Estimates of solutions of the Cauchy problem for equations of Schrödinger type in the spaces \(L_ p^{\alpha}\) and \(B^{\alpha}_{pq}\). I
type: Item
instance of: scholarly article
title: Estimates of solutions of the Cauchy problem for equations of Schrödinger type in the spaces \(L_ p^{\alpha}\) and \(B^{\alpha}_{pq}\). I
zbMATH Open document ID: 0693.35020
DOI: 10.1007/BF00966607
author: P. Shlekis
published in: Lithuanian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K30
zbMATH DE Number: 4134541
zbMATH Keywords: Schrödinger type equations
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Besov spaces
MaRDI profile type: Publication
cites work: HOLOMORPHIC FUNCTIONS WITH POSITIVE IMAGINARY PART IN THE FUTURE TUBE. II
cites work: Q5334545
cites work: Q5594793
cites work: Q5554601
cites work: Equazioni di Schrödinger e delle onde per l'operatore di Laplace iterato in \(L^ p (R^ n)\)
cites work: Q5552946
cites work: The Cauchy problem for systems in L\(_p\) and L\({_p,\alpha}\)
cites work: Q5598649
cites work: Q3923074
cites work: Q5560568
cites work: Q3913845
full work available at URL: https://doi.org/10.1007/bf00966607
OpenAlex ID: W4251586788
description: scientific article; zbMATH DE number 4134541