Exact Behavior of Singularities of Protter’s Problem for the 3-D Wave Equation

Popivanov, Nedyu; Popov, Todor

doi:10.1007/978-3-7091-6033-6_16

Cited by 9 publications

(18 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 3.6. Let us point out the difference between the cases in R 3 (see [23]) and R 4 here. In both of them the study of Protter problem P1 is based on the properties of the special Legendre functions.…”

Section: Remark 35mentioning

confidence: 97%

“…Many authors studied these problems using different methods, e.g., WienerHopf method, special Legendre functions, a priori estimates, nonlocal regularization, etc. (see [22] and references there, further [2,7,14,19,23,24]). On the other hand, in R 4 another analogue of the classical Darboux problem in the same domain Ω is given (Bazarbekov [4]).…”

Section: Introductionmentioning

confidence: 97%

“…Some other questions require to describe the exact order of singularity and to find some a priori estimates for the singular solutions. The answers for some Protter problems were given, after deriving necessary and sufficient conditions for the existence of solutions with fixed order of the singularity (see [23] in R 3 , [24] in R 4 and Remarks 3.5 and 3.6).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic expansions of singular solutions for (3+1)-D Protter problems

Popivanov

Popov

Scherer³

2007

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. The smoothness of the right-hand side function of the wave equation is decisive for the behavior of the solution of the boundary value problem. It is shown that for each n ∈ N there exists such a right-hand side function from C n , for which the uniquely determined generalized solution has a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. The present article describes asymptotic expansions of the generalized solutions in negative powers of the distance to this singularity point. Some necessary and sufficient conditions for existence of regular solutions, or solutions with fixed order of singularity, are derived and additionally some a priori estimates for the singular solutions are given.

show abstract

Section: Remark 35mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic expansions of singular solutions for (3+1)-D Protter problems

Popivanov

Popov

Scherer³

2007

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [12], they suggested to study the Protter problems in the frame of generalized solutions with possible big singularities. Today it is well-known that the Protter problems have singular generalized solutions, even for smooth right-hand sides [12,[14][15][16][17]. Different aspects of Protter problems and several their generalizations (including some applications in the industrial explosion process) are studied by many authors (see Aldashev and Kim [18], Choi and Park [19], Aldashev [20], and references therein).…”

Section: History Of the Problem And Motivationmentioning

confidence: 99%

“…More precisely, let us look for a solution to problem PK of the form (16). Using the spherical coordinates ( , , , ) ∈ R 4 , 0 ≤ < , 0 ≤ < 2 , > 0 with…”

Section: The Two-dimensional Darboux-goursat Problems Corresponding Tmentioning

confidence: 99%

Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

Popivanov

Hristov

Nikolov

et al. 2017

Advances in Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics. We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity.

show abstract