Existence and uniqueness of a generalized solution of the mixed problem for an equation of plate oscillation type in a noncylindrical domain

1993 ◽  
Vol 63 (1) ◽  
pp. 98-101
Author(s):  
Ya. I. Sidelnik
Keyword(s):  
Mixed Problem ◽  
Existence And Uniqueness ◽  
Generalized Solution ◽  
Noncylindrical Domain ◽  
Oscillation Type

scholarly journals A MIXED PROBLEM WITH INTEGRAL CONDITION FOR A DEGENERATIVE EQUATION OF THE HYPERBOLIC TYPE

2017 ◽  
Vol 17 (8) ◽  
pp. 29-36
Author(s):  
S.V. Kirichenko
Keyword(s):  
Mixed Problem ◽  
Existence And Uniqueness ◽  
Integral Condition ◽  
Generalized Solution ◽  
Hyperbolic Type ◽  
Degenerate Equation

In this article, the mixed problem for the hyperbolic degenerate equation with an integral condition is considered. The existence and uniqueness of a generalized solution are proved.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

scholarly journals Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

2020 ◽  
Vol 40 (6) ◽  
pp. 725-736
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Time Dependent ◽  
Energy Solution ◽  
Noncylindrical Domain ◽  
Finite Energy Solution ◽  
Initial Boundary Value

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

scholarly journals On the Existence and Uniqueness ofRv-Generalized Solution for Dirichlet Problem with Singularity on All Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
V. Rukavishnikov ◽  
E. Rukavishnikova
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Input Data ◽  
Generalized Solution ◽  
Two Dimensional ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
Dimensional Domain

The existence and uniqueness of theRv-generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.

scholarly journals Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

2020 ◽  
Vol 20 (4) ◽  
pp. 444-456
Author(s):  
Vitalii P. Kurdyumov ◽  
◽  
Avgust P. Khromov ◽  
Victoria A. Khalova ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Mixed Problem ◽  
Initial Velocity ◽  
Formal Solution ◽  
Fourier Method ◽  
Generalized Solution ◽  
Initial Position ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem. The classical solution is obtained for ψ(x) ∈ W1p (1 < p ≤ 2), and it is also shown that if ψ(x) ∈ Lp[0, 1] (1 ≤ p ≤ 2), the formal solution is a generalized solution of the mixed problem.

scholarly journals ON CERTAIN NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS OF THE FIRST KIND

2017 ◽  
Vol 17 (5) ◽  
pp. 29-36
Author(s):  
A.V. Duzheva
Keyword(s):  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Generalized Solution ◽  
Integral Conditions

In this article, we consider a nonlocal problem for hyperbolic equation with integral conditions of the first kind. The main goal of this article is to show the method which allows to reduce posed problem to the problem with integral condition of the second kind. Existence and uniqueness of generalized solution is proved.

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