On the initial value problem for a partial differential equation with operator coefficients

Mahmoud M El-Borai

doi:10.1155/s0161171280000075

Abstract

In the present work it is studied the initial value problem for an equation of the form urn:x-wiley:01611712:media:ijmm267936:ijmm267936-math-0001 where L is an elliptic partial differential operator and (Lj : j = 1, …, k) is a family of partial differential operators with bounded operator coefficients in a suitable function space. It is found a suitable formula for solution. The correct formulation of the Cauchy problem for this equation is also studied.

Highlights

(Lj where L is an elliptic partial differential operator and j i, k) is a family of partial differential operators with bounded operator coefficients in a suitable function space
It is convenient to introduce the following notations in order to complete the proof by considering the problem in a Banach space to be defined below
Let B denote the space of column vectors V, with the norm k where lfl f2 (x) dx

Summary

Introduction

(Lj where L is an elliptic partial differential operator and j i, k) is a family of partial differential operators with bounded operator coefficients in a suitable function space. It is assumed in equation (1.1) that the following conditions are satisfied; (aq(t), (a) The coefficients lql 2m) are continuous functions of t in [0,I]. N) L2(En) are linear bounded operators from L 2 E n (d) The operators (Aq,j(t), lql 2m, j i k), are strongly continuous in t[0,1].

Results

Conclusion