Skip to main content
Log in

Solvability of initial boundary value problems for non-autonomous evolution equations

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

The initial boundary value problems for linear non-autonomous first-order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change with t. We study existence, uniqueness and maximal regularity of solutions in Sobolev spaces. In contrast to the previous results, we use only the continuity assumption on the operators in the main part of the equation.

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

Access this article

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

References

  1. Acquistapace, P., Terreni, B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sere. Mat. Univ. Padova 78, 47–107 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Acquistapace, P.: Maximal regularity for abstract linear non-autonomous parabolic equations. J. of Funct. Anal. 60, 168–210 (1985)

    Article  MathSciNet  Google Scholar 

  3. Amann, H.: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4, 417–430 (2004)

    Article  MathSciNet  Google Scholar 

  4. Amann, H. : Nonautonomous parabolic equations involving measures. J. of Math. Sciences. 30, 4780–4802 (2005)

    Article  MathSciNet  Google Scholar 

  5. Amann, H.: Linear and quasilinear parabolic problems. 1. Birkhäuser Verlag, Basel-Boston-Berlin (1995)

    Chapter  Google Scholar 

  6. Arendt, W., Dier, D., Laasri, H., Ouhabaz, E.M.: Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms (2014). https://hal.archives-ouvertes.fr/hal-00797181v1

  7. Arendt, W., Chill, R., Fornaro, S., Poupaud, C.: \(L_{p}\)-maximal regularity for non-autonomous evolution equations. J. Diff. Equat. 237, 1–26 (2007)

    Article  Google Scholar 

  8. Butti, A. On the Evolution Operator for a Class of Non-autonomous Abstract Parabolic Equations. J. of Math. Anal. Appl. 170, 115–137 (1992)

    Article  MathSciNet  Google Scholar 

  9. Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéiares et équations différentielles opérationnelles. J. Math. Pures Appl. 54, 305–387 (1975)

    MathSciNet  MATH  Google Scholar 

  10. Denk, R., Hieber, M., Prüss, J., R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. 166. Mem. Amer. Math. Soc. (2003)

  11. Denk, R., Hieber, M., Prüss, J.: Optimal \(L_{p}-L_{q}\)-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 93–224 (2007)

    Article  Google Scholar 

  12. Denk, R., Krainer, T.: \(R\)-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators. Manuscripta Math. 124, 319–342 (2007)

    Article  MathSciNet  Google Scholar 

  13. Di Giorgio, D., Lunardi, A., Schnaubelt, R., Optimal regularity and Fredholm properties of abstract parabolic operators in \(L_{p}\) spaces on the real line. Proc. of the London Math. Soc. 91, 703–737 (2005)

    Article  MathSciNet  Google Scholar 

  14. Engel, K.-J., Klöss, B., Nagel, R., Fijavž, B., Sikolya, E.: Maximal controllability for boundary control problems. Appl. Math. Optim. 62, 205–227 (2010)

    Article  MathSciNet  Google Scholar 

  15. Engel, K.-J., Fijavž, B.: Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media. 12(2), 319–337 (2017)

    Article  MathSciNet  Google Scholar 

  16. Fackler, S.: J.-L. Lions’ problem concerning maximal regularity of equations governed by non-autonomous forms. Annales de l’Institut Henri Poincare (C). Non Linear Analysis 34, 699–709 (2017)

  17. Gallarati, C., Veraar, M.: Maximal regularity for non-autonomous equations with measurable dependence on time. Potential Analysis 46, 527–567 (2017)

    Article  MathSciNet  Google Scholar 

  18. Greiner, G.: Perturbing the boundary conditions of a generator. Houston J. of Math. 13, 213–229 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Grisvard, P.: Commutative de deux functeurs d’interpolation et applications. J. Math. pures et appliq. 45, 143–206 (1966)

    MATH  Google Scholar 

  20. Grisvard, P.: Equations differentielles abstraites. Ann. Scient. Ec. Norm. Sup. \(4^{e}\) series 2(3), 311–395 (1969)

