Topics in nonlinear PDE theory 1000-1M21NRC
We will take a closer look at a canonical semilinear partial differential equation - the Lane-Emden equation - and its flow - the semilinear heat equation. While working with this well-researched model our aim is to discuss in some detail a selection of fundamental questions such as existence, uniqueness and regularity of solutions and to explain a set of core techniques used in this branch of analysis of PDEs. In particular we will encounter elements of:
- fixed point theory,
- mountain pass lemma,
- Friedrich’s abc method,
- smoothing estimates,
- space-time L^p L^q estimates and Marcinkiewicz interpolation theorem,
- heat semigroup techniques,
- scaling symmetries,
- energy method - variational identities,
- Liouville theorems.
Along classical results we will also encounter open problems which are a subject of ongoing research.
Main fields of studies for MISMaP
Type of course
Mode
Self-reading
Prerequisites (description)
Learning outcomes
Student:
- understands what kind of questions are being asked in the context of analysis of nonlinear partial differential equations.
- Is able to identify analytical tools suitable to address the associated research tasks and knows how to apply them to a model semilinear equation.
Assessment criteria
- attendance,
- oral exam.
Bibliography
Pavol Quittner, Philippe Souplet: Superlinear Parabolic Problems: Blow-up, Global Existence and
Steady States
Haim Brezis, Thierry Cazenave: A nonlinear heat equation with singular initial data, Journal d’analyze Mathematique, Vol. 68 (1996)
Fred Weissler: Existence and nonexistence of global solutions for a semilinear heat equation, Israel Journal of Mathematics, Vol. 38, Nos. 1-2, 1981.
Yoshikazu Giga, Robert V. Kohn: Asymptotically Self-similar Blow-up of Semilinear Heat Equations, Communications on Pure and Applied Mathematics, Vol. XXXVIII, 297-319 (1985)
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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