# Questions tagged [heat-equation]

The heat-equation tag has no usage guidance.

182
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### Representation of heat kernel in general domains

I'm looking for results on the representation of the heat kernel in general domains. If $e^{-\Delta_{\Omega} t}$ denotes the heat semigroup in $\Omega$, we have to
$$ (e^{-\Delta_{\Omega} t}f)(x) = \...

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### Sharp Dirichlet heat kernel estimates in exterior domains?

I am right now working on some linear parabolic problems studying the behaviour of its solutions for large initial data. To do this, I have needed to use some estimates of the Dirichlet and Neumann ...

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### increasing property of the heat equation on the interval

$Q_T^l=\{(x,t); 0<x<l, 0<t\leq T\}$,
$u_l\in C^{2,1}(Q_T^l)\cap C(\bar Q_T^l)$ is the solution of $\frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l$...

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### Unique solutions to the heat equation on $\mathbb{R}^3$

Pierre-Gilles Lemarie-Rieusset, The Navier-Stokes Problem in the 21st Century treats the heat equation on $\mathbb{R}^3$ for time $t\geq 0$, and proves uniqueness of suitably smooth solutions by a ...

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### What do higher order diffusion terms do?

I have been trying to learn to work with the Python module FiPy, which is supposed to solve PDEs of the form
$$
\frac {\partial(\rho \phi)} {\partial t} - [\nabla\cdot(\Gamma_i\nabla)]^n\phi - \nabla \...

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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
...

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### 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\...

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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 ...

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### $L^\infty-L^\infty$ 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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### Is there any way this property of semigroups can be satisfied?

Suppose you have the heat semigroup $(S(t))_{t>0}$, such that
$$S(t)u(x) = (4\pi t)^{-n/2}\int_{\mathbb{R}^n}e^{-|x-y|^2/4t}u(y)dy.$$
The semigroup has the property that
$$S(t)S(s)u(x) = S(t+s)u(x)....

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### A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"

Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...

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### $\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains

I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary,
$$
\left\{
\begin{...

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1
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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 ...

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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 $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh(s)-\cosh(x)}}.$$...

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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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### Compactness for initial-to-final map for heat equation

Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation
$$\...

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### Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...

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### Covariance of an exclusion process

In Erhard and Hairer's recent paper, they say that the covariance of exclusion process is given by the discrete Heat Kernel (page 61, paragraph following equation 4.11). I have not been able to make ...

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### $L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...

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### Li-Yau inequality on $\mathbb R^2$ for functions that are somewhat close to $1$

Let $u:\mathbb R^2\times \mathbb R_{>0}\to \mathbb R_{>0}$ be a positive solution to the heat equation on $\mathbb R^2$ ($u_{xx}+u_{yy}=u_t$, no constants). The Li-Yau inequality in this case ...

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### How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?

I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...

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### Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality

I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...

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### Estimates for the heat equation with inhomogeneous boundary condition

EDIT2: I believe the estimate required is false. There is some evidence that I have added to this post. I only believed it was true because it seemed like it was used in certain papers. However, in ...

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### Lower Gaussian estimates for Dirichlet heat kernel on manifolds

Let $(M,g)$ be a Riemannian $n$-manifold with $Ric_g\ge -Kg$, $\Omega\subset M$ be an open subset. We can define Dirichlet heat kernel on $\Omega$, $p_{\Omega}(y,t,y',t')$ as the minimal fundamental ...

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### Martingales associated with heat equation

I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...

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### Uniqueness of solution to heat equation when initial condition is a generalized function

Let $u(t,x)$ be a solution to the heat equation $$\partial_t = \partial_{xx} \quad (t,x) \in [0,T) \times [-1,1]$$ subject to the initial/boundary conditions
$$u(0,x) = f(x), \quad x \in [-1,1], \\
u(...

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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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### How to solve a differential equation in the form $\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$?

How to find the general solution of a differential equation with a shift, in the following form?
$$\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$$
where $\...

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### Laplacian coupled with another equation over a two-dimensional rectangular region

I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is:
$$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$
where it is known that $t(x=0)=t_i$.
The ...

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### heat kernel Asymptotic expansion on manifolds with boundary or manifolds with conical singularities

This is a similar question to Heat Kernel Asymptotics on Manifold with Boundary. But I have some further questions.
Let $(M,g)$ be a closed Riemannian manifold, the heat kernel $p(x,y,t)$ of Laplace-...

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### Heat Equation Boundary Value Problem - alternative expressions for solution

Let $B_t$ be a Brownian motion, with with density function $f(t,x)dx = P(B_t \in dx)$. Then $f$ solves the heat equation $\partial_t = \frac{1}{2} \partial_{xx}f(t,x)$. Let for a fixed $u > 0$, $\...

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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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### Gaussian bounds with exponential decay for discrete (graph) Dirichlet heat kernel

Let $\Omega$ be a finite, connected subset of $\mathbb{Z}^n$, $W_t$ a standard random walk on $\mathbb{Z}^n$ started at $x$, and $T_\Omega$ the first time at which $W_t$ leaves $\Omega$; consider
$$
P^...

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### McKean-Singer formula in Heat Kernels and Dirac Operators book

I'm reading "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. The setting is: $E \to M$ is a $\mathbb{Z}_2$-graded vector bundle on a compact Riemannian manifold $M$
and $D :...

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1
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### Explicit solution of a free boundary problem for heat equation

Consider the free boundary problem
$$
\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\
u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\
u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...

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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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1
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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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1
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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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### Solution to Heat Equation By Projection

Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\...

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### Biharmonic heat flow on compact manifolds

Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...

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### Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients

Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...

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### Reference request: Long-term behaviour of the heat equation for bounded initial data

Let us consider the heat equation
\begin{align*}
\frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\
u(0,x) & = f(x)
\end{align*}
on the whole space $\mathbb{R^d}$. If $f \in L^p := L^...

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### The complex roots of $\left(\Gamma(\kappa s)+2^{1-s} \pi^{-s}\cos\left(\frac{\pi s}{2} \right)\Gamma(s)\,\Gamma(\kappa(1-s))\right)$

Question:
With $\kappa \in \mathbb{R}, \kappa \ge 1$, the function:
$$\Upsilon(s,\kappa) = \Gamma(\kappa s)\zeta(s)+\Gamma\left(\kappa(1-s)\right) \zeta(1-s)$$
could be rewritten as:
$$\Upsilon(s,\...

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### A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...

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### One dimensional heat equation with boundary conditions

Consider the heat equation
$$u_t = u_{xx}$$
for $t \ge 0$, $0 \le x \le L$, given boundary conditions
$$u(0,t) = u(L,t) = f(t)$$
and an initial condition
$$u(x,0) = g(x)$$
for some continuous ...

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1
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### Long time existence for heat flow in Corlette-Donaldson Theorem

I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14)
https://arxiv.org/pdf/1402.4203.pdf
For completeness, the statement is as follows.
...