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Unconditional uniqueness of solution for \(\dot{H}^{s_c}\) critical 4th order NLS in high dimensions. (English) Zbl 1393.35223

Summary: In this paper, we study the unconditional uniqueness of solution for the Cauchy problem of \(\dot{H}^{s_c}\) (\(0 \leq s_c < 2\)) critical nonlinear fourth-order Schrödinger equations \(\mathrm{i}\partial_t u+\Delta^2 u-\epsilon u=\lambda| u|^\alpha u\). By employing paraproduct estimates and Strichartz estimates, we prove that unconditional uniqueness of solution holds in \(C_t (I;\dot{H}^{s_c}(\mathbb R^d))\) for \(d\leq 11\) and \(\min \{1^-,\tfrac{8}{d - 4}\}\geq \alpha >\frac{-(d-4)+\sqrt{(d-4)^2+64}}{4}\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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