On a regular cauchy problem for the Euler-Poisson-Darboux equation

Davis, Ruth M.

doi:10.1007/bf02411877

Cited by 13 publications

(9 citation statements)

References 7 publications

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“…These are all of the values of a for which the Huygens principle is valid for the regular Cauchy problem related to the EPD equation. This result is known from [17].…”

Section: Olevskiȋ I E O Tsupporting

confidence: 51%

“…7). The corresponding problem in the Euclidean space E n was studied in [17,18]. Our approach is new also in that case (Sec.…”

Section: Olevskiȋ I E O Tmentioning

confidence: 89%

“…This problem has been solved in [17] for ϕ ≡ 0. That solution has been found by using a modification of M. Riesz's method applied to non-selfadjoint equation (4.1) and the result turned out to be rather cumbersome.…”

Section: 1mentioning

confidence: 99%

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An Operator Approach to the Cauchy Problem for the Euler-Poisson-Darboux Equation in Spaces of Constant Curvature

Olevskii

2004

Integral Equations and Operator Theory

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We develop an operator approach to the problem named in the title. It is based on operator treatment of some relations between Bessel functions, and hypergeometric functions as well. Our operator version of the DelsartePovzner formulas also plays an essential role in the paper. Introduction.These notes illustrate an operator reading of some formulas of analysis which gives these formulas a new meaning so that they become an heuristic source of non-trivial results.We have in mind some relations concerning Bessel and hypergeometrical functions and the Delsarte-Povzner formulas.Read anew, they make it possible to find, among other things a solution of the singular and regular Cauchy problems for the generalized Euler-Poisson-Darboux (EPD) equation in the spaces S n of constant curvature k = 0.We explain the essence of our approach on the well-investigated [1-6] singular Cauchy problem for the EPD equationThe solution of this problem can be represented symbolically using Bessel functions of the Laplacian. In order to give an explicit meaning to this representation, we notice that for a certain value of the parameter a the solution can be obtained as the average M (t, x; f ) of f over the sphere of radius t centered at x [7,8].On the other hand a Sonin formula gives a connection between Bessel functions with different values of the indices. We introduce an operator version of the Sonin formula which allows one to easily obtain the solution for any a. For a detailed presentation see Sec. 2.

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“…These are all of the values of a for which the Huygens principle is valid for the regular Cauchy problem related to the EPD equation. This result is known from [17].…”

Section: Olevskiȋ I E O Tsupporting

confidence: 51%

“…7). The corresponding problem in the Euclidean space E n was studied in [17,18]. Our approach is new also in that case (Sec.…”

Section: Olevskiȋ I E O Tmentioning

confidence: 89%

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An Operator Approach to the Cauchy Problem for the Euler-Poisson-Darboux Equation in Spaces of Constant Curvature

Olevskii

2004

Integral Equations and Operator Theory

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“…The case n =3, α >0 was considered by Diaz and Ludford using Hadamard's method, which in this case did not lead to divergent integrals. Young, studied the case of any

α \in double-struckR

in terms of the techniques of Davis, which used to solve the regular Cauchy problem and Diaz and Ludford . For the nonlinear case, Keller showed that the nonexistence of global solution with arbitrary φ ( x ) and certain source term for n =2 and α >1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…then there exists a uniquely global solution u ∈ C([0, +∞), L q (R n )) to problems (4)(5) with a decay at infinity of time direction…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane

Zhang

2018

Math Methods in App Sciences

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In this paper, we study the well posed‐ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane. By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter m∈(0,1) and establish local and global existences of solution to the semilinear equation. Meanwhile, we derive the existence of solutions to semilinear generalized Euler‐Poisson‐Darboux equation with a negative parameter α∈(−1,0).

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On singular and regular cauchy problems

Diaz

1956

Comm Pure Appl Math

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Singular Cauchy Problem for the Euler-Poisson-Darboux EquationConsider the partial differential equation Cauchy problem for (1) consists in the determination of a solution of (l), for t > 0, meeting the following initial conditions on the "singular" plane t = 0:where g is a given function. For k any real number, this singular Cauchy problem was first solved by A. Weinstein [7], who employed what he termed the "method of recurrence" and a generalized method of descent. Let, for the time being, the solution of the Cauchy problem (l), (2) be denoted by Z J~. Weinstein shows that for k = -1, -3, * * -the Cauchy problem for uk can be reduced to a Cauchy problem for u k + 2 by means of the following recurrence formulas:

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On a regular cauchy problem for the Euler-Poisson-Darboux equation

Cited by 13 publications

References 7 publications

An Operator Approach to the Cauchy Problem for the Euler-Poisson-Darboux Equation in Spaces of Constant Curvature

An Operator Approach to the Cauchy Problem for the Euler-Poisson-Darboux Equation in Spaces of Constant Curvature

The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane

On singular and regular cauchy problems

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