[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

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

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aida, M., Tsujikawa, T., Efendiev, M., Yagi, A., Mimura, M.: Lower estimate of the attractor dimension for a chemotaxis growth system. J. Lond. Math. Soc. 74(2), 453–474 (2006)

    Article  MathSciNet  Google Scholar 

  2. Andreu, F., Mazón, J.M., Simondon, F., Toledo, J.: Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source. Math. Ann. 314(4), 703–728 (1999)

    Article  MathSciNet  Google Scholar 

  3. Andreu, F., Mazón, J.M., Simondon, F., Toledo, J.: Blow-up for a class of nonlinear parabolic problems. Asymptot. Anal. 29(2), 143–155 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Aronson, D., Weinberger, H.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)

    Article  MathSciNet  Google Scholar 

  5. Aronson, D.G.: The Porous Medium Equation, pp. 1–46. Springer, Berlin (1986)

    MATH  Google Scholar 

  6. Ball, J.M.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. Oxf. 28, 473–486 (1977)

    Article  MathSciNet  Google Scholar 

  7. Bandle, C., Brunner, H.: Blowup in diffusion equations: a survey. J. Comput. Appl. Math. 97(1–2), 3–22 (1998)

    Article  MathSciNet  Google Scholar 

  8. Cao, X., Zheng, S.: Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source. Math. Methods Appl. Sci. 37(15), 2326–2330 (2014)

    Article  MathSciNet  Google Scholar 

  9. Fellner, K., Latos, E., Pisante, G.: On the finite time blow-up for filtration problems with nonlinear reaction. Appl. Math. Lett. 42, 47–52 (2015)

    Article  MathSciNet  Google Scholar 

  10. Ferreira, R., Groisman, P., Rossi, J.D.: Numerical blow-up for the porous medium equation with a source. Numer. Methods Part. Differ. Equ. 20(4), 552–575 (2004)

    Article  MathSciNet  Google Scholar 

  11. Ferreira, R., Groisman, P., Rossi, J.D.: Numerical blow-up for a nonlinear problem with a nonlinear boundary condition. Math. Models Methods Appl. Sci. 12(04), 461–483 (2002)

    Article  MathSciNet  Google Scholar 

  12. Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo 13, 109–124 (1966)

    Google Scholar 

  13. Galaktionov, V .A.: A boundary value problem for the nonlinear parabolic equation \(u_{t}=\Delta u^{\sigma +1}+u^{\beta }\). Differ. Uravn. 17(5), 836–842 (1981)

    MATH  Google Scholar 

  14. Galaktionov, V.A.: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. R. Soc. Edinb. Sect. A 124(3), 517–525 (1994)

    Article  MathSciNet  Google Scholar 

  15. Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Samarskii, A.A.: Unbounded solutions of the Cauchy problem for the parabolic equation \(u_t=\nabla (u^\sigma \nabla u)+u^\beta \). Dokl. Akad. Nauk SSSR 252(20), 1362–1364 (1980)

    MathSciNet  Google Scholar 

  16. Galaktionov, V.A., Shmarev, S.I., Vázquez, J.L.: Behaviour of interfaces in a diffusion–absorption equation with critical exponents. Interfaces Free Bound. 2(4), 425–448 (2000)

    Article  MathSciNet  Google Scholar 

  17. Galaktionov, V.A., Vázquez, J.L.: Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison. Commun. Partial Differ. Equ. 19(7–8), 1075–1106 (1994)

    Article  MathSciNet  Google Scholar 

  18. Grant, C.: Theory of Ordinary Differential Equations. CreateSpace Independent Publishing Platform (2014)

  19. Gurtin, M.E., MacCamy, R.C.: On the diffusion of biological populations. Math. Biosci. 33(1–2), 35–49 (1977)

    Article  MathSciNet  Google Scholar 

  20. Kielhöfer, H.: Halbgruppen und semilineare Anfangs-Randwertprobleme. Manuscr. Math. 12(2), 121–152 (1974)

    Article  MathSciNet  Google Scholar 

  21. Kobayashi, K., Sirao, T., Tanaka, H.: On the growing up problem for semilinear heat equations. J. Math. Soc. Jpn. 29(3), 407–424 (1977). 07

    Article  MathSciNet  Google Scholar 

  22. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Mathematics and its Applications (Soviet Series), vol. 7. D. Reidel Publishing Co., Dordrecht (1987)

    Book  Google Scholar 

  23. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. In Translations of Mathematical Monographs, volume 23. American Mathematical Society (1988)

  24. Latos, E., Tzanetis, D.: Existence and blow-up of solutions for a semilinear filtration problem. Electron. J. Differ. Equ. 178, 1–20 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Levine, H.A.: The role of critical exponents in blowup theorems. SIAM Rev. 32(2), 262–288 (1990)

