Remarks on Fourier multipliers and applications to the wave equation

Abstract

Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [A. Bényi, K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, preprint, April 2007 (available at ArXiv:0704.0833v1)]. The same arguments may apply to obtain local wellposedness for the NLKG equation.

Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces

“…In fact, there are some recent works which have been devoted to the study of the well-posedness for a class of nonlinear evolution equation in modulation spaces; cf. [2,3,5,7,8,9,20,24,25,26,31,39,40,41,42]. Our main goal of this paper is to study the global well-posedness of 4NLS in modulation spaces M 3+1/2 2,1…”

The Cauchy problem for the vibrating plate equation in modulation spaces

“…Indeed, the related Fourier multiplier T τ , having symbol τ (ξ) = e πit|ξ | 2 , is unbounded on W (F L p , L q ), when p = q, see Proposition 2.7. Hence, in contrast to what happens for other equations such as the wave equation [7,Theorem 4.4], there is no wellposedness of (1) in Wiener amalgam spaces.…”

“…The case p = q gives W (F L p , L p ) = M p (see (7)) and we come back to modulation spaces. Indeed, the related Fourier multiplier T τ , having symbol τ (ξ) = e πit|ξ | 2 , is unbounded on W (F L p , L q ), when p = q, see Proposition 2.7.…”

“…These latter are then combined with classical fixed point arguments to obtain local wellposedness in modulation spaces for (1). Contrary to what happens for other equations such as the wave equation [7,Theorem 4.4], there is no local wellposedness of (1) in the framework of Wiener amalgam spaces. Precisely, we shall show that the latter already fails for the homogeneous case F ≡ 0.…”

“…Here the tools employed follow the pattern of similar Cauchy problems studied for other equations such as the Schrödinger, wave and Klein-Gordon equations [3,4,7]. Let us a quote [22][23][24] as inspiring works on this topic.…”