Abstract
Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened.
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Acknowledgements
DJA is partially supported by CNPq 311138/2019-5 and grant 2019/0014 Paraiba State Research Foundation (FAPESQ). RT is partially supported by FCT - Fundação para a Ciência e a Tecnologia, I.P., through projects PTDC/MAT-PUR/28686/2017 and UTAP-EXPL/MAT/0017/2017, as well as by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.
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Araújo, D.J., Teymurazyan, R. An optimal Liouville theorem for the porous medium equation. Arch. Math. 118, 427–433 (2022). https://doi.org/10.1007/s00013-022-01706-4
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DOI: https://doi.org/10.1007/s00013-022-01706-4