Norm Inflation for Nonlinear Schrödinger Equations in Fourier–Lebesgue and Modulation Spaces of Negative Regularity

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

“…We first consider the case X p,q s " p w p,q s . If the initial data u 0,N satisfy (11), Corollary 1 guarantees the existence of the solution to (3) and the power series expansion in F L 1 up to time TR ! 1 (as R " 1).…”

“…where ϕpξq " ξ 1`ξ 2 , D x " 1 i B x and ϕpD x q is the Fourier multiplier operator defined by p w p,q s pMq " $ ' ' ' ' & ' ' ' ' % F L q s pMq (Fourier-Lebesgue spaces) if p " q M 2,q s pMq (modulation spaces) if p " 2 M 2,2 s pMq " W 2,2 s pMq " H s pMq (Sobolev spaces) if p " q " 2 F L q s pMq " M p,q s pMq " W p,q s pMq if M " T d . These time-frequency spaces are proven to be very fruitful in handling various problems in analysis and have gained prominence in nonlinear dispersive PDEs, e.g., [7][8][9][10][11][12][13][14][15]. We now state our main theorem.…”

Wellposedness of NLS in Modulation Spaces

“…As a consequence of polarization identities for real symmetric multilinear operators [32], in order to show an estimate of the form (7), or more generally for some Banach space Y ,…”

“…We will work in modulation spaces which do not admit homogeneous scaling, and are also above the scaling critical exponent for NLS. As a result, the bounds ( 6) and (7) will depend on the time variable T . This will show that a solution exists with guaranteed time of existence depending on f D , and will result in a blow-up alternative later.…”

“…This will show that a solution exists with guaranteed time of existence depending on f D , and will result in a blow-up alternative later. Lemma 9 Let (8) be quantitatively wellposed in D, X = X T , and assume that the constants in (6) respectively (7) are…”