# Tagged Questions

Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

1answer
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### Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
1answer
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### Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
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### Curvatures preserved under the Kahler-Ricci flow

Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not ...
0answers
73 views

### Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...
0answers
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### Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$ we have \sum_{i,j=1}^{n}a_{i}a_{j}\...
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### Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$

Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
2answers
292 views

### Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
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1answer
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### Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function. The classical set up is the following : If $(M,g)$ is a riemannian manifold then we could consider the ...
1answer
227 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$\begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases}$$ ...
1answer
222 views