Weak solutions of functional differential inequalities with first-order partial derivatives

Zdzisław Kamont

doi:10.1186/1029-242x-2011-15

Abstract

The article deals with functional differential inequalities generated by the Cauchy problem for nonlinear first-order partial functional differential equations. The unknown function is the functional variable in equation and inequalities, and the partial derivatives appear in a classical sense. Theorems on weak solutions to functional differential inequalities are presented. Moreover, a comparison theorem gives an estimate for functions of several variables by means of functions of one variable which are solutions of ordinary differential equations or inequalities. It is shown that there are solutions of initial problems defined on the Haar pyramid.Mathematics Subject Classification: 35R10, 35R45.

Highlights

1 Introduction Two types of results on first-order partial differential or functional differential equations are taken into considerations in the literature
We consider initial problems which are local with respect to spatial variables
The Haar pyramid is a natural domain on which solutions of differential or functional differential equations or inequalities are considered

Summary

Introduction

Two types of results on first-order partial differential or functional differential equations are taken into considerations in the literature. They can be used for investigations of solutions to (1), (2). Suppose that Assumption H0 and H[s] are satisfied and (1) the estimate (14) holds on Ω for w ≤ w , (2) z(t, x) ≤ z(t, x)for (t, x) Î E0 and z(0, x) < z(0, x) for x Î [-b, b], (3) for each x Î [-b, b] the functional differential inequality (15) is satisfied for almost all t Î I[x].

Proof Let

Under these assumptions we have

Proof Let us write