On a regularized porous medium equation

Giuseppe Maria Coclite; Lorenzo di Ruvo

doi:10.3934/dcdss.2022166

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2022. Doi: 10.3934/dcdss.2022166

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On a regularized porous medium equation

Giuseppe Maria Coclite^1, , and
Lorenzo di Ruvo^2,

1.
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, via E. Orabona 4, 70125 Bari, Italy
2.
Dipartimento di Matematica, Università di Bari, via E. Orabona 4, 70125 Bari, Italy

^*Corresponding author: Giuseppe Maria Coclite
^*Corresponding author: Giuseppe Maria Coclite

Received: April 2022

Revised: September 2022

Early access: September 2022

The authors are members of the Gruppo Nazionale per lAnalisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). GMC has been partially supported by the Research Project of National Relevance Multiscale Innovative Materials and Structures granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017J4EAYB and the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP - D94I18000260001)

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Abstract

We study a porous medium equation with dissipative and hyperdiffusive effects and no sign restrictions on the diffusion coefficient.

Keywords:

Local existence,
uniqueness,
stability,
the wave equation in dilatant granular materials,
Cauchy problem.

Mathematics Subject Classification: Primary: 35G25; Secondary: 35K55.

Citation:

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References

[1]	G. M. Coclite and L. di Ruvo, Existence results for the Kudryashov–Sinelshchikov–Olver equation, Proc. R. Soc. Edinb. Sect. A, 151 (2021), 425-450. doi: 10.1017/prm.2020.23.
[2]	G. M. Coclite and L. di Ruvo, Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects, Z. Angew. Math. Phys., 72 (2021), Article number: 68. doi: 10.1007/s00033-021-01506-w.
[3]	G. M. Coclite and L. di Ruvo, The porous medium equation with capillary pressure effects, Preprint.
[4]	M. E. Taylor, Partial Differential Equations I. Basic Theory, vol. 115 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.
[5]	J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.