2025 (Current Year) Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Nonlinear Diffusion Equations
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Jin Takahashi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue / 3-4 Fri
- Class
- -
- Course Code
- MCS.M431
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
Linear and nonlinear diffusion equations are typical example of parabolic partial differential equations and appear in various kinds of mathematical models of diffusion phenomena. In this course, we give mathematical methods for analyzing the behavior of solutions. In particular, we study analytical methods based on the scaling invariance structure of equations.
Course description and aims
The goal is to understand methods for analyzing the behavior of solutions of the linear diffusion equation (heat equation). Based on the method, we aim for analyzing nonlinear diffusion equations (porous medium equation and semilinear heat equation) in practice.
Keywords
Diffusion phenomena, heat equation, porous medium equation, semilinear heat equation
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
I will explain mathematical methods for analyzing the behavior of solutions of linear and nonlinear diffusion equations in a lecture format. Report assignments will be announced in several classes.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction to diffusion equation (heat equation), heat kernel | Understand the content of the class |
| Class 2 | Asymptotic behavior of solutions, proof of asymptotic formula by representation formula | Understand the content of the class |
| Class 3 | Scaling invariance, constructing sequence of functions based on scaling | Understand the content of the class |
| Class 4 | Compactness I, preparation for convergence of sequence of functions | Understand the content of the class |
| Class 5 | Compactness II, convergence of subsequence | Understand the content of the class |
| Class 6 | Characterization of limit function I, weak solution | Understand the content of the class |
| Class 7 | Characterization of limit function II, proof of asymptotic formula by scaling | Understand the content of the class |
| Class 8 | Introduction to porous medium equation, Barenblatt solution | Understand the content of the class |
| Class 9 | Asymptotic behavior of solutions I | Understand the content of the class |
| Class 10 | Asymptotic behavior of solutions II | Understand the content of the class |
| Class 11 | Introduction to semilinear heat equation, comparison principle | Understand the content of the class |
| Class 12 | Global-in-time solutions and blow-up solutions | Understand the content of the class |
| Class 13 | Blow-up rate of solutions | Understand the content of the class |
| Class 14 | Advanced topics | Understand the content of the class |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
Not specified.
Reference books, course materials, etc.
Nonlinear Partial Differential Equations (Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal), Basel: Birkhäuser, 2010.
Avner Friedman, Shoshana Kamin, The asymptotic behavior of gas in an n-dimensional porous medium. Trans. Am. Math. Soc. 262, 551--563 (1980).
Yoshikazu Giga, Robert V. Kohn, Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36, 1--40 (1987).
Evaluation methods and criteria
By reports.
Related courses
- MCS.T211 ： Applied Calculus
- MCS.T301 ： Vector and Functional analysis
- MCS.T304 ： Lebesgue Interation
- MCS.T311 ： Applied Theory on Differential Equations
Prerequisites
I require that you are familiar with calculus. I do not assume that you have knowledge of functional analysis, Lebesgue integration or (partial) differential equations.