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,px, tq P M ˆ 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 pTq established by Bona– Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs pRq. Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces M2,1 s pRq for s ě 0