Nonlinear scattering theory at low energy

From MaRDI portal

Publication:1155228

Jump to:navigation, search

DOI10.1016/0022-1236(81)90063-XzbMath0466.47006MaRDI QIDQ1155228

Walter A. Strauss

Publication date: 1981

Published in: Journal of Functional Analysis (Search for Journal in Brave)

zbMATH Keywords

scattering operator semilinear Schrödinger and Klein-Gordon equations

Mathematics Subject Classification ID

Semigroups of nonlinear operators (47H20) Scattering theory for PDEs (35P25) (2)-body potential quantum scattering theory (81U05) Scattering theory of linear operators (47A40)

Related Items

On global solutions to the initial–boundary value problem for the nonlinear Schrödinger equations in exterior domains ⋮ Regularity of solutions of critical and subcritical nonlinear wave equations ⋮ Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities ⋮ OptimalL^∞decay for solutions to the wave equation with a potential ⋮ On the asymptotic behavior of nonlinear Schrödinger equations with magnetic effect ⋮ Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions ⋮ Lifespan estimates of solutions to semilinear wave equations with damping term on the exterior domain ⋮ Existence and nonexistence criteria for a system of biharmonic wave inequalities in an exterior domain of ℝN ⋮ A blow-up result for a nonlinear wave equation on manifolds: the critical case ⋮ A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations ⋮ Blowup for a damped wave equation with mass and general nonlinear memory ⋮ Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture ⋮ Global existence and scattering for nonlinear Schrödinger equations with time-dependent damping ⋮ Critical exponent of Fujita type for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime ⋮ The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping ⋮ On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space ⋮ The rescaling method for some quasilinear wave equations with the divergence forms of the nonlinearity ⋮ Attractors of nonlinear Hamiltonian partial differential equations ⋮ Nonexistence of global solutions for time fractional wave equations in an exterior domain ⋮ Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities ⋮ Global well-posedness for the radial, defocusing, nonlinear wave equation for 3 < p < 5 ⋮ A general blow-up result for a degenerate hyperbolic inequality in an exterior domain ⋮ Asymptotic behavior of solutions of a generalized Boussinesq type equation ⋮ Global well-posedness and self-similarity for semilinear wave equations in a time-weighted framework of Besov type ⋮ Convergence dans $L^p (R^{n+1})$ de la solution de l'équation de Klein-Gordon vers celle de l'équation des ondes ⋮ H m -modified wave operator for nonlinear Hartree equation in space of dimensionn≥2 ⋮ Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds ⋮ L∞C(ℝⁿ)-decay of classical solutions for nonlinear Schrödinger equations ⋮ NUMERICAL ASPECTS OF NONLINEAR SCHRÖDINGER EQUATIONS IN THE PRESENCE OF CAUSTICS ⋮ THE GLOBAL CAUCHY PROBLEM FOR THE NLS WITH HIGHER ORDER ANISOTROPIC DISPERSION ⋮ Continuous dependence of solutions to the strongly damped nonlinear Klein{Gordon equation ⋮ Unnamed Item ⋮ Life-Span of Solutions to Critical Semilinear Wave Equations ⋮ Inverse scattering for the nonlinear schrodinger equation ⋮ Weak concentration and wave operator for a 3D coupled nonlinear Schrödinger system ⋮ On 2D nonlinear Schrödinger equations with data on \(\mathbb{R} \times \mathbb{T}\) ⋮ Critical exponent for a nonlinear wave equation with damping ⋮ The rescaling method for some critical quasilinear wave equations with the divergence form of the nonlinearity ⋮ Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space ⋮ Unnamed Item ⋮ Weighted decay estimates for the wave equation ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Modified scattering for the mixed initial-boundary problem for the nonlinear Klein–Gordon equation ⋮ Unnamed Item ⋮ Global Existence for then-Dimensional Semilinear Tricomi-Type Equations ⋮ Global Existence of Solutions to Nonlinear Wave Equations ⋮ Unnamed Item ⋮ CONCENTRATION PHENOMENON FOR THE L² CRITICAL AND SUPER CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS IN ENERGY SPACES ⋮ SCATTERING OPERATOR FOR NONLINEAR KLEIN–GORDON EQUATIONS ⋮ Bessel (Riesz) potentials on banach function spaces and their applications I theory ⋮ On Stability of Small Solitons of the 1-D NLS with a Trapping Delta Potential ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Dispersive estimates for nonlinear Schrödinger equations with external potentials ⋮ Isometric decomposition operators, function spaces \(E_{p,q}^{\lambda}\) and applications to nonlinear evolution equations ⋮ Long time behavior for the equation of finite-depth fluids ⋮ Global existence for semilinear wave equations with the critical blow-up term in high dimensions ⋮ Conformal conservation law, time decay and scattering for nonlinear wave equations ⋮ Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data ⋮ New blow-up results for nonlinear boundary value problems in exterior domains ⋮ Scattering