Skip to main content
SpringerLink
Log in

On the principle of linearized stability in interpolation spaces for quasilinear evolution equations

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We give a proof for the asymptotic exponential stability in admissible interpolation spaces of equilibrium solutions to quasilinear parabolic evolution equations.

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

Similar content being viewed by others

Notes

  1. We use \(C^{k-}\), \(1\le k\in {\mathbb {N}}\) to denote the space of functions which possess a locally Lipschitz continuous \((k-1)\)th derivative. Similarly, \(C^\vartheta \) with \(\vartheta \in (0,1)\) denotes local Hölder continuity.

  2. That is, \(\mathcal {D}:=\{(t,v^0)\,:\, 0\le t<t^+(v^0)\,,\,v^0\in O_\alpha \}\) is open in \([0,\infty )\times E_\alpha \) and the function

    $$\begin{aligned} v:\mathcal {D}\rightarrow E_\alpha \,,\quad (t,v^0)\mapsto v(t;v^0) \end{aligned}$$

    is continuous with \(v(0;v^0)=v^0\) and

    $$\begin{aligned} v(t;v(s;v^0))=v(t+s;v^0)\quad for\, 0\le s< t^+(v^0)\, \text {and}\, 0\le t<t^+(v(s;v^0)). \end{aligned}$$

  3. Given \(\theta \in (0,1)\), \(S\subset E_\theta \), and \(\delta >0\), we denote by \(\mathbb {B}_{E_\theta }(S, \delta )\) the set \(\{x\in E_\theta \,:\, \mathrm {dist}_{E_\theta }\big (x,S\big )<\delta \}\).

References

  1. Acquistapace, P., Terreni, B.: On quasilinear parabolic systems. Math. Ann. 282, 315–335 (1988)

    Article  MathSciNet  Google Scholar 

  2. Amann, H.: Gewöhnliche Differentialgleichungen, de Gruyter Lehrbuch [de Gruyter Textbook]. Walter de Gruyter & Co., Berlin (1983)

    Google Scholar 

  3. Amann, H.: Quasilinear evolution equations and parabolic systems. Trans. Am. Math. Soc. 293, 191–227 (1986)

    Article  MathSciNet  Google Scholar 

  4. Amann, H.: Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations. Nonlinear Anal. 12, 895–919 (1988)

    Article  MathSciNet  Google Scholar 

  5. Amann, H.: Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems. Differ. Integr. Equ. 3, 13–75 (1990)

    MathSciNet  MATH  Google Scholar 

  6. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Mathmatics, pp. 9–126. Teubner, Stuttgart (1993)

    Google Scholar 

  7. Amann, H.: Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics. Abstract Linear Theory, Birkhäuser Boston, Inc., Boston (1995)

    Chapter  Google Scholar 

  8. Amann, H.: Maximal regularity and quasilinear parabolic boundary value problems. In: Recent Advances in Elliptic and Parabolic Problems, pp. 1–17. World Scientific Publishing, Hackensack (2005)

  9. Angenent, S.B.: Nonlinear analytic semiflows. Proc. R. Soc. Edinburgh Sect. A 115, 91–107 (1990)

    Article  MathSciNet  Google Scholar 

  10. Cheng, C.H.A., Granero-Belinchón, R., Shkoller, S.: Well-posedness of the Muskat problem with \(H^2\) initial data. Adv. Math. 286, 32–104 (2016)

    Article  MathSciNet  Google Scholar 

  11. Clément, P., Simonett, G.: Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations. J. Evol. Equ. 1, 39–67 (2001)

    Article  MathSciNet  Google Scholar 

  12. Córdoba, A., Córdoba, D., Gancedo, F.: Interface evolution: the Hele–Shaw and Muskat problems. Ann. Math. (2) 173, 477–542 (2011)

    Article  MathSciNet  Google Scholar 

  13. Da Prato, G.: Fully nonlinear equations by linearization and maximal regularity, and applications. In: Partial Differential Equations and Functional Analysis, vol. 22 of Progr. Nonlinear Differential Equations Application, pp. 80–92. Birkhäuser Boston, Boston (1996)

  14. Da Prato, G., Grisvard, P.: Equations d’évolution abstraites non linéaires de type parabolique. Ann. Mat. Pura Appl. (4) 120, 329–396 (1979)

    Article  MathSciNet  Google Scholar 

  15. Da Prato, G., Lunardi, A.: Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in banach space. Arch. Ration. Mech. Anal. 101, 115–141 (1988)

