D. Jaruszewska-Walczak

Hyperbolic Differential Functional Equations with Unbounded Delay

abstract:
The present paper deals with the existence theory of initial and initial boundary value problems for the first order partial functional differential equations with unbounded delay. Strongly coupled quasilinear functional differential systems in the Schauder canonic form and nonlinear equations are considered. We give sufficient conditions for the existence, uniqueness and continuous dependence on data of generalized or classical solutions. In the case of quasilinear systems we apply the method of bicharacteristics. Existence theorems for nonlinear initial problems which are global with respect to spatial variables and for nonlinear mixed problems are proved by using the method of successive approximations. Results for nonlinear initial problems on the Haar pyramid are based on the fixed point method.

Mathematics Subject Classification: 35R10, 35L60, 35A07, 35B30

Key words and phrases: Initial problems, mixed problems, generalized solutions, classical solutions, bicharacteristics, successive approximations, Volterra condition, Schauder canonic form, Haar pyramid