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The Cauchy problem with modified initial data for the generalized Euler- Poisson-Darboux equation. (English. Russian original) Zbl 0544.35076

Math. USSR, Sb. 48, 141-157 (1984); translation from Mat. Sb., Nov. Ser. 120(162), No. 2, 147-163 (1983).
The author considers the degenerate equation \[ \phi(y- \tau(x))\partial^ 2u/\partial x\partial y+a_ 1(x,y)\partial u/\partial x+b_ 1(x,y)\partial u/\partial y+c_ 1(x,y)u=f_ 2(x,y) \] near the line \(y=\tau(x)\) with the data \[ \lim_{y-\tau(x)\to +0}((y- \tau(x))^{\beta}u-C)=\psi(x)\quad or\quad \lim_{y-\tau(x)\to +0}(y- \tau(x))^{\beta}u=\psi(x)\quad etc. \] Under an appropriate condition he proves an existence and uniqueness theorem.
Reviewer: V.Isakov

MSC:

35Q05 Euler-Poisson-Darboux equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L80 Degenerate hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35R25 Ill-posed problems for PDEs
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