On the solvability of a spatial problem of Darboux type for the wave equation

Kharibegashvili, S.

doi:10.1007/bf02255987

Cited by 16 publications

(8 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Alternatively, different multidimensional analogues of the classical Darboux problem for the wave equation are considered in [20][21][22], while for some related semilinear equations and systems see [23]. The existence of bounded or unbounded solutions for the wave equation in R 3 and R 4 , as well as for the Euler-PoissonDarboux equation, has been studied in [4,10,17,19,[24][25][26].…”

Section: Historical Remarks On the Main Resultsmentioning

confidence: 99%

“…In Corollaries 9 and 10 the emphasis is on the extreme cases: when all orthogonality conditions (22) are fulfilled or, alternatively, when none of them are satisfied. In both Abstract and Applied Analysis 5 cases the exact behavior of the solution is given.…”

Section: Corollary 10 the Generalized Solution Of Problem 2 With A Rmentioning

confidence: 99%

“…However, the explicit asymptotic expansion here has no analogue in [10], where only the behavior of the singularities is given. Additionally, if the orthogonality conditions (22) are fulfilled, Corollary 9 states that the generalized solution is in fact bounded, while the estimates in [10, Theorem 1.1] still allow the solution to have some logarithmic singularities.…”

Section: Corollary 10 the Generalized Solution Of Problem 2 With A Rmentioning

confidence: 99%

See 2 more Smart Citations

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

Popivanov

Popov

Tesdall

2014

Abstract and Applied Analysis

View full text Add to dashboard Cite

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 vertex of 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 to . Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

show abstract

Section: Historical Remarks On the Main Resultsmentioning

confidence: 99%

Section: Corollary 10 the Generalized Solution Of Problem 2 With A Rmentioning

confidence: 99%

Section: Corollary 10 the Generalized Solution Of Problem 2 With A Rmentioning

confidence: 99%

See 1 more Smart Citation

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

Popivanov

Popov

Tesdall

2014

Abstract and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…A comparison of various recent results for Protter problems is made in [6]. For the wave equation some other multidimensional versions of the planar Darboux problem are studied in [4,5,14]. For Protter problems with lower order terms or with more general boundary condition on Σ 0 see [8,9] and references therein.…”

Section: Consider the Wave Equation Inmentioning

confidence: 98%

Analysis of singularities of solutions to Protter problems for the wave equation

Popivanov¹,

Popov²

2014

AIP Conference Proceedings

View full text Add to dashboard Cite

“…On the other hand, in R 4 another analogue of the classical Darboux problem in the same domain Ω is given (Bazarbekov [4]). Some different statements on Darboux type problems in R 3 are known also [5,16,17]. Together with P1 its adjoined problem P1 * has been studied for existence of bounded or unbounded solutions for the wave equation in R 3 and R 4 as well as for the Euler-Poisson-Darboux equation [2,11,14,15,18,19,24].…”

Section: Introductionmentioning

confidence: 99%

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

Popivanov

Popov

Scherer³

2007

Journal of Mathematical Analysis and Applications

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

On the solvability of a spatial problem of Darboux type for the wave equation

Cited by 16 publications

References 2 publications

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

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

Analysis of singularities of solutions to Protter problems for the wave equation

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

Contact Info

Product

Resources

About