A boundary value problem for quasilinear hyperbolic systems with a retarded argument. (English) Zbl 0658.35085

Kamont, Z.; Turo, J.

Ann. Pol. Math. 47, 347-360 (1987).

Consider the boundary value problem for systems of quasilinear hyperbolic differential equations with retarded argument: \[ (1)\quad \sum A_{ij}(x,y,z(x,y))[D_ xz_ j(x,y)+\sum^{m}_{k=1}\ell_{ik}(x,y,z(x,y),(z\circ \alpha)+(x,y)D_{yk}z_ j(x,y)]= \] \[ =f_ i(x,y,z(x,y),z\circ \beta)(x,y)), \] i\(=1,2,...,n\), \((x,y)\in D_{\alpha}\), \(y=(y_ 1,...,y_ m)\in R^ m\), \(m\geq 1\), \((z\circ \alpha)(x,y)=z(\alpha (x,y))\). P. Bassanini [Boll. Unione Mat. Ital., VI. Ser. B 1, 225- 250 (1982; Zbl 0488.35056)] studied the system (1). Basing on this work, a theorem of existence, uniqueness and continuous dependence on boundary data is proved for a.e. solutions of problem (1).

Reviewer: J.H.Tian

MSC:

35R10	Functional partial differential equations
35L70	Second-order nonlinear hyperbolic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

quasilinear; retarded argument; existence; uniqueness; continuous dependence