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Global solutions of semilinear first order partial functional differential equations with mixed conditions. (English) Zbl 1232.35184
Summary: We consider the initial-boundary value problem for a semi-linear partial functional differential equation of the first order, \[ \partial_t z(t,x)+\partial_x z(t,x) f(t,x) = F(t,x,z_{\alpha(t,x)}). \] Using the method of characteristics and the sequence of successive approximations, we prove the global existence, uniqueness and continuous dependence on data of classical \(C^1\) solutions of the problem. This approach covers equations with deviating variables, namely the above one, with right-hand-side \(\bar{F}(t,x,z(\alpha_0(t,x), \alpha_1(t,x), \dots, \alpha_n(t,x)))\), as well as integro-differential equations, where the last variable of \(\bar{F}\) is the integral \(\int_D z\left(\alpha_0(t,x)+s, \alpha_1(t,x)+y_1, \dots, \alpha_n(t,x)+y_n\right) ds\,dy\). The Gâteaux differential of the solution operator with respect to the initial-boundary data, or with respect to the pair: the data and the right-hand-side, is found.
35R10 Partial functional-differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35R09 Integro-partial differential equations
35F16 Initial-boundary value problems for linear first-order PDEs