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Properties of solutions to porous medium problems with different sources and boundary conditions

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Abstract

In this paper, we study nonnegative and classical solutions \(u=u(\mathbf{x},t)\) to porous medium problems of the type

where \(\Omega \) is a bounded and smooth domain of \({\mathbb {R}}^N\), with \(N\ge 1\), \(I=(0,t^*)\) is the maximal interval of existence of u, \(m>1\) and \(u_0(\mathbf{x})\) is a nonnegative and sufficiently regular function. The problem is equipped with different boundary conditions and depending on such boundary conditions as well as on the expression of the source g, global existence and blow-up criteria for solutions to (\(\Diamond \)) are established. Additionally, in the three-dimensional setting and when blow-up occurs, lower bounds for the blow-up time \(t^*\) are also derived.

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Acknowledgements

The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. GV is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and is partially supported by the research project Integro-differential Equations and Non-Local Problems, funded by Fondazione di Sardegna (2017). The research of TL is supported by NNSF of P. R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), and NSF of Shandong Province (Grant No. ZR2016JL021), DSRF of Linyi University (Grant No. LYDX2015BS001) and the AMEP of Linyi University. The authors would like to thank Professor Cornelis van der Mee for his fruitful remarks during the revision process of this paper.

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Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, People’s Republic of China

    Tongxing Li

  2. Dipartimento di Matematica e Informatica, Università di Cagliari, V. le Merello 92, 09123, Cagliari, Italy

    Nicola Pintus & Giuseppe Viglialoro

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  1. Tongxing Li
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  2. Nicola Pintus
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Correspondence to Giuseppe Viglialoro.

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Li, T., Pintus, N. & Viglialoro, G. Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70, 86 (2019). https://doi.org/10.1007/s00033-019-1130-2

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