for semilinear wave equations with small data in three space dimensions ⋮ A scale of critical exponents for semilinear waves with time-dependent damping and mass terms ⋮ Analyticity of the scattering operator for semilinear dispersive equations ⋮ Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions ⋮ Numerical simulation of blow-up of a 2D generalized Ginzburg-Landau equation ⋮ Attractors of Hamilton nonlinear PDEs ⋮ From the Gross-Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data ⋮ Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations ⋮ Decay and scattering of small solutions of a generalized Boussinesq equation ⋮ The global Cauchy problem for the NLS and NLKG with small rough data ⋮ Neumann problem for the nonlinear Klein-Gordon equation ⋮ The scattering theory for the nonlinear wave equation with small data. II ⋮ Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions ⋮ Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions ⋮ Nonlinear Schrödinger equations with exceptional potentials ⋮ Decay estimates for fourth order wave equations ⋮ Blow-up of solutions to semilinear wave equations with variable coefficients and boundary ⋮ Modified scattering for the critical nonlinear Schrödinger equation ⋮ Concerning the Strauss conjecture for the subcritical and critical cases on the exterior domain in two space dimensions ⋮ On a nonlinear scattering model ⋮ Almost global solutions of semilinear wave equations with the critical exponent in high dimensions ⋮ Cubic nonlinear Dirac equation in a quarter plane ⋮ The Cauchy problem for nonlinear Schrödinger equations in modulation spaces ⋮ Asymptotic completeness of scattering in the nonlinear Lamb system for nonzero mass ⋮ On the new critical exponent for the nonlinear Schrödinger equations ⋮ Global existence for a coupled wave system related to the Strauss conjecture ⋮ Blow up of solutions to nonlinear wave equation in \(2D\) exterior domains ⋮ Global solutions for 2D quadratic Schrödinger equations ⋮ Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation ⋮ Small data global existence for the semilinear wave equation with space-time dependent damping ⋮ Global existence for solutions of \(\square u=A| u| ^ p\) ⋮ Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential ⋮ Global nonexistence results for a class of hyperbolic systems ⋮ Regularity of solutions to nonlinear dispersive equations ⋮ Remark on the global existence and large time asymptotics of solutions for the quadratic NLS ⋮ A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results ⋮ Nonlinear scattering theory at low energy: Sequel ⋮ Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2 ⋮ Klein-Gordon equations on modulation spaces ⋮ \(L^p\)-\(L^{\acute p}\) estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential ⋮ Critical blowup for systems of semilinear wave equations in low space dimensions ⋮ On asymptotic behavior of solutions to Korteweg-de Vries type equations related to vortex filament with axial flow ⋮ Long range scattering for nonlinear Schrödinger equations in one space dimension ⋮ Blow-up of solutions to critical semilinear wave equations with variable coefficients ⋮ Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations ⋮ Finite time blow up to critical semilinear wave equation outside the ball in 3-D ⋮ Scattering theory for the radial \(\dot{H}^{\frac{1}{2}}\)-critical wave equation with a cubic convolution ⋮ Scattering operator for the fourth order nonlinear Schrödinger equation ⋮ Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity ⋮ Global well-posedness of a system from quantum hydrodynamics for small data ⋮ Global behavior for Whitham type equations on a finite interval ⋮ Global dynamics of solutions with group invariance for the nonlinear Schrödinger equation ⋮ Scattering and blow-up for the focusing nonlinear Klein-Gordon equation with complex-valued data ⋮ Global well-posedness of the Euler-Korteweg system for small irrotational data ⋮ Rapidly decaying solutions of the nonlinear Schrödinger equation ⋮ Small solutions to nonlinear Schrödinger equations ⋮ On the global solution problem for semilinear generalized Tricomi equations. I ⋮ Blow up for systems of wave equations in exterior domain ⋮ Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension \(n\geq{}2\) ⋮ Quadratic derivative nonlinear Schrödinger equations in two space dimensions ⋮ Small data solutions for the Euler-Poisson-Darboux equation with a power nonlinearity ⋮ On the long time behavior of a generalized KdV equation ⋮ Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities ⋮ On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations ⋮ Blow-up of solutions of some nonlinear wave equations ⋮ Long-time behaviour of time-dependent density functional theory ⋮ Weighted \(L^2-L^2\) estimate for wave equation in \(\mathbb{R}^3\) and its applications ⋮ Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping ⋮ \(L^1-L^p\) estimates for radial solutions of the wave equation and application ⋮ A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type ⋮ Scattering theory for semilinear wave equations