scholarly journals Compact imbedding of weighted Sobolev space defined on an unbounded domain. II.

1988 ◽  
Vol 113 (3) ◽  
pp. 293-308
Author(s):  
Bohumír Opic
Keyword(s):  
Sobolev Space ◽  
Unbounded Domain ◽  
Weighted Sobolev Space ◽  
Compact Imbedding

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

2021 ◽  
pp. 1-13
Author(s):  
Kita Naoyasu ◽  
Sato Takuya
Keyword(s):  
Sobolev Space ◽  
Global Solution ◽  
Initial Value Problem ◽  
Trivial Solution ◽  
Decay Estimate ◽  
Weighted Sobolev Space ◽  
The Other ◽  
Initial Value ◽  
Decay Of Solutions ◽  
Dissipative Nonlinearity

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

scholarly journals A parabolic system in a weighted Sobolev space

2007 ◽  
Vol 34 (2) ◽  
pp. 169-191
Author(s):  
Adam Kubica ◽  
Wojciech M. Zajączkowski
Keyword(s):  
Sobolev Space ◽  
Parabolic System ◽  
Weighted Sobolev Space

Sampling in a weighted Sobolev space

2012 ◽  
Vol 350 (21-22) ◽  
pp. 941-944
Author(s):  
Nestor G. Acala ◽  
Noli N. Reyes
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space

scholarly journals Existence of Solution of Space–Time Fractional Diffusion-Wave Equation in Weighted Sobolev Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Kangqun Zhang
Keyword(s):  
Wave Equation ◽  
Sobolev Space ◽  
Fractional Derivatives ◽  
Weighted Sobolev Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Existence Of Solution ◽  
Multiplier Theorem ◽  
Diffusion Wave ◽  
Loss Of Regularity

In this paper, we consider Cauchy problem of space-time fractional diffusion-wave equation. Applying Laplace transform and Fourier transform, we establish the existence of solution in terms of Mittag-Leffler function and prove its uniqueness in weighted Sobolev space by use of Mikhlin multiplier theorem. The estimate of solution also shows the connections between the loss of regularity and the order of fractional derivatives in space or in time.

A Compact Imbedding Theorem for Functions without Compact Support

1971 ◽  
Vol 14 (3) ◽  
pp. 305-309 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Sobolev Space ◽  
Compact Support ◽  
Unbounded Domains ◽  
Bounded Domains ◽  
Complete Continuity ◽  
Compact Imbedding ◽  
Compactness Theorems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Made In

The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort1to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings2are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.

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