Estimates of solutions of the Cauchy problem for equations of Schrödinger type in the spaces \(L_ p^{\alpha}\) and \(B^{\alpha}_{pq}\). I
Consider the Cauchy problem for Schrödinger type equations \[ (1)\quad i(du/dt)=P(D)u,\quad t>0;\quad u(0)=u_ 0 \] where \(P(D)=\sum_{| \alpha | \leq m}a_{\alpha}D^{\alpha}(a_{\alpha}\in R\), \(D^{\alpha}=i^{-| \alpha |} \partial^{| \alpha |}/\partial x_ 1^{\alpha_ 1}...\partial x_ n^{\alpha_ n}\), \(| \alpha | =\alpha_ 1+\alpha_ 2+...+\alpha_ n)\) is differential operator with constant coefficients. The problem on the existence of the following estimates for the solutions of the Cauchy problem is studied \[ \| u(t)\|_{L^ 0_ p}\leq C(\alpha,t)\| u_ 0\|_{L_ p^{\alpha}},\quad t\geq 0,\quad \forall u_ 0\in S \] \(\| u(t)\|_{B^ 0_{pq}}\leq C(\alpha,t)\| u_ 0\|_{B^{\alpha}_{pq}}\), \(t\geq 0\), \(\forall u_ 0\in S\) where \(L_ p^{\alpha}=L_ p^{\alpha}(R^ n)\), \(1<p<\infty\), \(\alpha\geq 0\) are Liouville spaces and \(B^{\alpha}_{pq}=B^{\alpha}_{pq}(R^ n)\), \(1<p<\infty\), \(1<q\leq \infty\), \(\alpha\geq 0\) are Besov spaces.
scientific article; zbMATH DE number 4134541
scientific article