Fundamentals of Semi-Group Theory

Abstract

As a motivation for the study of semi-groups of operators, we shall consider the equation of heat-conduction for an infinite rod

$$\frac{{\partial \omega }}{{\partial t}} = \frac{{{\partial ^2}\omega }}{{\partial {x^2}}}\quad \left( { - \infty < x < \infty ;\,t > 0} \right)$$

((1.0.1))

with the given initial temperature distribution

$$\omega \left( {x,0} \right) = f\left( x \right)$$

((1.0.2))

Here we restrict our discussion to the case where f(x) belongs to the space of all bounded, uniformly continuous real-valued functions defined on the real axis, in notation: f ∈ U⊂B(E₁).