# Tagged Questions

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### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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### Decay of Solutions to the Heat equation

Consider the heat equation $$(\partial_t + \Delta + V)u = 0$$ on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential. Consider the semigroup ...
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### Heat transfer: boundary conditions with fluid velocity

The following equation is considered: $$\frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f.$$ I have difficulties in formulating boundary conditions for this equation. If ...
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### Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...
The context: for my research I am currently looking at parabolic systems of the type  \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional assumptions. In ...