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Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces. (English) Zbl 1067.35116
Summary: Well-posedness of the Cauchy problem for the semilinear Schrödinger equation \[ \begin{aligned} \partial_t u=i\Delta u+N(u,\overline u), \quad & x\in\mathbb R^d,\;t\in\mathbb R,\\ u(x,0)=u_0(x),\quad & x\in\mathbb R^d, \end{aligned} \] with quadratic nonlinear terms is studied. By making use of Besov spaces we can improve the regularity assumption on the initial data. When the nonlinear term is \(c_1u^2+ c_2\overline u^2\), our results are as follows: When \(d=1\) or 2, for any initial data \(u_0\in H^{-3/4}(\mathbb R^d)\) there exists a unique local-in-time solution \(u\in B_{2,(2,1),-|\xi|^2}^{(-3/4,1/2)} (\mathbb R^d\times I_T)\). When \(d\geq 3\) for any small data \(u_0\in H^\rho(\mathbb R^d)\), where \(\rho(z)=z^{d/2-2} \log(2+z)\), there exists a unique local-in-time solution \(u\in B_{2,(2,1),-|\xi|^2}^{(\rho,1/2)} (\mathbb R^d\times I_T)\), and for any \(u_0\in H^s (\mathbb R^d)\), \(s>d/2-2\), there exists a unique local-in-time solution \(u\in B_{2,(2,1),-|\xi|^2}^{(s,1/2)}(\mathbb R^d \times I_T)\). Here \(I_T=(-T, T)\). We also have results for the equation with the nonlinear term \(c_3u \overline u\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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