# Tagged Questions

For questions related to the solution and analysis of the heat equation.

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### Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
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### Heat Kernel in CD(k,N)/RCD(k,N) spaces

I'd like to know if some results are known concerning the heat kernel in CD(k,N) or RCD(k,N) spaces. I am mainly interested in two points: (gaussian) estimates asymptotic behavior when time goes to ...
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### Solve the initial value problem for this inhomogeneous heat equation.

I'm trying to solve this IVP for heat equation, $$u_t-\frac{1}{4}u_{xx}=e^{-t}~~\text{ in }-\infty<x<\infty,~t>0,$$ $$u(x,0)=x^2.$$ By the superposition principle, the solution should equal ...
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### Does anyone recognise this non-linear diffusion equation?

I'm doing some work on modelling cell migration, I've derived this particular form of a non-linear diffusion equation to describe the mean behaviour of a stochastic model I'm studying. I was wondering ...
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### Understanding the notation $\nabla u \otimes \nabla u$

On a Riemannian manifold $(M,g)$ let $u = u(t,x)$ the solution to the heat equation $\partial_t u = \frac 12 \Delta u$. The Laplace-Beltrami operator etc. are taken with respect to the metric $g$. I'...
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### A comparison principle for a nonlinear parabolic PDE

We know the following comparison principle holds for the diffusion equation: Suppose that $u(x,t)$ and $v(x,t)$ satisfy \begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \...
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### Explicit heat kernels

For quite general domains, the Dirichlet heat kernel has a representation via the eigenfunctions of the corresponding Dirichlet problem. This form is usually not easy to analyse so I was wondering - ...
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### Is the solution to the heat equation always $C^k$, no matter what the boundary condition is?
Let $\Omega$ be a bounded Lipschitz domain (can be smooth if necessary). Consider the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ $$\text{some Robin boundary condition (B) for u}$$ We know ...