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Takahashi Lab
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The lab uses mathematics to investigate the singularities of solutions to equations involving diffusion.
AboutAbout the lab
Singularities of solutions to nonlinear partial differential equations
The Takahashi Lab uses mathematical methods to study the singularities of solutions to partial differential equations involving diffusion.
It is particularly interested in a class of equations known as nonlinear parabolic equations, which involve time evolution.
The lab deals with cases in which solutions depart from ordinary behavior, such as blow-up solutions and singular solutions.
Key terms
- Partial differential equation
- An equation that expresses the relationships among the ways a quantity changes with position and time
- Nonlinear parabolic equation
- Among equations describing the diffusion of heat or matter, one in which the change is not a simple proportional relationship
- Blow-up solution
- A solution whose value grows without bound as time passes
ResearchResearch themes
The lab uses mathematical methods to study the singularities of solutions to partial differential equations involving diffusion. In particular, it is interested in equations with time evolution that belong to the class known as nonlinear parabolic equations.
*Illustrative imageWhen Solutions of the Heat Equation Blow Up
When a strongly nonlinear term is added to the equation describing the spread of heat, the value of a solution can "blow up" to infinity in finite time. The lab mathematically investigates how solutions break down, including the rate of blow-up and cases in which blow-up takes infinite time.
*Illustrative imageSolutions of Equations with Moving Sharp Points
In the p-Laplace equation, which describes flow and diffusion, points where the slope turns sharply pointed can appear in the shape of a solution. The lab constructs solutions in which such a sharp point moves over time, and clarifies the behavior of the singularity.
MembersFaculty
- Associate ProfessorJin TakahashiPublications ↗
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Heat” Nicoguaro. Based on File:Heat eqn.gif by en:User:Oleg Alexandrov / CC BY 4.0 (cropped)
- When Solutions of the Heat Equation Blow Up: “Heat cylinder” Nicoguaro / CC BY 4.0 (cropped)
- Solutions of Equations with Moving Sharp Points: “Bézier curve with cusp” KSmrq / CC BY-SA 3.0 (cropped)
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