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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 …
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 …
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 …
Matthias Ludewig's user avatar
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 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, …