Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity - Divyang G. Bhimani; Saikatul Haque (Mathematics, 2021)
We consider the Benjamin–Bona–Mahony (BBM) equation of the form <semantics>ut+ux+uux−uxxt=0,(x,t)∈M×R</semantics> where <semantics>M=T</semantics> or <semantics>R.</semantics> 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 <semantics>Hs(T)</semantics> established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in <semantics>Hs(R).</semantics> Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces <semantics><msubsup>Ms2,1</msubsup>(R)</semantics> for <semantics>s≥0</semantics>.