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

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 .

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.

scholarly journals Singular and Degenerate Boundary Value Problems Related to the Electricity Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Maria-Magdalena Boureanu ◽  
Andaluzia Matei
Keyword(s):  
Sobolev Space ◽  
Mathematical Models ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Weighted Sobolev Space ◽  
Nonlinear Boundary Conditions ◽  
Uniqueness Result ◽  
Physical Phenomena ◽  
Mathematical Literature ◽  
Weak Solvability

The present paper draws attention to the weak solvability of a class of singular and degenerate problems with nonlinear boundary conditions. These problems derive from the electricity theory serving as mathematical models for physical phenomena related to the anisotropic media with “perfect” insulators or “perfect” conductors points. By introducing an appropriate weighted Sobolev space to the mathematical literature, we establish an existence and uniqueness result.

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