Evolution equations describe the time evolution of dynamical systems by an ordinary differential equation in a Banach space. We first investigate linear and autonomous (time invariant) problems. Based on this theory, we then consider nonlinear (more precisely, semilinear) equations. The solutions in the linear case are represented by a one-parameter semigroup of linear operators. In the semilinear case, the solution is given by the formula of variation of constants. For such operator semigroups there is a quite complete theory, which allows to study the properties of the underlying dynamical system. This approach essentially relies on functional analytic methods and results.
We treat the basic existence theorem for linear and semilinear autonomous evolution equations. In this framework, we then investigate qualitative properties of the solutions (e.g., the longterm behavior). These results can be applied to the diffusion, the wave and the (nonlinear) Schrödinger equation.
Knowledge of the lecture Functional Analysis is required. The necessary parts from the lecture Spectral Theory will be recalled (without proofs).
Further informations concerning this lecture you find in the Studierendenportal of the KIT at the URL
- 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.
(More literature will be given in the lecture.)