    Article  MathSciNet  Google Scholar 

  26. Li, Z., Peletier, L.: A comparison principle for the porous media equation with absorption. J. Math. Anal. Appl. 165(2), 457–471 (1992)

    Article  MathSciNet  Google Scholar 

  27. Marras, M., Piro, S., Viglialoro, G.: Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions. Kodai Math. J. 37(3), 532–543 (2014)

    Article  MathSciNet  Google Scholar 

  28. Payne, L., Philippin, G., Piro, S.V.: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition. II. Nonlinear Anal. Theory Methods Appl. 73(4), 971–978 (2010)

    Article  MathSciNet  Google Scholar 

  29. Payne, L., Philippin, G., Schaefer, P.: Blow-up phenomena for some nonlinear parabolic problems. Nonlinear Anal. Theory Methods Appl. 69(10), 3495–3502 (2008)

    Article  MathSciNet  Google Scholar 

  30. Payne, L., Philippin, G., Schaefer, P.: Bounds for blow-up time in nonlinear parabolic problems. J. Math. Anal. Appl. 338(1), 438–447 (2008)

    Article  MathSciNet  Google Scholar 

  31. Payne, L., Schaefer, P.: Lower bounds for blow-up time in parabolic problems under Dirichlet conditions. J. Math. Anal. Appl. 328(2), 1196–1205 (2007)

    Article  MathSciNet  Google Scholar 

  32. Payne, L., Schaefer, P.: Blow-up in parabolic problems under Robin boundary conditions. Appl. Anal. 87(6), 699–707 (2008)

    Article  MathSciNet  Google Scholar 

  33. Payne, L.E., Philippin, G.A., Proytcheva, V.: Continuous dependence on the geometry and on the initial time for a class of parabolic problems I. Math. Methods Appl. Sci. 30(15), 1885–1898 (2007)

    Article  MathSciNet  Google Scholar 

  34. Peletier, L., Terman, D.: A very singular solution of the porous media equation with absorption. J. Differ. Equ. 65(3), 396–410 (1986)

    Article  MathSciNet  Google Scholar 

  35. Philippin, G.A., Proytcheva, V.: Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems. Math. Methods Appl. Sci. 29(3), 297–307 (2006)

    Article  MathSciNet  Google Scholar 

  36. Porretta, A., Souplet, P.: Analysis of the loss of boundary conditions for the diffusive Hamilton–Jacobi equation. Ann. I. H. Poincare An. 34(7), 1913–1923 (2017)

    Article  MathSciNet  Google Scholar 

  37. Porretta, A., Souplet, P.: The profile of boundary gradient blowup for the diffusive Hamilton–Jacobi equation. Int. Math. Res. Not. 2017(17), 5260–5301 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984)

    Book  Google Scholar 

  39. Sacks, P.E.: Global behavior for a class of nonlinear evolution equations. SIAM J. Math. Anal. 16(2), 233–250 (1985)

    Article  MathSciNet  Google Scholar 

  40. Schaefer, P.: Lower bounds for blow-up time in some porous medium problems. Proc. Dyn. Syst. Appl. 5, 442–445 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Schaefer, P.: Blow-up phenomena in some porous medium problems. Dyn. Syst. Appl. 18, 103–110 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Souplet, P.: Finite time blow-up for a non-linear parabolic equation with a gradient term and applications. Math. Methods Appl. Sci. 19(16), 1317–1333 (1996)

    Article  MathSciNet  Google Scholar 

  43. Vázquez, J.: The Porous Medium Equation: Mathematical Theory. Clarendon Press, Oxford Mathematical Monographs (2007)

  44. Viglialoro, G.: Blow-up time of a Keller–Segel-type system with Neumann and Robin boundary conditions. Differ. Integral Equ. 29(3–4), 359–376 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Viglialoro, G.: Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source. Nonlinear Anal. Real World Appl. 34, 520–535 (2017)

    Article  MathSciNet  Google Scholar 

  46. Viglialoro, G., Woolley, T.: Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete Contin. Dyn. Syst. Ser. B. 23(8), 3023–3045 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Winkler, M.: Chemotaxis with logistic source: very weak global solutions and their boundedness properties. J. Math. Anal. Appl. 348(2), 708–729 (2008)

    Article  MathSciNet  Google Scholar 

  48. Winkler, M.: Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384(2), 261–272 (2011)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

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

Authors
  1. Tongxing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicola Pintus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giuseppe Viglialoro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Viglialoro.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-019-1130-2

Keywords

Mathematics Subject Classification

Search

Navigation