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Nonlocal Nonlinear Evolution Equations

Termine

Termine
Di
12:00 - 14:00 Uhr
Vorlesung
MA 850
Prof. Dr. Albert Erkip
Mi
16:00 - 18:00 Uhr
Vorlesung
MA 841
Prof. Dr. Albert Erkip
Sekretariat
MA 568
Alexandra Schulte

Beschreibung

In this series of lectures we aim to look at several classical examples as well as some recent results in relation to nonlocal evolution type equations.

The starting point will be to compare different approaches for proving the existence of solutions to initial value problems. To solve a nonlinear problem one usually starts by constructing a sequence of approximate solutions; hopefully the sequence or a subsequence of it will converge to a solution. Typical examples of the phenomena are observed in the case of Picard's Theorem versus Peano's Theorem for ODE's. These two theorems can also be regarded as fixed-point schemes corresponding to the Banach and the Schauder fixed-point theorems. Starting with a short discussion of ODE’s we will then concentrate on nonlinear PDE’s.

The plan for the lectures is as follows (each item may take several lectures):  

  1. A short look at ODE's: The Banach and the Schauder fixed point theorems.  Existence theorems for ODE’s.
  2. Local well posedness for initial value problems in PDE:

    1. Banach space valued ODE’s: Solving the initial value problem for the Improved Boussinesq and the BBM (Benjamin-Bona-Mahoney) equations.
    2. The semi group approach: Linear and nonlinear semigroups. Solving the initial value problem for the Boussinesq and the KdV equations.

  3. Examples of nonlocal equations and local well posedness of the corresponding initial value problems.
  4. Existence of global solutions. Energy estimates. Blow up results. Thresholds for global existence versus blow up.
  5. Variational methods and weak solutions.

Diese Veranstaltung bringt 10 ECTS.

Voraussetzungen

Differentialgleichungen I

Übungscheinkriterien und Prüfungsmodalitäten

Wöchentliche Übungsblätter sind zu bearbeiten, weitere Informationen werden in der Übung gegeben. Im Anschluss an die Vorlesung werden Termine für mündliche Prüfungen angeboten.

Literatur

Wird in der Vorlesung bekanntgegeben.

Zusatzinformationen / Extras

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