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Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

  • Tianling Jin , Xavier Ros-Oton EMAIL logo and Jingang Xiong
Published/Copyright: March 26, 2024

Abstract

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time T * . More precisely, we show that solutions are C 2 , α ( Ω ¯ ) in space, with α = 1 m , and C in time (uniformly in x Ω ¯ ), for t > T * . Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence O ( t - 1 - γ ) , and we prove that the convergence holds in the C 1 , α ( Ω ¯ ) topology.

Award Identifier / Grant number: 801867

Award Identifier / Grant number: 101123223

Award Identifier / Grant number: 12122120

Award Identifier / Grant number: 12325104

Award Identifier / Grant number: 12271028

Funding statement: Tianling Jin was partially supported by NSFC grant 12122120, and Hong Kong RGC grants GRF 16306320 and GRF 16303822. Xavier Ros-Oton was partially supported by the European Research Council (ERC) under the Grant Agreements No 801867 and No 101123223, the AEI project PID2021-125021NA-I00 (Spain), the MINECO grant RED2018-102650-T (Spain), and the Spanish State Research Agency, through the María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). Jingang Xiong was partially supported by NSFC grants 12325104 and 12271028.

Acknowledgements

All authors would like to thank Matteo Bonforte, Beomjun Choi and Juan Luis Vázquez for interesting discussions and comments. Finally, we thank the anonymous referee for his/her careful reading of the paper and for invaluable suggestions that greatly improved the presentation of the paper.

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Received: 2022-11-11
Revised: 2024-02-19
Published Online: 2024-03-26
Published in Print: 2024-04-01

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