with small data in two space dimensions ⋮ Semilinear wave equation on compact Lie groups ⋮ On the asymptotic behavior of solutions of generalized Korteweg-de Vries equations ⋮ Small data blow-up of \(L^2\) or \(H^1\)-solution for the semilinear Schrödinger equation without gauge invariance ⋮ Spatial behavior for NLS and applications to scattering ⋮ Nonlinear small data scattering for the generalized Korteweg-de Vries equation ⋮ Large time asymptotics of solutions to the generalized Korteweg-de Vries equation ⋮ Nonexistence of global solutions to semilinear wave equations in high dimensions ⋮ Decay estimates for the critical semilinear wave equation ⋮ Analyticity of scattering for the \(\phi ^ 4\) theory ⋮ Small data blowup for systems of semilinear wave equations with different propagation speeds in three space dimensions ⋮ Inverse scattering on the line for the nonlinear Klein-Gordon equation with a potential ⋮ Nonlinear Schrödinger equations and sharp interpolation estimates ⋮ On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations ⋮ Global existence of small solutions to nonlinear evolution equations ⋮ Nonlinear small data scattering for the wave and Klein-Gordon equation ⋮ Small solutions to nonlinear Schrödinger equations in the Sobolev spaces ⋮ The global Cauchy problem for the nonlinear Klein-Gordon equation ⋮ The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces ⋮ Scattering and complete integrability in the massive \(\phi ^ 4\) theory ⋮ Sharp \(L^r\) asymptotics of the small solutions to the nonlinear Schrödinger equations ⋮ Finite-time blow-up of solutions to semilinear wave equations ⋮ Uniqueness in the inverse scattering problem for Hartree type equation ⋮ Low energy scttering for nonlinear Klein-Gordon equations ⋮ Global solution for the \(3D\) quadratic Schrödinger equation of \(Q(u,\bar{u}\)) type ⋮ Wave and scattering operators for the nonlinear Klein-Gordon equation on a quarter-plane ⋮ Blow up for initial-boundary value problem of wave equation with a nonlinear memory in 1-D ⋮ A remark on the blowing up of solutions to Nakao's problem ⋮ Final state problem for Korteweg–de Vries type equations ⋮ Finite time blow up for critical wave equations in high dimensions ⋮ Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions ⋮ Global Existence for Two Dimensional Quadratic Derivative Nonlinear Schrödinger Equations ⋮ The Schrödinger Equation with a Potential in Rough Motion ⋮ On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case ⋮ Sharp threshold of global existence and blow-up of the combined nonlinear Klein-Gordon equation ⋮ Nonexistence for nonlinear hyperbolic inequalities in an annulus ⋮ Revisit to Fritz John's paper on the blow-up of nonlinear wave equations ⋮ Self-similar solutions of semilinear wave equation with variable speed of propagation ⋮ Multichannel nonlinear scattering for nonintegrable equations ⋮ Threshold odd solutions to the nonlinear Schrödinger equation in one dimension ⋮ Blow up of solutions to semilinear wave equations with critical exponent in high dimensions ⋮ First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities ⋮ Global existence of small solutions for the fourth-order nonlinear Schrödinger equation ⋮ On the Strauss index of semilinear Tricomi equation ⋮ Lifespan of classical solutions to semilinear Neumann-wave equations ⋮ On traveling waves of the nonlinear Schrödinger equation escaping a potential well ⋮ \(\mathbf{L^2}\)-decay rate for the critical nonlinear Schrödinger equation with a small smooth data ⋮ Blow-up result for a semilinear wave equation with a nonlinear memory term ⋮ Critical exponent for a class of semilinear damped wave equations with decaying in time propagation speed ⋮ A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type ⋮ Blow-up and lifespan estimates of solutions to semilinear Moore-Gibson-Thompson equations ⋮ On semilinear Tricomi equations with critical exponents or in two space dimensions ⋮ Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term ⋮ Strauss exponent for semilinear wave equations with scattering space dependent damping ⋮ Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions ⋮ Blowup results and concentration in focusing Schrödinger-Hartree equation ⋮ Lifespan estimates for critical semilinear wave equations and scale invariant damped wave equations in exterior domain in high dimensions ⋮ The critical exponent(s) for the semilinear fractional diffusive equation ⋮ Optimal small data scattering for the generalized derivative nonlinear Schrödinger equations ⋮ Scattering for the quadratic Klein-Gordon equations ⋮ Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part ⋮ Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications ⋮ Nonexistence of global solutions for the semilinear Moore - Gibson - Thompson equation in the conservative case ⋮ Lifespan estimates for semilinear wave equations with space dependent damping and potential ⋮ Lifespan estimates of solutions for a coupled system of wave equations with damping terms and negative mass terms \(\ast\) ⋮ The low energy scattering for nonlinear Schrödinger