Existence results for a class of semi-linear initial value problems

Abstract

The main result of this thesis is an existence result for parabolic semi-linear problems. This is done by reformulating the semi-linear problem as an abstract Cauchy problem ut(t) = Au(t) + f(t; u(t)), t > 0 u(0) = u0 (1) for u0 2 X, where X is a Banach space. We then develop and use the theory of compact semigroups to prove an existence result. In order to make this result applicable, we give a characterization of compact semigroups in terms of its resolvent operator and continuity in the uniform operator topology. Thus, using the theory of analytic semigroups, we are able to determine under what conditions on A a solution to (1) exists. Furthermore, we consider the asymptotic behaviour and regularity of such solutions. By developing perturbation theory, we are easily able to apply our existence result to a larger class of problems. We then demonstrate these results with an example. This work is signi cant in providing a novel approach to a group of previously established results. The content can be considered pure mathematics, but it is of signi cant importance in real world situations. The structure of the thesis, and the choice of certain de nitions, lends itself to be easily understood and interpreted in the light of these real world situations and is intended to be easily followed by an applied mathematician. An important part of this process is to develop the problem in a real Hilbert space and then to consider the complexi cation of the problem in order to reset it in a complex Hilbert space, in which we can apply the theory of analytic semigroups. A large number of real world problems fall into the class of problems discussed here, not only in biology as demonstrated, but also in physics, chemistry, and elsewhere.