    Article  MathSciNet  Google Scholar 

  21. Haase, M.: The Functional calculus for sectorial operators. Operator Theory: Adv. and Appl. 169 Birkhauser Verlag, Basel-Boston-Berlin (2006)

  22. Hieber, M., Monniaux, S., Heat kernels and maximal \(L_{p}-L_{q}\) estimates: The nonautonomous case. J. Fourier Anal. Appl. 328, 467–481 (2000)

    Article  Google Scholar 

  23. Hieber, M., Monniaux, S.: Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations. Proc. of the AMS 128, 1047–1053 (1999)

    Article  MathSciNet  Google Scholar 

  24. Kunstmann, P.C., Weis, L., Maximal \(L_{p}\) regularity for parabolic equations, Fourier multiplier theorems and \(H^{\infty }\) functional calculus. In: Iannelli, M., Nagel, R., Piazzera S. (Eds.) Proceedings of the Autumn School on Evolution Equations and Semigroups, Levico Lectures 69, pp. 65–320. Springer-Verlag. Heidelberg (2004)

  25. Laasri, H., Agadir, O.E.: Stability for non-autonomous linear evolution equations with \(L_{p}\)-maximal regularity. Czechoslovak Math. J. 63 (138), 887–908 (2013)

    Article  MathSciNet  Google Scholar 

  26. Laasri, H.: Regularity properties for evolution family governed by non-autonomous forms (2017). arXiv:1706.06340

  27. Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progr. Nonlinear Differential Equations Appl. 16. Birkhauser, Basel (1995)

  28. Meyries, M., Schnaubelt, R.: Interpolation, embeddings and traces for anisotropic fractional Sobolev spaces with temporal weights. J. Funct. Anal. 262, 1200–1229 (2012)

    Article  MathSciNet  Google Scholar 

  29. Ouhabaz, M.: Maximal regularity for non-autonomous evolution equations governed by forms having less regularity. Arch. der Math., 105, 79–91 (2015)

    Article  MathSciNet  Google Scholar 

  30. Prüss, J., Schnaubelt, R.: Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl., 256 (2001), 405–430

    Article  MathSciNet  Google Scholar 

  31. Pruss, J., Simonett, G.: Maximal regularity for evolution equations in weighted \(L_p\)-spaces. Arch. Math. 82, 415–431 (2004)

    Article  MathSciNet  Google Scholar 

  32. Pyatkov, S.G., Uvarova, M.V.: Some Properties of Solutions of the Cauchy Problem for Evolution Equations. Diff. Equat. 48, 379–389 (2012)

    Article  MathSciNet  Google Scholar 

  33. Rudin, W.: Functional analysis. McGrow-Hill Company, New York (1973)

    MATH  Google Scholar 

  34. Tanabe, H.: Functional Analytic Methods for Partial Differential Equations. Marcel Dekker, Inc. New York (1997)

    MATH  Google Scholar 

  35. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, 18. North-Holland Publishing, Amsterdam (1978)

  36. Triebel, H.: Theory of Function Spaces. II. Birkhauser Verlag, Basel (1992)

  37. Uvarova, M.V.: On some nonlocal boundary value problems for evolution equations. Sib. Adv. Math. 21, 211–231 (2011)

    Article  MathSciNet  Google Scholar 

  38. Yagi, A.: Abstract Parabolic Evolution Equations and their Applications. Springer-Verlag. Berlin Heidelberg (2010)

    Book  Google Scholar 

  39. Weis, L.: Operator-valued Fourier multiplier theorems and maximal \(L_{p}\)-regularity. Math. Ann. 319(4), 735–758 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author was supported by the Russian Foundation for Basic Research (Grant 18-01-00620) and by the Act 211 of the Government of the Russian Federation, contract No. 02.A03.21.0011.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. G. Pyatkov.

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

Pyatkov, S.G. Solvability of initial boundary value problems for non-autonomous evolution equations. J. Evol. Equ. 20, 39–58 (2020). https://doi.org/10.1007/s00028-019-00516-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-019-00516-6

Keywords

Mathematics Subject Classification

Navigation