# Dimensional Bounds for Ancient Caloric Functions on Graphs

@article{Hua2019DimensionalBF, title={Dimensional Bounds for Ancient Caloric Functions on Graphs}, author={Bobo Hua}, journal={International Mathematics Research Notices}, year={2019} }

We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi’s theorem [9] on manifolds to graphs: for a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.

## 8 Citations

Ancient Caloric Functions on Graphs With Unbounded Laplacians

- MathematicsInternational Mathematics Research Notices
- 2020

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We extend Colding and Minicozzi’s theorem [12] on…

Harmonic functions of polynomial growth on infinite penny graphs

- MathematicsJournal of the London Mathematical Society
- 2022

For an infinite penny graph, we study the finite‐dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically…

Time analyticity of ancient solutions to the heat equation on graphs

- Mathematics
- 2019

We study the time analyticity of ancient solutions to heat equations on graphs. Analogous to Dong and Zhang [DZ19], we prove the time analyticity of ancient solutions on graphs under some sharp…

Discrete Harmonic Functions on Infinite Penny Graphs

- Mathematics
- 2020

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincare…

Counting Ancient Solutions on A Strip with Exponential Growth

- Mathematics
- 2020

We study the ancient solutions of parabolic equations on an infinite strip. We show that any polynomial growth ancient solution for a class of parabolic equations must be constant. Furthermore, we…

A note on time analyticity for ancient solutions of the heat equation

- MathematicsProceedings of the American Mathematical Society
- 2019

It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times…

Time analyticity for the heat equation and Navier-Stokes equations

- MathematicsJournal of Functional Analysis
- 2020

Time Analyticity for Inhomogeneous Parabolic Equations and the Navier–Stokes Equations in the Half Space

- Mathematics
- 2020

We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal…

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