# Questions tagged [heat-equation]

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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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### 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 ...
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. ...
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### 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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### 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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### 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}...
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### 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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### 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 $S(t)$ ...