Penning, F.D.2024-11-272024-11-2721/11/251984http://hdl.handle.net/2263/99577Thesis (DSc)--University of Pretoria, 1984.Let Q be an open set in JRn; we assume Q to be bounded and to have an (n - 1) dimensional, infinitely differentiable boundary r such that Q is locally on one side of r. For each t E [0,T] we define the second order differential operator A(t) by A(t) = r ap(x,t)aPu with jpj..;;2 and Q = Q x ( 0 , T) . a E C00 (Q) p We also define the first order boundary operator B(t) by n B(t)u = r b.(x,t)a.u + b 0 (x,t)u with j=l J J r x (0 ,T). We assume A and B to satisfy the well-known compatibility relations of the theory of elliptic equations. In this thesis we consider the stability of the problem A(t)u(x,t) + atu(x,t) = f(x,t) in Q B(t)u(x,t) q(x,t) on r under small changes in the coefficients and right hand sides. We obtain conditions under which the solution of a perturbed problem converges to the solution of a fixed problem as mentioned above, when the coefficients and right-hand sides of the perturbed problem converges to that of the fixed problem. The function spaces in which the convergence takes place is defined in paragraph 2 of chapter 1. In chapters 2, 3 and 4 we use the method of semigroups and evolution operators to study the stability of the problem in which f and g = O. The case where A and B are dependant only on space variables are studied in chapter 2. In chapter 3 the operator A is also time dependant and in chapter 4 the operators A and B are both space and time dependant. The study of the non-homogeneous case is done in chapter 5 by the method of parabolic evolution operatorsafr© 2024 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.SteuringsteorieEvolusievergelykingsUCTDThesis