×

A mixed problem for the generalized Euler-Poisson-Darboux equation in an exceptional case. (English. Russian original) Zbl 0656.35109

Math. Notes 40, 548-551 (1986); translation from Mat. Zametki 40, No. 1, 87-92 (1986).
We investigate a certain mixed problem for the equation \[ (1)\quad L_{\alpha}u\equiv (x_ 0\partial^ 2_ 0+\alpha \partial_ 0-B)u=0, \] where \(B=\) \(\sum^{m}_{i=1}\)(x\({}_ i\partial_ i^ 2+\beta_ i\partial_ i)\), \((\alpha,\beta_ 1,...,\beta_ m)=const\), and also \[ (2)\quad \alpha \leq 1,\quad \alpha \in {\mathbb{Z}},\quad \beta_ i>0,\quad \beta_ i\in {\mathbb{R}},\quad i=1,...,m. \] We prove the uniqueness of the solution and write out its explicit representation. We note that in Q for \(\alpha\leq 0\), \(\alpha\in {\mathbb{Z}}\), \(\beta_ i=1/2\), \(i=1,...,m\), in the variables \(\eta_ i=2\sqrt{x_ i}\), \(i=0\), 1,...,m, Eq. (1) coincides with the exceptional case of the Euler-Poisson-Darboux equation \((k=1- \alpha)\).

MSC:

35Q05 Euler-Poisson-Darboux equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35C05 Solutions to PDEs in closed form
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Blum, ?The Euler-Poisson-Darboux equation in the exceptional cases,? Proc. Am. Math. Soc.,5, No. 4, 511-520 (1954). · Zbl 0056.09302 · doi:10.1090/S0002-9939-1954-0063543-0
[2] S. A. Tersenov, Introduction to the Theory of Equations Degenerate on the Boundary [in Russian], Novosibirsk Gos. Univ., Novosibirsk (1973). · Zbl 0289.35044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.