Given a square matrix A, the first order system d/dt u(t) = A u(t) can be solved by the matrix exponential: u(t) = exp(t A) u(0). In particular, as a function of t this gives rise to a one-parameter group.
Similarly, partial differential equations (e.g., the heat equation) fit within this framework if the matrix A is replaced by a suitable operator. For the heat equation, it is the Laplace operator. However, since the Laplace operator is not bounded, the series for the exponential will not converge and this raises the question how the exponential should be defined in this case. Moreover, solutions of the heat equation exist in general only for positive times and the solution operator can form at best a semigroup.
This course aims at developing a solid mathematical foundation to this approach. As a byproduct, we obtain a very general solution concept for certain evolution equations and a framework to characterize long-term behavior of the solutions.
K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations. Springer. 2000.
A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer. 1983.
wöchentlich, ab 21.04.2022, 08:00 - 09:30 (13 Termine)