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Initial-boundary value problems of nonlinear Euler-Poisson-Darboux equations. (Chinese) Zbl 0686.35094

The initial-boundary value problem of the nonlinear Euler-Poisson-Darboux equation \[ L[u]=\partial^ 2_{tt}u+(k/t)u-\partial^ 2_{xx}u- \partial^ 2_{yy}u=F(x,y,u), \]
\[ u(x,y,0)=0,\quad \partial_ tu(x,y,0)=0,\quad u(0,y,t)=g(y,t) \] is considered in this paper. The existence and the uniqueness of the solution \(u(x,y,t)\in C^ 2(D(X,Y,T)\cap \{x\geq 0,t\geq 0\})\) is obtained, where D(X,Y,T) is a characteristic cone.
To get the purpose, a formulation of a functional integral equation for the problem is derived and a fixed point theorem in Banach space is applied to it.
Reviewer: Z.Ding

MSC:

35Q05 Euler-Poisson-Darboux equations
35L70 Second-order nonlinear hyperbolic equations
47H10 Fixed-point theorems
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