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

Abstract

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

Families of propagating soliton solutions for (3+1)-fractional Wazwaz-BenjaminBona-Mahony equation through a novel modification of modified extended direct algebraic method

“…Numerous analytical methods have been developed to deal with this issue. These techniques include the Adomian decomposition method [9], Khater method [10], Variable separable method [12], (G'/G)-expansion method [13], integral transforms method [14], Exp-function method [15], Differential transform method [16], homotopy perturbation method [17], and EDAM [18].…”

“…The KdV equation functions as an existence equation, whereas the BBM equation is classified as a regularity equation [14]. In [15,16], the theory of the BBM model, such as its uniqueness, stability, & consistency, is examined. In recent years, the BBM model has also been the subject of extensive study.…”