Questions tagged [heat-equation]

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0answers
34 views

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 ...
3
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0answers
56 views

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 ...
4
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214 views

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 ...
2
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1answer
71 views

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 ...
1
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0answers
136 views

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, ...
4
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3answers
326 views

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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0answers
74 views

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 ...
2
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0answers
41 views

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 ...
2
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1answer
68 views

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 $\...
2
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0answers
89 views

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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147 views

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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35 views

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 $u > 0$, $\...
1
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1answer
43 views

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 ...
2
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1answer
98 views

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^...
5
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1answer
181 views

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 :...
2
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1answer
82 views

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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0answers
52 views

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: ...
5
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1answer
232 views

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 ...
3
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1answer
203 views

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} = \...
2
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1answer
273 views

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 ...
3
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1answer
376 views

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 ...
2
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0answers
102 views

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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144 views

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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80 views

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 ...
5
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1answer
353 views

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 ...
5
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3answers
236 views

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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97 views

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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64 views

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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3answers
730 views

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 ...
5
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1answer
248 views

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. ...
1
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1answer
77 views

A question about positivity preserving property of semigroup of Laplacian

Sry this the second question from the following article, I am asking in this week. At page 6 (126), 3th line, of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say ...
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120 views

Diffusion equation solution using Laplace transform [closed]

Consider the operator $$ L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t} $$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
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0answers
156 views

Convolution and approximate heat kernel

I have the following question: Let $A, B\in C^\infty((0,\infty)\times M\times M)$, where $M$ is a compact manifold. Define \begin{align} (A*B)(t,x,y) = \int_0^t\int_M A(t-s,x,z)B(s,z,y)dzds \end{align}...
2
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0answers
213 views

Feynman-Kac formula for domains with boundary

As far as I know (I am not an expert), any solution of the heat equation on $\mathbb{R}^n$ or on a closed Riemannian manifold with the given initial condition can be presented in terms of stochastic ...
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0answers
160 views

Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$. It is known that $S(t)$ ...
3
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2answers
203 views

Gaussian upper heat kernel bounds on closed Riemannian manifolds

Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists short time upper Gaussian heat kernel bounds of the following kind: $$ h(t, x, y) ...
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0answers
94 views

May the heat kernel of a connection Laplacian vanish?

Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...
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175 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D Laplace equation: $$\nabla^{2} T_w = 0$$ where $\nabla^{2}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [0,...
3
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0answers
135 views

large time behavior for the Neumann problem for the heat equation

I am interested in the large time asymptotic behavior of parabolic equations. For instance, let $\Omega$ be a regular bounded open subset of $\mathbb{R}^n$ . Let $g \in C(]0,+\infty[ \times \partial\...
2
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1answer
268 views

Bounded solution for parabolic equation

Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\...
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0answers
228 views

Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
3
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0answers
97 views

Ratio of solutions to two heat equations

Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
2
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0answers
148 views

Fundamental solution of parabolic PDE with variable coefficients

Let us consider the parabolic operator $$ \mathcal{L} = \partial_t - \nabla_x \cdot(a(x)\nabla_x) $$ over a bounded domain $\Omega\subset\mathbb{R}^d$. The coefficient matrix $a(x)$ is elliptic and ...
3
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0answers
105 views

The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....
2
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1answer
456 views

Heat kernel on Riemannian manifold

The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a ...
4
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1answer
116 views

Quick question on the constants involved in heat kernel upper bounds

Let $M$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the Laplace-Beltrami operator on $M$. It is known that for small $t$, let's say, $0< t < t_0(M, g)$, the heat ...
13
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1answer
404 views

One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
3
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1answer
154 views

Gradient blowup for the 1-dim heat equation near an irregular boundary point

I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point. Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...
0
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1answer
154 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
0
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1answer
335 views

Heat kernel and convergence

Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...