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 \(u(x,t)\in B^{A,\#}_{A'} (\mathbb R\times I_T)\); \(A=(-1/4,1/2)\). \(A'=2,1,-|\xi|^2\).


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
Full Text: DOI