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Course Overview
The school will be organized around three minicourses and several survey lectures.
Each minicourse will be accompanied by tutorial(s), where practical exercises and open problems
related to the topics of the lectures will be discussed.
In addition, there will be an opportunity for PhD students and young researchers to give short
presentations on the topic of their research.
Minicourses will be run in the morning while survey lectures, tutorials and participants presentations
will be arranged in the afternoons.
One of the afternoons will be reserved for the organized trip to Gower Peninsula and
Rhossili Bay, where participants could enjoy (weather permitting!) one of the most beautiful
coastal scenarios in Wales and the UK as well as a traditional pub dinner.
Detailed description of the minicourses
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Juan Luis Vázquez (Madrid, Spain)
Theory of Fast Diffusions
(slides used in the lectures)
The minicourse is aimed at presenting some of the recent progress in the mathematical theory
of nonlinear diffusion processes by focusing on the model called the fast diffusion equation.
Introduction to Linear and Nonlinear Diffusion.
Main models in problems of viscous fluids, phase change, water infiltration, population dynamics, and plasma physics.
The Fast Diffusion Equation and the Porous Medium Equation. Main features. Slow and fast propagation.
Existence of different types of solutions. Well-posedness. Existence in optimal classes of data. Cases of non-uniqueness and non-existence.
The role of critical exponents. Regularity of the solutions: smoothing effects and continuity.
Asymptotic behavior of the solutions for large time. Extinction. Special Selfsimilar solutions.
The geometrical models. The evolution flows: Yamabe (n=3)and Ricci (n=2).
New lines of research: p-Laplacians and fast diffusion.
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Marek Fila (Bratislava, Slovakia)
Blow-up of Solutions of Semi-linear Parabolic Equations
(notes used in the lectures)
For some parabolic equations, solutions may not exist globally but may become unbounded in finite time.
This phenomenon is called "blow-up" and it has been intensively studied in connection with various fields of science such as plasma physics,
combustion theory and population dynamics. We shall first discuss sufficient conditions on nonlinearities and initial functions which guarantee blow-up.
Then we shall pay attention to the structure of singularities that appear as solutions blow up.
More precisely, we describe the rate of blow-up, the location of blow-up points and the blow-up profile.
The last part of the course will be devoted to the question what happens after blow-up.
We shall discuss semi-linear and quasi-linear equations including the porous medium and evolutionary p-Laplace equations with sources.
In particular, we focus on the question how does the degeneracy of the equation affect the
blow-up behavior mentioned above.
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Victor Galaktionov (Bath, UK)
Cauchy Problem for Thin Films and other Nonlinear Parabolic PDEs
(slides used in the lectures: 1, 2, 3,
4-5
)
The course is concerned with mathematical concepts that are necessary to define a proper solution of the Cauchy problems
for higher-order degenerate parabolic PDEs. The main model is the classic fourth-order thin film equation (TFE-4) from lubrication theory
that describes distributions of thin film drops on rigid surfaces. This equation does not have any monotone or potential operator
so the classic theory of monotone operators does not apply. The concept of proper solutions and extended semigroup theory is based
on the idea of regular approximations of the TFE-4. It turns out that in a wide range of parameters, the solutions of maximal regularity
are those that change sign infinitely many times. All the results are illustrated by numerical experiments on the basis of MatLab
that produce a number of Figures justifying mathematical theory.
Survey Lectures
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