Cauchy Problem for a Semilinear Nonstrictly Hyperbolic Equation on a Half-Plane in the Case of a Single Characteristic

Abstract

In the upper half-plane, we consider a semilinear nonstrictly hyperbolic partial differential equation of the nth-order, which arises in modeling of wave propagation in a layered medium, in the study of k-out-of-n systems, in classical field t heory. T he e quation h as a single characteristic with a multiplicity equal to n. The operator in the equation is a sum of linear and nonlinear parts. The linear part of this operator is a composition of the first-order differential operator with constant coefficients. The nonlinear part depends on independent variables and unknown function. The equation is equipped with the Cauchy conditions. We find the solution of this problem in the upper half-plane in an implicit analytical form in the case of two independent variables under some smoothness conditions on the initial data, the right-hand side, and nonlinearity, and sign and growth conditions on the nonlinearity.