Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [heat-equation]

For questions related to the solution and analysis of the heat equation.

The heat equation is a particular parabolic partial differential equation used to describe the temperature or heat distribution of a system over time. It can be written most generally as

$$\frac{\partial u}{\partial t} - \alpha \nabla^2 u = 0$$

where $\nabla^2$ is the Laplace operator, and $\alpha$ is a positive constant describing thermal diffusivity (which is usually normalized to $1$).

There are a number of common solution techniques, including separation of variables and Fourier series, as well as using a Green's function to find a fundamental solution.

Reference: Heat equation.

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The heat kernel as a distance metric on manifolds

I recently came across Varadhan's formula (see e.g. [1], [2], [3], [4], [5]): $$ {d_{\text{g}}(x,y)^2}{} = -\lim_{t \rightarrow 0} 4 t \log K_t(x,y) $$ where $d_\text{g}$ is the geodesic distance and $K_t$ is the heat kernel (i.e. the…
user3658307
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Energy in the heat equation.

Before getting to question, some background. Let $u(x,t)$ be the temperature in a laterally insulated rod of length $L$, at position $x$ and time $t$. The temperature satisfies the heat equation $\partial_t u = \alpha \, \partial^2_x u$, where…
Curiosity
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Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: $$K(t,x,x)\sim (4\pi…
asd
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Interpretation of an integral transform from the wave equation to the heat equation

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ \mathbb{R}^n\times(0,\infty)\\u=g,\ \partial_tu=0\ \mathrm{on}\…
Maximilian M.
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Intuition behind the paradox of instantaneous heat propagation

On this MIT lecture, the difference between the heat equation and the wave equation includes signal travelling infinitely fast in the heat equation, while it has finite speed in the wave equation: I guess I don't get the idea of "signal" because in…
Antoni Parellada
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Theoretical link between the graph diffusion/heat kernel and spectral clustering

The graph diffusion kernel of a graph is the exponential of its Laplacian $\exp(-\beta L)$ (or a similar expression depending on how you define the kernel). If you have labels on some vertices, you can get labels on the rest of the vertices by a…
highBandWidth
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The Heat Equation in Brezis' book

I am reading the heat equation in Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis, and having some concerns about the proof, whose screenshot is as attached below. The notation $u\in L^2 (0, \infty;…
Justin Lien
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Heat Equation on Manifold

Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold. So is there any analogy to heat equation or wave equation on manifold? And what book is recommended for…
gaoxinge
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Why heat equation is not time-reversible? (Time arrow in mathematics)

Inspired by a question I asked here, I am rethinking about a question: Why heat equation is not time-reversible? I don't know too much about PDE and physics but I guess there should be some "time arrow" in mathematics. Consider the following…
wh0
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Uniqueness of the heat equation for initial data in $L^\infty$

For $1 \leq p < \infty$ and initial-data $u _ 0 \in L ^p ( \mathbb{R} ^d)$ due to P. Li (Uniqueness of $L^1$ solutions for the Laplace equation and the heat equation on Riemannian manifolds) there exists a unique solution $u$ of the heat equation…
Zahlenteufel
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The backwards heat equation is not well posed

I have some sort of issue on this problem. Here it is: Show that the backwards heat equation, $\frac {\partial u}{\partial t} = -k \frac {\partial^2 u}{\partial x^2}$, subject to $u(0,t) = u(L,t) =…
user179766
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Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in D(A)$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function $U:[0,\infty)\to X$ given by $U(t)=T(t)U_0$ is a…
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Noether's theorem in the critical heat equation

I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met recall the setting. We are studying the following…
Falcon
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What physical information does the mean value property of heat equation convey?

I'm reading through the Evans' book on PDE, the chapter on heat equation. The definitions are the same as here. I see that mean value property of heat equation is useful for proving maximum principle and various uniqueness results, but I'm curious…
ante.ceperic
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heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way close to the heat kernel in $\mathbb{R}^n$: that…
Cristopher Moore
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