Abstract
This paper deals with the blow-up phenomena of classical solutions to porous medium problems, defined in a bounded domain of \(\mathbb{R}^{n}\), with n ≥ 1. We distinguish two situations: in the first case, no gradient nonlinearity is present in the reaction term contrarily to the other case. Specifically, some theoretical and general results concerning the mathematical model, existence analysis and estimates of the blow-up time t ∗ of unbounded solutions to these problems are summarized and discussed. More exactly, for both problems, explicit lower bounds of t ∗ if blow-up occurs are derived in the case n = 3 and in terms of an auxiliary function. On the other hand, in order to compute the real blow-up times of such blowing-up solutions and discuss their properties, a general resolution method is proposed and used in some two-dimensional examples.
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Acknowledgements
The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The author also gratefully acknowledges Sardinia Regional Government for the financial support (P.O.R. Sardegna, F.S.E. 2007–2013). This work is also supported by the research group IFQM315-Análisis Teórico y Numérico de Modelos de las Ciencias Experimentales, of the Department of Mathematics of the University of Cadiz (Spain).
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Appendix
Appendix
For completeness of the reader, we emphasize some details used in the proof of Theorem 3.
Proposition 1
Let the coefficients ci (i = 1,…,6) of Theorem 3 satisfy
Then there exits at least a ξ ∈ (0,∞) such that
Proof
For any ξ ∈ (0, ∞), function \(\varPhi (\xi ):= c_{5}\xi + c_{6}\xi ^{1-\frac{2(m+d)} {3\delta } }\) attains its minimum at the point
Therefore, since
and (32) holds, relation (33) is proven.
In addiction, let us give this
Remark 6
Relation (32) can be explicitly written as
being
Therefore, once m, p, q and Ω are fixed in (P 2), relation (32) is satisfied for k 2 (respectively, k 1) big (respectively, small) enough. Since k 2 is the coefficient associated to − | ∇u | q, which contrasts the explosion, and k 1 the one associated to u p, which stimulates it, this effect of coefficients k 1 and k 2 is coherent in terms of estimate (8). In fact, t ∗ increases when constants c 7 and c 8 decreasing, which in turn decrease with k 2 (respectively k 1) increasing (respectively, decreasing).
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Viglialoro, G. (2016). Explicit Blow-Up Time for Two Porous Medium Problems with Different Reaction Terms. In: Ortegón Gallego, F., Redondo Neble, M., Rodríguez Galván, J. (eds) Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-32013-7_9
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