On solutions of a system of hereditary and self-referred partial-differential equations

Abstract

In this paper, we present the local and global solutions of a system of hereditary and self-referred partial-differential equations. Namely, by the assumption on the Lipschitz continuity of the initial conditions u0, v0, Theorem 1 states the existence of local solutions of the problem (1.3–1.4); furthermore, under the assumption that those initial conditions are non-negative, non-decreasing, bounded, and lower semi-continuous functions, Theorem 2 gives global solution which is also a non-negative, non-decreasing, bounded, and lower semi-continuous function (in variable x of even for any time t).