Estimates of solutions of the Cauchy problem for Schrödinger type equations in the spaces \(L_ p^{\alpha}\) and \(B^{\alpha}_{pq}\). IV
type: Item
instance of: scholarly article
title: Estimates of solutions of the Cauchy problem for Schrödinger type equations in the spaces \(L_ p^{\alpha}\) and \(B^{\alpha}_{pq}\). IV
zbMATH Open document ID: 0678.35010
DOI: 10.1007/BF00972126
author: P. Shlekis
published in: Lithuanian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The following Cauchy problem for the Schrödinger type equations: \(i(du/dt)=P(D_ 1,D_ 2)u,\) \(t>0\), \(u(0)=u_ 0\in S(R^ 2)\), where P is a polynom with real constant coefficients, \(D_ k=i(\partial /\partial x_ k),\) \(k=1,2\); is considered. For a certain class of differential operators \(P(D_ 1,D_ 2)\) necessary and sufficient conditions for the existence of the bounds \(\| u(t)\|_{\Lambda^ 0_ p}\leq C_{\alpha}(t)\| u_ 0\|_{\Lambda_ p^{\alpha}},\) \(t\geq 0\), \(\forall u_ 0\in S(R^ 2)\), where \(\Lambda_ p^{\alpha}\) is either a Liouville space \(L_ p^{\alpha}(R^ 2)\) or a Besov space \(B^{\alpha}_{pq}(R^ 2)\), are received. The asymptotic behavior of the function \(C_{\alpha}(t)\), as \(t\to \infty\), is studied.
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4111143
zbMATH Keywords: Fourier multiplier
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Schrödinger type equations
zbMATH Keywords: existence
zbMATH Keywords: bounds
zbMATH Keywords: Liouville space
zbMATH Keywords: Besov space
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
cites work: Q3913845
description: scientific article; zbMATH DE number 4111143