A Limit Theorem for the Time-Dependent Evolution Equation

Abstract

The topic of the present lecture is slightly at variance with those treated in the lecture series of this course. The main theme has been the abstract differential equation

$$ \begin{array}{*{20}c} {\frac{{{\text{du}}}} {{{\text{dt}}}} = {\text{A}}\left( {\text{t}} \right){\text{u}}\,\,\,{\text{,}}} \hfill & {{\text{u}} \in {\text{X}},} \hfill & {{\text{t}} \geqslant 0} \hfill \\ \end{array} $$

((1))

where X is a Banach space, and where A(t) is a family of unbounded linear operators.

The main problem has been the solution of (1) under very weak conditions on A(t). Under suitable conditions, which we shall specify below, the solution u(t) of (1) which for t=0 satisfies the initial condition u(0) = f, where f ϵ X, can be expressed in the form

$$ {\text{u}}\left( {\text{t}} \right)\,\,{\text{ = }}\,\,{\text{P}}\left( {{\text{t,}}\,{\text{0}}} \right){\text{f}}\,\,\,{\text{,}} $$

((2))

where P(t, s) is a family of bounded linear operators satisfying the evolution equation

$$ \begin{array}{*{20}c} {{\text{P}}\left( {{\text{t,}}\,{\text{s}}} \right){\text{P}}\left( {{\text{s,}}\,{\text{r}}} \right)\,\,{\text{ = }}\,\,{\text{P}}\left( {{\text{t,}}\,{\text{r}}} \right)\,{\text{,}}} \hfill & {} \hfill \\ {} \hfill & {{\text{t,}}\,{\text{s,}}\,{\text{r}}\,\, \geqslant \,\,0} \hfill \\ {{\text{P}}\left( {{\text{t,}}\,{\text{t}}} \right)\,\,{\text{ = }}\,\,{\text{1}}{\text{.}}} \hfill & {} \hfill \\ \end{array} $$

((3))

Thus, we shall understand the expression “solving the initial value problem for the equation (1)” as meaning that a family of bounded linear operators P can be determined which satisfies (3) and which gives a solution of (1) by formula (2).