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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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