## The initial value problem for the 1-D semilinear Schrödinger equation in Besov spaces.(English)Zbl 1061.35137

The authors study the semilinear Schrödinger equation
$\partial_t u=i \partial_x^2 u+N(u,\overline u)$ with $$u(x,0)=u_0(x)$$, $$x\in \mathbb R$$. The solution $$u$$ is expressed by
$u(x,t)=W(t)u_0(x)+ \int_{[0, t]} W(t-t') N(u,\overline u) (x,t')\,dt',$
$$\{W(t)f\}(x,t)= F_x^{-1}e^{itP (\varepsilon)} F_x(x,t)$$, $$P(\xi)=\pm \xi^2$$. They define the Besov type spaces $$B_{A'}^A (\mathbb R^{d+1})$$; $$A=(\rho,b)$$, $$A'=p,q,P, B_{A'}^{A,\#} (\mathbb R^{d+1})$$; $$A=(s,b)=(z^s,b)$$, $$A'=2$$, $$q,P, B^\rho_{p,q} (\mathbb R^d)$$ and $$B^{s,\#}_{2,q}(\mathbb R^d)$$ of the tempered distributions. They use $$\widehat f_{jk,P} (\xi,\tau)=\varphi_j(|\xi|)$$. $$\varphi_k(\tau- P (\xi)) \widehat f(\xi,\tau)$$ etc., $$\{\varphi_j\}$$: a sequence of $$C_0^\infty$$-functions. Bilinear estimates hold in $$B^A_A (\mathbb R^{d+ 1})$$ and $$B^{A,\#}_{A'}(\mathbb R^{d+1})$$.
Main Theorem: (I) If $$N(u, \overline u)=c_1u^2+c_2\overline u^2$$ and $$u_0\in B^{-3/4}_{2,1} (\mathbb R)$$, then there exist $$T=T(\| u_0\|)>0$$ and a unique solution $$u(x,t)=W(t)u_0(x)+v(x,t)$$ in $$\mathbb R\times I_T$$, $$I_T=(-T,T)$$. Here $$v\in B^A_A (\mathbb R\times I_T)$$; $$A=(\rho,1/2)= (z^s\log(2+z), 1/2)$$, $$A'= 2$$, $$1,-|\xi|^2$$, and $$s\geq-3/4$$.
(II) If $$N(u,\overline u)= c_3u\overline u$$ and $$u_0\in B^{-1/4, \#}_{2,1}(\mathbb R)$$, then there exist $$T=T(\| u_0 \|)>0$$ and a unique solution $$u(x,t)\in B^{A,\#}_{A'} (\mathbb R\times I_T)$$; $$A=(-1/4,1/2)$$. $$A'=2,1,-|\xi|^2$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35G25 Initial value problems for nonlinear higher-order PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46F05 Topological linear spaces of test functions, distributions and ultradistributions
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