Skip to main content
SpringerLink
Log in

Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly non-unique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Akagi, G.: Local solvability of a fully nonlinear parabolic equation. Kodai Math. J. 37, 702–727 (2014)

    Article  MathSciNet  Google Scholar 

  2. Akagi, G.: Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces. J. Evol. Equ. 11, 1–41 (2011)

    Article  MathSciNet  Google Scholar 

  3. Akagi, G., Kimura, M.: Unidirectional evolution equations of diffusion type. J. Differ. Equ. 266, 1–41 (2019)

    Article  MathSciNet  Google Scholar 

  4. Arai, T.: On the existence of the solution for \(\partial \varphi (u^{\prime }(t)) + \partial \psi (u(t)) \ni f(t)\). J. Fac. Sci. Univ. Tokyo Sec. IA Math. 26, 75–96 (1979)

    MATH  Google Scholar 

  5. Aso, M., Frémond, M., Kenmochi, N.: Phase change problems with temperature dependent constraints for the volume fraction velocities. Nonlinear Anal. 60, 1003–1023 (2005)

    Article  MathSciNet  Google Scholar 

  6. Aso, M., Kenmochi, N.: Quasivariational evolution inequalities for a class of reaction–diffusion systems. Nonlinear Anal. 63, e1207–e1217 (2005)

    Article  Google Scholar 

  7. Barbu, V.: Existence theorems for a class of two point boundary problems. J. Differ. Equ. 17, 236–257 (1975)

    Article  MathSciNet  Google Scholar 

  8. Barenblatt, G.I., Prostokishin, V.M.: A mathematical model of damage accumulation taking into account microstructural effects. Eur. J. Appl. Math. 4, 225–240 (1993)

    Article  MathSciNet  Google Scholar 

  9. Bertsch, M., Bisegna, P.: Blow-up of solutions of a nonlinear parabolic equation in damage mechanics. Eur. J. Appl. Math. 8, 89–123 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Bertsch, M., Dal Passo, R., Nitsch, C.: A system of degenerate parabolic nonlinear PDE’s: a new free boundary problem. Interfaces Free Bound 7, 255–276 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Bonetti, E., Schimperna, G.: Local existence for Frémond’s model of damage in elastic materials. Contin. Mech. Thermodyn. 16, 319–335 (2004)

    Article  MathSciNet  Google Scholar 

  12. Bonetti, E., Schimperna, G., Segatti, A.: On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differ. Equ. 218, 91–116 (2005)

    Article  MathSciNet  Google Scholar 

  13. Bonfanti, G., Luterotti, F.: Well-posedness results and asymptotic behavior for a phase transition model taking into account microscopic accelerations. J. Math. Anal. Appl. 320, 95–107 (2006)

    Article  MathSciNet  Google Scholar 

  14. Bonfanti, G., Frémond, M., Luterotti, F.: Global solution to a nonlinear system for irreversible phase changes. Adv. Math. Sci. Appl. 10, 1–24 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Bonfanti, G., Frémond, M., Luterotti, F.: Local solutions to the full model of phase transitions with dissipation. Adv. Math. Sci. Appl. 11, 791–810 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Brézis, H.: Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. In: Mathematical Studies, vol. 5. North-Holland, Amsterdam/New York (1973)

  17. Brézis, H.: Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis, pp. 101–156. Academic Press, New York (1971)

    Chapter  Google Scholar 

  18. Colli, P.: On some doubly nonlinear evolution equations in Banach spaces. Jpn. J. Ind. Appl. Math. 9, 181–203 (1992)

    Article  MathSciNet  Google Scholar 

  19. Colli, P., Visintin, A.: On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15, 737–756 (1990)

    Article  MathSciNet  Google Scholar 

  20. Frémond, M.: Non-smooth Thermomechanics. Springer, Berlin (2002)

    Book  Google Scholar 

  21. Gianazza, U., Savaré, G.: Some results on minimizing movements. Rend. Accad. Naz. Sci. XL, Mem. Mat. 112, 57–80 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Gianazza, U., Gobbino, M., Savaré, G.: Evolution problems and minimizing movements. Rend. Mat. Acc. Lincei IX 5, 289–296 (1994)

    Article  MathSciNet  Google Scholar 

  23. Gurtin, M.E.: Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1996)

    Article  MathSciNet  Google Scholar 

  24. Kachanov, L.M.: Introduction to Continuum Damage Mechanics. Martinus Nijhoff, The Hague (1986)

    Book  Google Scholar 

  25. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. In: Pure and Applied Mathematics, vol. 88. Academic Press Inc, New York (1980)

