In this lecture we consider the shape and stability of a stationary solution of semilinear parabolic equations or systems which is also called reaction-diffusion equations or systems. The stripe pattern of a zebra and various patterns of shell can be described as these equations. These equations have a long history of research more than 80 years. Although these equations are time evolution equations, we consider stationary solutions. Since these equations appear as mode equations in physics, chemistry, and biology, we would like to study stable stationary solutions which can happen in nature. Problems are as follows for example: Does nontrivial stable steady state exist? If it exists, then what shape is it? Since a reaction-diffusion system is difficult to study, we reduce the system to a shadow system. We see that the shape of the stable steady state of a certain shadow system has a deep relation with the so-called hot spots conjecture, and show the conjecture under a certain assumption. We prove the nonlinear hot spots conjecture, which is a generalization of the standard hot spots conjecture, for some specific domains.
In the study of the stability of steady states of reaction-diffusion system we see that various general theorems in functional analysis are powerful tools.
・You can determine types of reaction-diffusion systems, which are for example cooperation-diffusion systems, competition-diffusion systems, activator-inhibitor systems, etc.
・You can derive a linearized eigenvalue problem from a original equation.
・You will be familiar with various properties of eigenfunctions.
・You will be familiar with various properties of nodal sets.
・You can give a precise statement of the nonlinear hot spots conjecture.
semilinear parabolic equation, semilinear elliptic equation, reaction-diffusion system, shadow system, stability, shape, linearized eigenvalue problem, hot spots conjecture, nonlinear hot spots conjecture
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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This is a standard lecture course. There will be an assignment at the last lecture.
Course schedule | Required learning | |
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Class 1 | Examples of reaction-diffusion equations and systems are given. We recall known results and see how the solution behavior depends on the type of the equation. We show that every stable steady state of a scalar reaction-diffusion equation is constant if the Neumann boundary condition is imposed and if the domain is convex. We also see that the same result holds for a certain class of reaction-diffusion systems. When the system does not belong to this class, a nontrivial stable steady state may exist. We will see that it actually occurs for activator-inhibitor systems. We introduce the shadow system, and then we consider the shape of the stable steady state. Using techniques used there, we show that the hot spots conjecture holds for a certain class of planar convex domains. | Details will be provided during each class session. |
None required.
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Assignments (100%)
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