Norm inflation with infinite loss of regularity at general initial data for nonlinear wave equations in Wiener amalgam and Fourier amalgam spaces

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

“…We plan to address the norm-inflation (the stronger phenomenon than the mere ill-poseness) and even the worst situation of norm inflation with infinite loss of regularity for (1.1) in our future works. We note that recently similar questions we have already addressed for Hartree, nonlinear Schrödinger, BBM and wave equations in [2,3,4]. We also expect to develop well-posedness theory for (1.1) in w p,q space in the future.…”

“…survey article [14]. Recently, in [17] authors have studied KdV in modulation spaces, and in [2,3,4] authors have studied ill-poedesness for wave, BBM and NLS in Fourier amalgam spaces, see also [16,13]. We refer to [1] by Bejenaru-Tao for abstract well-posedness and ill-posedness theory.…”

Estimates for generalized wave‐type Fourier multipliers on modulation spaces and applications

“…Fourier amalgam spaces ω𝑠 𝑝,𝑞 have similar properties to modulation spaces, such as embedding, algebraic property (cf. Lemma 2.4 in [4]). The generalized wave-type multipliers are also bounded on ω𝑠 𝑝,𝑞 for any 1 ⩽ 𝑝, 𝑞 ⩽ ∞, 𝑠 ∈ ℝ (cf.…”

“…The linear parts of the solution 𝑊(𝑡)𝑢 1 , 𝑊 ′ (𝑡)𝑢 0 is similar to that in Theorem 6. We only need to get the following nonlinear estimates: [4] proved that the norm inflation occurs with infinite loss of regularity at any initial data in 𝑀 𝑠 2,𝑞 when 𝑠 < 0, which implies the ill-posedness in 𝑀 𝑠 2,𝑞 when 𝑠 < 0. Actually, their results hold for any Fourier amalgam spaces ω𝑠 𝑝,𝑞 with negative regularity, which is coincident with the modulation space 𝑀 𝑠 𝑝,𝑞 when 𝑝 = 2.…”

“…Remark The restriction $$s\geqslant 0$$ in Theorems 6 and 7 is sharp in some cases. In fact, for the classical wave equation ($$h(\xi )=\left|\xi \right|, \phi (r)=r$$), Bhimani and Haque in [4] proved that the norm inflation occurs with infinite loss of regularity at any initial data in $$M_{2,q}^{s}$$ when $$s&lt;0$$, which implies the ill‐posedness in $$M_{2,q}^{s}$$ when $$s&lt;0$$. Actually, their results hold for any Fourier amalgam spaces $$\widehat{\omega } _{p,q}^{s}$$ with negative regularity, which is coincident with the modulation space $$M_{p,q}^{s}$$ when $$p=2$$.…”