equation ⋮ Decay estimates for a class of wave equations ⋮ Global existence for the one-dimensional second order semilinear hyperbolic equations ⋮ The matrix nonlinear Schrödinger equation with a potential ⋮ \(L^2\)-decay estimate for the dissipative nonlinear Schrödinger equation in the Gevrey class ⋮ A test function method for evolution equations with fractional powers of the Laplace operator ⋮ Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms ⋮ Local existence, global existence, and scattering for the nonlinear Schrödinger equation ⋮ Blowup behaviour for the nonlinear Klein-Gordon equation ⋮ Nonexistence results for the hyperbolic-type equations on graded Lie groups ⋮ CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS ⋮ Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms ⋮ Inverse scattering for the non-linear Schrödinger equation: Reconstruction of the potential and the non-linearity ⋮ Global existence and decay in multi-component reaction-diffusion-advection systems with different velocities: oscillations in time and frequency ⋮ A note on \(L^p - L^q\) estimates for semilinear critical dissipative Klein-Gordon equations ⋮ New blow-up phenomena for hyperbolic inequalities with combined nonlinearities ⋮ Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations ⋮ On the existence or absence of global solutions for the multidimensional version of the second Darboux problem for some nonlinear hyperbolic equations ⋮ Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity ⋮ On the nonexistence of global solutions of the characteristic Cauchy problem for a nonlinear wave equation in a conical domain ⋮ Wave operator for the system of the Dirac-Klein-Gordon equations ⋮ Global existence of small solutions for a quadratic nonlinear fourth-order Schrödinger equation in six space dimensions ⋮ Lifespan estimates for 2-dimensional semilinear wave equations in asymptotically Euclidean exterior domains ⋮ On the global weak solution problem of semilinear generalized Tricomi equations. II ⋮ Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions ⋮ On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations ⋮ Wave and inverse wave operators for the quadratic nonlinear Schrödinger equations in 3D ⋮ Oscillatory asymptotic expansion for semilinear wave equation ⋮ On the solvability of the Cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part ⋮ Energy scattering for a Klein-Gordon equation with a cubic convolution ⋮ SHARP ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR SCHRÖDINGER EQUATIONS WITH REPULSIVE INTERACTIONS ⋮ Lifespan of classical solutions to quasilinear wave equations outside of a star-shaped obstacle in four space dimensions ⋮ CONVERGENCE TO SCATTERING STATES IN THE NONLINEAR SCHRÖDINGER EQUATION ⋮ Final state problem for the cubic nonlinear Klein–Gordon equation ⋮ Lower bounds for the life span of solutions of nonlinear wave equations in higher dimensions ⋮ NUMERICAL SIMULATIONS OF THE ENERGY-SUPERCRITICAL NONLINEAR SCHRÖDINGER EQUATION ⋮ Lower bounds of asymptotics in time of solutions to nonlinear Schrödinger equations in 3D ⋮ Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities ⋮ First integrals for nonlinear dispersive equations ⋮ The Strauss conjecture on Kerr black hole backgrounds ⋮ Blow-up phenomena of semilinear wave equations and their weakly coupled systems ⋮ Some non-existence results for the semilinear Schrödinger equation without gauge invariance ⋮ Finite time blowup of solutions to semilinear wave equation in an exterior domain ⋮ Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains ⋮ Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential ⋮ A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension ⋮ Local smooth solutions of the nonlinear Klein-Gordon equation ⋮ Scattering for the radial defocusing cubic nonlinear wave equation with initial data in the critical Sobolev space ⋮ Lower bound estimate for the dissipative nonlinear Schrödinger equation ⋮ Global existence and scattering for quadratic NLS with potential in three dimensions ⋮ Blow-up results for higher-order evolution differential inequalities in exterior domains ⋮ Strichartz estimate and nonlinear Klein-Gordon equation on nontrapping scattering space ⋮ Dispersive decay for the nonlinear magnetic Schrödinger equation ⋮ Blow-up of solutions to semilinear wave equations with a time-dependent strong damping ⋮ Wave and scattering operators for the nonlinear matrix Schrödinger equation on the half-line with a potential ⋮ Asymptotic behavior of solutions to a dissipative nonlinear Schrödinger equation with time-dependent harmonic potentials ⋮ Blowup for semilinear wave equation with space-dependent damping and combined nonlinearities ⋮ Lifespan estimates for a class of semilinear wave equations with time- and space-dependent coefficients on the power nonlinearity ⋮ Global in time self-interacting Dirac fields in the de Sitter space

Cites Work

This page was built for publication: Nonlinear scattering theory at low energy

Retrieved from "https://portal.mardi4nfdi.de/w/index.php?title=Publication:1155228&oldid=13220011"