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# Questions tagged [heat-equation]

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### Comparison principle for porous medium equation in Fourier variables

Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u)$. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
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### Noether's theorem in the critical heat equation

I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me. I am watching a serie of lectures on "Blow up solution ...
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### The heat equation for complex time

Let $\Delta$ be a Laplacian or an elliptic operator over a manifold, can the heat equation be defined for complex time? Can we define: $$e^{-z \Delta}$$ for $Re(z)>0$ ? Also can the Ricci flow be ...
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### Fractional partial derivatives and integrals of De Bruijn's $H_t(z)$-function. Does a simpler form exist for the $z$ derivative/integral?

$\newcommand\KummerU{\text{KummerU}}$ $\newcommand\Hypergeom{\text{Hypergeom}}$ During 2018/2019, the polymath 15 project managed to successfully reduce the upper bound of the De Bruijn-Newman ...
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For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\... 2 votes 0 answers 61 views ### Representation of heat kernel in general domains I'm looking for results on the representation of the heat kernel in general domains. If$e^{-\Delta_{\Omega} t}$denotes the heat semigroup in$\Omega$, we have to $$(e^{-\Delta_{\Omega} t}f)(x) = \... 2 votes 2 answers 114 views ### Sharp Dirichlet heat kernel estimates in exterior domains? I am right now working on some linear parabolic problems studying the behaviour of its solutions for large initial data. To do this, I have needed to use some estimates of the Dirichlet and Neumann ... 0 votes 0 answers 47 views ### increasing property of the heat equation on the interval Q_T^l=\{(x,t); 0<x<l, 0<t\leq T\}, u_l\in C^{2,1}(Q_T^l)\cap C(\bar Q_T^l) is the solution of \frac{\partial u_l}{\partial t}-a^2\frac{\partial^2u_l}{\partial x^2}=f(x,t), (x,t)\in Q_T^l... 3 votes 1 answer 233 views ### Unique solutions to the heat equation on \mathbb{R}^3 Pierre-Gilles Lemarie-Rieusset, The Navier-Stokes Problem in the 21st Century treats the heat equation on \mathbb{R}^3 for time t\geq 0, and proves uniqueness of suitably smooth solutions by a ... 2 votes 1 answer 216 views ### What do higher order diffusion terms do? I have been trying to learn to work with the Python module FiPy, which is supposed to solve PDEs of the form$$ \frac {\partial(\rho \phi)} {\partial t} - [\nabla\cdot(\Gamma_i\nabla)]^n\phi - \nabla \... 2 votes 1 answer 178 views ### 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 ... 2 votes 0 answers 91 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 0 answers 59 views ### 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 ... 3 votes 0 answers 95 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 ... 1 vote 0 answers 61 views ### 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 $$S(t)S(s)u(x) = S(t+s)u(x).... 0 votes 0 answers 100 views ### A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation" Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126." On page 125, at the end of the proof of Theorem 4.3, I abstract ... 4 votes 0 answers 85 views ### \mathcal{C}^1(\overline{\Omega}) gradient bounds for the Dirichlet problem of the heat equation on general domains I am studying the heat equation on a general bounded domain \Omega \subset \mathbb{R}^+ \times \mathbb{R}^n with continuously differentiable Dirichlet data \phi on the boundary,$$ \left\{ \begin{... 1 vote 1 answer 173 views ### 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 1 answer 107 views ### 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 1 answer 217 views ### 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 ... 2 votes 1 answer 234 views ### 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 $$\... 0 votes 0 answers 159 views ### Has this form of the heat equation been solved for the radiation boundary condition Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ... 2 votes 0 answers 43 views ### Covariance of an exclusion process In Erhard and Hairer's recent paper, they say that the covariance of exclusion process is given by the discrete Heat Kernel (page 61, paragraph following equation 4.11). I have not been able to make ... 4 votes 0 answers 141 views ### L^\infty solutions for parabolic Neumann problem (heat equation) Consider the heat equation on a (smooth) domain in \mathbb{R}^n with homogeneous Neumann BCs:$$u_t - \Delta u = f\partial_\nu u = 0u|_{t=0} = u_0$$where f \in L^p(0,T;L^r(\Omega)) and ... 7 votes 0 answers 186 views ### 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 ... 3 votes 0 answers 93 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 votes 0 answers 76 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 votes 0 answers 295 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 votes 1 answer 167 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 vote 0 answers 348 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, ... 3 votes 3 answers 1k 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(... 1 vote 0 answers 91 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 votes 0 answers 47 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 ... 3 votes 1 answer 77 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$\...
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 ...
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-...