In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator $mathcal{B}_tf$ converges pointwise to its initial data $fin H^s(mathbb{R})$ as $tto 0$ provided $sgeqfrac{1}{4}$ -- more precisely -- on the one hand, by constructing a counterexample in $mathbb{R}$ we discover that the optimal convergence index $s_{c,1}=frac14$; on the other hand, we find that the Hausdorff dimension of the disconvergence set for $mathcal{B}_tf$ is begin{align*} alpha_{1,mathcal{B}}(s)&=begin{cases} 1-2s& text{as} frac{1}{4}leq sleqfrac{1}{2}; 1 & text{as} 0<s<frac{1}{4}. end{cases} end{align*} Moreover, Theorem 1.3 presents a higher dimensional lift of Theorems 1.1- 1.2 under $f$ being radial.
تحميل البحث