Steuringsteorie vir evolusievergelykings

Abstract

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 operators