Issue 6, 2013 Pages 2423-2450. Some minor misprints have been corrected

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^\infty$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrodinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality., Comment: 14 pages

We study global dynamics for the focusing nonlinear Klein-Gordon equation with the energy-critical nonlinearity in two or higher dimensions when the energy equals the threshold given by the ground state of a mass-shifted equation, and prove that the solutions are divided into scattering and blowup. In short, the Kenig-Merle scattering/blowup dichotomy extends to the threshold energy in the case of mass-shift for the critical nonlinear Klein-Gordon equation., Comment: 16 pages

We study equivariant maps corresponding to the classical Skyrme model and the Adkins-Nappi model, for which we prove global existence and scattering in critical Sobolev-Besov spaces., Comment: 12 pages

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing energy sub-critical nonlinearity, associated with a family of solitary waves which is generated from any radial stationary solution by the action of all Lorentz transforms and spatial translations. The construction is based on the graph transform (or Hadamard) approach, which requires less spectral information on the linearized operator, and less decay of the nonlinearity, than the Lyapunov-Perron method employed previously in this context. The only assumption on the stationary solution is that the kernel of the linearized operator is spanned by its spatial derivatives, which is known to hold for the ground states. The main novelty of this paper lies with the fact that the graph transform method is carried out in the presence of modulation parameters corresponding to the symmetries., Comment: 38 pages

In this paper we obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case arXiv:1005.4894, the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein-Gordon equation, similar to those of Mizumachi arXiv:math/0605031. The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.

We extend our previous result on the focusing cubic Klein-Gordon equation in three dimensions to the non-radial case, giving a complete classification of global dynamics of all solutions with energy at most slightly above that of the ground state., Comment: 40 pages

We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground states and gets away from them in the energy space, we can classify its behavior into four cases, according to whether it blows up in finite time or scatters to zero, in forward or backward time direction. We prove that initial data for each case constitute a non-empty open set in the energy space. This is an extension of the recent results by the latter two authors on the subcritical nonlinear Klein-Gordon and Schr\"odinger equations, except for the part of the center manifolds. The key step is to prove the "one-pass" theorem, which states that the transition from the scattering region to the blow-up region can take place at most once along each trajectory. The main new ingredients are the control of the scaling parameter and the blow-up characterization by Duyckaerts, Kenig and Merle., Comment: 26 pages

Consider a nonlinear Schr\"odinger equation in $R^3$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity. An essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states, around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space.

We extend our previous result on the nonlinear Klein-Gordon equation to the nonlinear Schrodinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which contains the ground states and separates the phase space locally into two connected regions exhibiting blow-up and scattering to 0, respectively. The special solutions found by Duyckaerts and Roudenko appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with the Klein-Gordon case, the proof combines the hyperbolic dynamics near the ground states with the variational structure away from them. The main technical ingredient in the proof is a "one-pass" theorem which precludes "almost homoclinic orbits", i.e., those solutions starting in, then moving away from, and finally returning to, a small neighborhood of the ground states. The main new difficulty compared with the Klein-Gordon case is the lack of finite propagation speed. We need the radial Sobolev inequality for the error estimate in the virial argument. Another major difference from the Klein-Gordon case is the need to control two modulation parameters., Comment: 47 pages. Simplified the virial argument in the scattering region (Sect. 4.2)

We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with large radial data in the energy space. This equation admits a unique positive stationary solution, called the ground state. In 1975, Payne and Sattinger showed that solutions with energy strictly below that of the ground state are divided into two classes, depending on a suitable functional: If it is negative, then one has finite time blowup, if it is nonnegative, global existence; moreover, these sets are invariant under the flow. Recently, Ibrahim, Masmoudi and the first author improved this result by establishing scattering to zero in the global existence case by means of a variant of the Kenig-Merle method. In this paper we go slightly beyond the ground state energy and give a complete description of the evolution. For example, in a small neighborhood of the ground states one encounters the following trichotomy: on one side of a center-stable manifold one has finite-time blowup, on the other side scattering to zero, and on the manifold itself one has scattering to the ground state, all for positive time. In total, the class of initial data is divided into nine disjoint nonempty sets, each displaying different asymptotic behavior, which includes solutions blowing up in one time direction and scattering to zero on the other, and also, the analogue of those found by Duyckaerts and Merle for the energy critical wave and Schr\"odinger equations, exactly with the ground state energy. The main technical ingredient is a "one-pass" theorem which excludes the existence of "almost homoclinic" orbits between the ground states., Comment: 34 pages, minor corrections

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for the $H^1$ critical wave and Schr\"odinger equations. Our result includes the $H^1$ critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger-Moser inequality. The main difficulty is lack of scaling invariance both in the linear and nonlinear terms., Comment: 56 pages, no figure, minor corrections, to appear in Analysis & PDE

We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in the whole space or in the exterior of a star shaped obstacle.

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation., Comment: 33 pages, submitted

Nonlinear Schr\"odinger (NLS) equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically., Comment: submitted December 2005, revised November 2006