# Questions tagged [heat-equation]

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### Feynman-Kac formula with non-zero boundary condition

Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by ...
65 views

### Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition

$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$ \begin{cases} \partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
1 vote
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### Where can I find a calculation of the asymptotic expansion of the heat kernel of the square of a Dirac operator?

Please note that I am not asking for a reference on asymptotic expansions in general (this was answered here), but rather for the concrete computation of the asymptotic expansion of a specific ...
79 views

### $L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
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1 vote
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### Heat kernel on hyperbolic space of variable curvature

I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $-\kappa=-1$. Now I am trying to do ...
1 vote
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### Verify that a given function is a fundamental solution to the heat equation on the hyperbolic plane [closed]

A few days ago (see this), I asked a question regarding the derivatives of the function $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh(s)-\cosh(x)}}.$$...
1 vote
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### Partial derivative of the heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can ...
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1 vote
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### question about the book "Holomorphic Morse Inequalities" by Marinescu-Ma

Could somebody please explain to me the proof of proposition 1.6.4 (page 52) in the book "Holomorphic Morse Inequalities" by Marinescu-Ma? I am completely lost. One point that is really ...
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### Mixed boundary value problems for Heat equation

This might be a very simple question, but basically I am looking for a good reference for studying the heat equation on Riemannian manifolds with boundary, specifically when data is put on lateral ...
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1 vote
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### Gaussian bounds for discrete (graph) Dirichlet heat kernel

(this is an attempt to refine a previous question; I was told that it would be better to create a new question than edit the previous one, I hope this is the correct ettiquete.) Let $\Omega$ be a ...
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1 vote
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### Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
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### Backward uniqueness for a heat equation with a drift

Consider heat equation with a drift (=reaction-diffusion equation) $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]$$ with periodic or ...
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### Ancient Heat equation and Liouville's theorem

I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation: \begin{equation} u_{t} = \...
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### The solutions of the heat equation from $0$ datum

Consider the initial value problem $$\partial_t u = \Delta u$$ $$u(0,x) = 0$$ for the heat equation in $\mathbb R^n$, where $u: [0,T] \times \mathbb R^n \to \mathbb R$ is a smooth solution up to the ...
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### Possible flaw in the proof of the Eells-Sampson theorem on harmonic maps in Nishikawa's book

Background: I am reading the book Variational Problems in Geometry by Seiki Nishikawa. The main purpose of this book is to prove the existence of harmonic maps $M\to N$ between two compact Riemannian ...
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### Heat-Flow on continuous differential forms and the Feller peroperty

Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...
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1 vote