On infinite dimensional systems of conservation laws of Keyfitz-Kranzer type

We prove existence and uniqueness of strong generalized entropy solution to the Cauchy problem for an infinite dimensional system of Keyfitz-Kranzer type, in which the unknown vector takes its value in an arbitrary Banach space. We study the Cauchy problem for an equation ut + (φ(‖u‖)u)x = 0 (1) with initial condition u(0, x) = u0(x). (2) Here the unknown vector u = u(t, x) is defined in a half-plane Π = R+ × R, R+ = (0,+∞) and takes its values in some real Banach space X equipped with norm ‖ · ‖. We suppose that the function φ(r) ∈ C(R+) and rφ(r) → 0 as r → 0 + . (3) The initial function u0(x) ∈ L (R, X), i.e. it is an essentially bounded strongly measurable function on R. Remark that in the case when X = L(R) equation (1) can be written as the following integral-differential equation ∂ ∂t u(t, x, λ) + ∂ ∂x [