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Results tagged with parabolic-pde
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user 16702
Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
2
votes
0
answers
103
views
Inhomogeneous heat kernel estimates
I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation
$$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$
where $\De …
- 9,853
4
votes
0
answers
95
views
One-dimensional harmonic map flow with low regularity
My question is the following:
What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in …
- 39.1k
5
votes
Accepted
Stochastic interpretation of heat kernel on fiber bundle
Let $P\longrightarrow M$ be a $G$ principal bundle endowed with a connection $1$-form $\omega$ (which has values in the Lie algebra $\mathfrak{g}$). If $X_t^x$ denotes Brownian motion on $M$ starting …
- 9,853
3
votes
1
answer
728
views
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 gene …
- 2,715
2
votes
0
answers
77
views
Well-posedness of a certain linear Cauchy-problem
I am interested in solutions to the linear Cauchy problem
$$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, …
- 9,853