Classical solutions of hyperbolic IBVPs with state dependent delays on a cylindrical domain

Abstract

We consider the initial boundary value problem for a nonlinear partial functional differential equation of the first order ∂tz(t,x)=f(t,x,V(z;t,x),∂xz(t,x)), where V is a nonlinear operator of Volterra type, mapping bounded subsets of the space of Lipschitz-continuously differentiable functions, into bounded subsets of the space of Lipschitz continuous functions with Lipschitz continuous spatial partial derivatives. Using the method of bicharacteristics and successive approximations, we prove the local existence, uniqueness and continuous dependence on data of classical solutions of the problem. This approach covers equations of the form ∂tz(t,x)=f(t,x,zα(t,x,z(t,x)),∂xz(t,x)), where (t,x)↦z(t,x) is the (multidimensional) Hale operator, and all the components of α may depend on (t,x,z(t,x)). More specifically, problems with deviating arguments and integro-differential equations are included.