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

Divyang G. Bhimani and Saikatul Haque
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Divyang G. Bhimani: Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
Saikatul Haque: Department of Mathematics, Harish-Chandra Research Institute, Allahabad 2110019, India

Mathematics, 2021, vol. 9, issue 23, 1-13

Abstract: We consider the Benjamin–Bona–Mahony (BBM) equation of the form u t + u x + u u x − u x x t = 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 H s ( T ) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in H s ( R ) . Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces M s 2 , 1 ( R ) for s ≥ 0 .

Keywords: BBM equation; ill-posedness; Fourier amalgam spaces; Wiener amalgam spaces; Fourier–Lebesgue spaces; modulation spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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