    Article  MathSciNet  Google Scholar 

  16. Drangeid, A.-K.: The principle of linearized stability for quasilinear parabolic evolution equations. Nonlinear Anal. 13, 1091–1113 (1989)

    Article  MathSciNet  Google Scholar 

  17. Escher, J., Laurençot, P., Walker, C.: Dynamics of a free boundary problem with curvature modeling electrostatic MEMS. Trans. Am. Math. Soc. 367, 5693–5719 (2015)

    Article  MathSciNet  Google Scholar 

  18. Guidetti, D.: Convergence to a stationary state and stability for solutions of quasilinear parabolic equations. Ann. Math. Pura Appl. (4) 151, 331–358 (1988)

    Article  MathSciNet  Google Scholar 

  19. Lunardi, A.: Analyticity of the maximal solution of an abstract nonlinear parabolic equation. Nonlinear Anal. 6, 503–521 (1982)

    Article  MathSciNet  Google Scholar 

  20. Lunardi, A.: Abstract quasilinear parabolic equations. Math. Ann. 267, 395–415 (1984)

    Article  MathSciNet  Google Scholar 

  21. Lunardi, A.: Asymptotic exponential stability in quasilinear parabolic equations. Nonlinear Anal. 9, 563–586 (1985)

    Article  MathSciNet  Google Scholar 

  22. Lunardi, A.: Global solutions of abstract quasilinear parabolic equations. J. Differ. Equ. 58, 228–242 (1985)

    Article  MathSciNet  Google Scholar 

  23. Lunardi, A.: On the local dynamical system associated to a fully nonlinear abstract parabolic equation. In: Nonlinear Analysis and Applications (Arlington, Tex., 1986), vol. 109 of Lecture Notes in Pure and Applied Mathematics, pp. 319–326. Dekker, New York (1987)

  24. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, vol. 16. Birkhäuser Verlag, Basel (1995)

    MATH  Google Scholar 

  25. Matioc, A.-V., Matioc, B.-V.: Well-posedness and stability results for a quasilinear periodic Muskat problem. J. Differ. Equ. 266, 5500–5531 (2019)

    Article  MathSciNet  Google Scholar 

  26. Matioc, B.-V.: Viscous displacement in porous media: the Muskat problem in 2D. Trans. Am. Math. Soc. 370, 7511–7556 (2018)

    Article  MathSciNet  Google Scholar 

  27. Matioc, B.-V.: Well-posedness and stability results for some periodic Muskat problems (2018)

  28. Matioc, B.-V.: The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12, 281–332 (2019)

    Article  MathSciNet  Google Scholar 

  29. Potier-Ferry, M.: The linearization principle for the stability of solutions of quasilinear parabolic equations. I. Arch. Ration. Mech. Anal. 77, 301–320 (1981)

    Article  MathSciNet  Google Scholar 

  30. Prüss, J.: Maximal regularity for evolution equations in \(L_p\)-spaces. Conf. Semin. Math. Univ. Bari 2002, 1–39 (2003)

    Google Scholar 

  31. Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations, vol. 105 of Monographs in Mathematics. Springer, Cham (2016)

    Book  Google Scholar 

  32. Prüss, J., Simonett, G., Wilke, M.: Critical spaces for quasilinear parabolic evolution equations and applications. J. Differ. Equ. 264, 2028–2074 (2018)

    Article  MathSciNet  Google Scholar 

  33. Prüss, J., Simonett, G., Zacher, R.: On convergence of solutions to equilibria for quasilinear parabolic problems. J. Differ. Equ. 246, 3902–3931 (2009)

    Article  MathSciNet  Google Scholar 

  34. Prüss, J., Simonett, G., Zacher, R.: On normal stability for nonlinear parabolic equations. Discrete Contin. Dyn. Syst. 612–621 (2009)

  35. Walker, C.: Age-dependent equations with non-linear diffusion. Discrete Contin. Dyn. Syst. 26, 691–712 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referees for carefully reading the manuscript and pointing out improvements.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93053, Regensburg, Germany

    Bogdan-Vasile Matioc

  2. Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167, Hannover, Germany

    Christoph Walker

Authors
  1. Bogdan-Vasile Matioc
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Walker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bogdan-Vasile Matioc.

Additional information

Communicated by Joachim Escher.

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

Matioc, BV., Walker, C. On the principle of linearized stability in interpolation spaces for quasilinear evolution equations. Monatsh Math 191, 615–634 (2020). https://doi.org/10.1007/s00605-019-01352-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-019-01352-z

Keywords

Mathematics Subject Classification

Search

Navigation