  26. Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23, 565–616 (2013)

    Article  MathSciNet  Google Scholar 

  27. Knees, D., Rossi, R., Zanini, C.: A quasilinear differential inclusion for viscous and rate-independent systems in non-smooth domains. Nonlinear Anal. Real World Appl. 24, 126–162 (2015)

    Article  MathSciNet  Google Scholar 

  28. Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Vol. I., Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften 181, Springer, New York (1972)

  29. Liu, Q.: Waiting time effect for motion by positive second derivatives and applications. Nonlinear Differ. Equ. Appl. NoDEA 21, 589–620 (2014)

    Article  MathSciNet  Google Scholar 

  30. Luterotti, F., Schimperna, G., Stefanelli, U.: Local solution to Frémond’s full model for irreversible phase transitions, Mathematical Models and Methods for Smart Materials (Cortona, 2001). Ser. Adv. Math. Appl. Sci. 62, 323–328 (2002)

    MATH  Google Scholar 

  31. Mielke, A., Rossi, R.: Existence and uniqueness results for a class of rate-independent hysteresis problems. Math. Models Methods Appl. Sci. 17, 81–123 (2007)

    Article  MathSciNet  Google Scholar 

  32. Mielke, A., Roubíček, T., Stefanelli, U.: \(\Gamma \)-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31, 387–416 (2008)

    Article  MathSciNet  Google Scholar 

  33. Mielke, A., Theil, F.: On rate-independent hysteresis models. Nonlinear Differ. Equ. Appl. NoDEA 11, 151–189 (2004)

    Article  MathSciNet  Google Scholar 

  34. Natalini, R., Nitsch, C., Pontrelli, G., Sbaraglia, S.: A numerical study of a nonlocal model of damage propagation under chemical aggression. Eur. J. Appl. Math. 14, 447–464 (2003)

    Article  MathSciNet  Google Scholar 

  35. Nitsch, C.: A nonlinear parabolic system arising in damage mechanics under chemical aggression. Nonlinear Anal. 61, 695–713 (2005)

    Article  MathSciNet  Google Scholar 

  36. Nitsch, C.: A free boundary problem for nonlocal damage propagation in diatomites. In: Free Boundary Problems, International Series of Numerical Mathematics, vol. 154, pp. 339–349. Birkhäuser, Basel (2007)

  37. Roubíček, T.: Nonlinear partial differential equations with applications. In: International Series of Numerical Mathematics, vol. 153. Birkhäuser, Basel (2005)

  38. Rocca, E., Rossi, R.: Entropic solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal. 47, 2519–2586 (2015)

    Article  MathSciNet  Google Scholar 

  39. Schimperna, G., Segatti, A., Stefanelli, U.: Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete Contin. Dyn. Syst. 18, 15–38 (2007)

    Article  MathSciNet  Google Scholar 

  40. Segatti, A.: Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete Contin. Dyn. Syst. 14, 801–820 (2006)

    Article  MathSciNet  Google Scholar 

  41. Senba, T.: On some nonlinear evolution equation. Funkcial Ekvac 29, 243–257 (1986)

    MathSciNet  MATH  Google Scholar 

  42. Simon, J.: Compact sets in the space \(L^{p}(0, T;B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

  43. Stefanelli, U.: The Brezis–Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47, 1615–1642 (2008)

    Article  MathSciNet  Google Scholar 

  44. Stefanelli, U.: On a class of doubly nonlinear nonlocal evolution equations. Differ. Integral Equ. 15, 897–922 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

GA is supported by JSPS KAKENHI Grant Numbers JP16H03946, JP16K05199, JP17H01095, JP18K18715 and by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation. SM acknowledges the support of the Austrian Science Fund (FWF) Project P27052-N25.

Author information

Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

    Goro Akagi

  2. Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstraße 1, 85764, Neuherberg, Germany

    Goro Akagi

  3. Technische Universität München, Zentrum Mathematik, Bolzmannstraße 3, 85748, Garching bei München, Germany

    Goro Akagi

  4. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Stefano Melchionna

Authors
  1. Goro Akagi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefano Melchionna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Melchionna.

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

Akagi, G., Melchionna, S. Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint. Nonlinear Differ. Equ. Appl. 26, 10 (2019). https://doi.org/10.1007/s00030-019-0551-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-019-0551-0

Mathematics Subject Classification

Keywords

Search

Navigation