On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-Order Degenerating Hyperbolic Equation

1997 ◽  
Vol 4 (4) ◽  
pp. 341-354
Author(s):  
S. Kharibegashvili
Keyword(s):  
Hyperbolic Equation ◽  
Weighted Space ◽  
Second Order ◽  
Darboux Problem ◽  
Negative Norm ◽  
Correct Formulation ◽  
Sobolev Weighted Space

Abstract A multidimensional version of the first Darboux problem is considered for a model second order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.

CLOSURE OF FINITE FUNCTIONS IN ONE WEIGHT SOBOLEV TYPE SPACE

2020 ◽  
Vol 69 (1) ◽  
pp. 12-17
Author(s):  
A. Adiyeva ◽  
◽  
A.O. Baiarystanov ◽  
Keyword(s):  
Weighted Space ◽  
Functional Space ◽  
Second Order ◽  
Weight Space ◽  
Sobolev Type ◽  
Space Theory ◽  
Smooth Functions ◽  
Type Space ◽  
Sobolev Weighted Space ◽  
Smooth Functional

The description of the closure of finite or smooth finite functions in functional spaces are classical tasks of functional space theory. This task is important in smooth functional spaces such as those of Sobolev, Nikolski, Besov and in their various generalizations. Usually, in a weightless space of smooth functions, the set of compactly finite functions, generally speaking, is not dense. But in the weighted space of smooth functions, for example, in the Sobolev weighted space, with strong degeneracy of the weight, many compactly finite functions can be dense. Therefore, an important issue is the problem of characterizing the closure of compactly finite functions in the weight space under consideration. Here we consider a weighted space of Sobolev type of the second order with three weights and it describes the closure of the set of functions with compact supports.

Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise

2019 ◽  
Vol 19 (06) ◽  
pp. 1950044
Author(s):  
Haijuan Su ◽  
Shengfan Zhou ◽  
Luyao Wu
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Sufficient Conditions ◽  
Second Order ◽  
Lattice System ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Random System ◽  
Multiplicative White Noise ◽  
Infinite Sequences

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system.

On the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

On a Spatial Problem of Darboux Type for a Second-Order Hyperbolic Equation

1995 ◽  
Vol 2 (3) ◽  
pp. 299-311
Author(s):  
S. Kharibegashvili
Keyword(s):  
Sobolev Space ◽  
Hyperbolic Equation ◽  
Unique Solvability ◽  
Second Order ◽  
Spatial Problem ◽  
Order Hyperbolic Equation ◽  
Second Order Hyperbolic Equation

Abstract The theorem of unique solvability of a spatial problem of Darboux type in Sobolev space is proved for a second-order hyperbolic equation.

Symmetrization of second-order vector hyperbolic equation with two space variables

1987 ◽  
Vol 27 (3) ◽  
pp. 307-312
Author(s):  
O. A. Bondarenko ◽  
V. M. Gordienko
Keyword(s):  
Hyperbolic Equation ◽  
Second Order
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