scholarly journals Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Nedyu Popivanov ◽  
Todor Popov ◽  
Allen Tesdall
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Generalized Solution ◽  
Generalized Solutions ◽  
Singular Solutions ◽  
Priori Estimates ◽  
Characteristic Cone ◽  
Equation Boundary ◽  
Necessary And Sufficient

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertexOof the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance toO. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

scholarly journals Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-33 ◽  
Author(s):  
Lubomir Dechevski ◽  
Nedyu Popivanov ◽  
Todor Popov
Keyword(s):  
Asymptotic Expansion ◽  
A Priori Estimates ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Generalized Solution ◽  
Adjoint Problem ◽  
Generalized Solutions ◽  
Singular Solutions ◽  
Classical Solutions ◽  
Darboux Problem

We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems inℝ2. In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right-hand side functions there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.

scholarly journals Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 292
Author(s):  
Anna Anop ◽  
Iryna Chepurukhina ◽  
Aleksandr Murach
Keyword(s):  
Boundary Conditions ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Generalized Solutions ◽  
Bounded Operators ◽  
Priori Estimates

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

scholarly journals On the solvability of parabolic and hyperbolic problems with a boundary integral condition

2002 ◽  
Vol 31 (4) ◽  
pp. 201-213 ◽  
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
A Priori Estimates ◽  
Hyperbolic Equations ◽  
A Priori ◽  
Integral Condition ◽  
Generalized Solution ◽  
Boundary Integral ◽  
Analysis Method ◽  
Hyperbolic Problems ◽  
Priori Estimates ◽  
Abstract Formulation

We prove the existence, uniqueness, and the continuous dependence of a generalized solution upon the data of certain parabolic and hyperbolic equations with a boundary integral condition. The proof uses a functional analysis method based on a priori estimates established in nonclassical function spaces, and on the density of the range of the linear operator associated to the abstract formulation of the studied problem.

scholarly journals Problem on vibration of a bar with nonlinearsecond-order boundary damping

2017 ◽  
Vol 21 (3) ◽  
pp. 9-20
Author(s):  
A.B. Beylin ◽  
L.S. Pulkina
Keyword(s):  
A Priori Estimates ◽  
Longitudinal Vibration ◽  
A Priori ◽  
Initial Boundary ◽  
Generalized Solution ◽  
Application Form ◽  
Boundary Damping ◽  
Initial Boundary Problem ◽  
Priori Estimates ◽  
Pseudohyperbolic Equation

In this paper, we study the initial-boundary problem with nonlinear dynam- ical boundary condition for the pseudohyperbolic equation. This problem repre- sents a mathematical model of longitudinal vibration in a thick short bar with dynamic nonlinear second-order boundary damping. The existence and unique- ness of a generalized solution are proved. The proof is based on a priori estimates and Galerkin procedure. This approach allows to construct approximation in the suitable for practical application form.

scholarly journals A PROBLEM WITH SECOND KIND INTEGRAL CONDITIONS FOR HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (1-2) ◽  
pp. 33-45
Author(s):  
L. S. Pulkina ◽  
A. E. Savenkova
Keyword(s):  
Hyperbolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Main Idea ◽  
Integral Condition ◽  
Generalized Solution ◽  
Integral Conditions ◽  
Priori Estimates ◽  
Nonlocal Integral Condition ◽  
Definition Of

In this paper, we consider a problem for one-dimensional hyperbolic equation with second kind integral conditions and prove unique solvability. To prove this statement we suggest a new approach. The main idea of it is that given nonlocal integral condition is equivalent with a different condition, nonlocal as well but this new condition enables us to introduce a definition of a generalized solution bazed on an integral identity and derive a priori estimates of a required solution in Sobolev space. This approach shows that integral conditions are closely connected with dynamical conditions.

scholarly journals PROBLEM WITH DYNAMIC BOUNDARY CONDITIONS FOR A HYPERBOLIC EQUATION

2017 ◽  
Vol 23 (1) ◽  
pp. 21-27
Author(s):  
V. A. Kirichek ◽  
L. S. Pulkina
Keyword(s):  
Boundary Condition ◽  
Hyperbolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Generalized Solution ◽  
Dynamic Boundary Condition ◽  
Initial Boundary Problem ◽  
Dynamic Boundary ◽  
Priori Estimates

We consider an initial-boundary problem with dynamic boundary condition for a hyperbolic equation in a rectangle. Dynamic boundary condition represents a relation between values of derivatives with respect of spacial variables of a required solution and first-order derivatives with respect to time variable. The main result lies in substantiation of solvability of this problem. We prove the existence and uniqueness of a generalized solution. The proof is based on the a priori estimates obtained in this paper, Galyorkin’s procedure and the properties of Sobolev spaces.

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