We consider nonlinear half-wave equations with focusing power-type nonlinearity i partial derivative tu = root-Delta u - vertical bar u vertical bar(p-1) u, with (t, x) is an element of R x R-d with exponents 1 < p < infinity for d = 1 and 1 < p < (d + 1)/(d - 1) for d >= 2. We study traveling solitary waves of the form u(t, x) = e(i omega t) Q(upsilon)(x - nu t) with frequency omega is an element of R, velocity nu is an element of R-d, and some finite-energy profile Q(upsilon) is an element of H-1/2(R-d), Q(upsilon) not equal 0. We prove that traveling solitary waves for speeds |upsilon| >= 1 do not exist. Furthermvore, we generalize the non-existence result to the square root Klein-Gordon operator root-Delta + m(2) and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds |upsilon| < 1. Finally, we discuss the energy-critical case when p = (d + 1)/(d - 1) in dimensions d >= 2.

On traveling solitary waves and absence of small data scattering for nonlinear half-wave equations

Visciglia, Nicola;Bellazzini , Jacopo;
2019-01-01

Abstract

We consider nonlinear half-wave equations with focusing power-type nonlinearity i partial derivative tu = root-Delta u - vertical bar u vertical bar(p-1) u, with (t, x) is an element of R x R-d with exponents 1 < p < infinity for d = 1 and 1 < p < (d + 1)/(d - 1) for d >= 2. We study traveling solitary waves of the form u(t, x) = e(i omega t) Q(upsilon)(x - nu t) with frequency omega is an element of R, velocity nu is an element of R-d, and some finite-energy profile Q(upsilon) is an element of H-1/2(R-d), Q(upsilon) not equal 0. We prove that traveling solitary waves for speeds |upsilon| >= 1 do not exist. Furthermvore, we generalize the non-existence result to the square root Klein-Gordon operator root-Delta + m(2) and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds |upsilon| < 1. Finally, we discuss the energy-critical case when p = (d + 1)/(d - 1) in dimensions d >= 2.
2019
Gueorguiev, Vladimir Simeonov; Visciglia, Nicola; Bellazzini, Jacopo; Lenzmann, Enno
File in questo prodotto:
File Dimensione Formato  
bellazzini2019.pdf

solo utenti autorizzati

Tipologia: Versione finale editoriale
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 323.08 kB
Formato Adobe PDF
323.08 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
1808.08134.pdf

accesso aperto

Tipologia: Documento in Pre-print
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 284.01 kB
Formato Adobe PDF
284.01 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1026402
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 3
social impact