Generalized solutions of mixed problems for first-order partial functional differential equations. (English) Zbl 1114.35146

Czernous, W.

Ukr. Mat. Zh. 58, No. 6, 804-828 (2006) and Ukr. Math. J. 58, No. 6, 904-936 (2006).

Summary: A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators.

MSC:

35R10	Partial functional-differential equations
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

first-order partial functional differential equation; generalized solution; mixed problem; continuous dependence; method of bicharacteristics; integral functional equations; method of successive approximations