Existence and regularity for semilinear parabolic evolution equations

Herbert Amann

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1984)

  • Volume: 11, Issue: 4, page 593-676
  • ISSN: 0391-173X

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Amann, Herbert. "Existence and regularity for semilinear parabolic evolution equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 11.4 (1984): 593-676. <http://eudml.org/doc/83948>.

@article{Amann1984,
author = {Amann, Herbert},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {semilinear; time dependent boundary conditions; existence; regularity; semigroup methods; interpolation spaces; mild solutions; abstract problem; classical solutions},
language = {eng},
number = {4},
pages = {593-676},
publisher = {Scuola normale superiore},
title = {Existence and regularity for semilinear parabolic evolution equations},
url = {http://eudml.org/doc/83948},
volume = {11},
year = {1984},
}

TY - JOUR
AU - Amann, Herbert
TI - Existence and regularity for semilinear parabolic evolution equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1984
PB - Scuola normale superiore
VL - 11
IS - 4
SP - 593
EP - 676
LA - eng
KW - semilinear; time dependent boundary conditions; existence; regularity; semigroup methods; interpolation spaces; mild solutions; abstract problem; classical solutions
UR - http://eudml.org/doc/83948
ER -

References

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  1. [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Zbl0314.46030MR450957
  2. [2] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), pp. 119-147. Zbl0109.32701MR147774
  3. [3] S. Agmon - A. Douglis - L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, II, Comm. Pure Appl. Math., 12 (1959), pp. 623-727 and 17 (1964), pp. 35-92. Zbl0093.10401MR162050
  4. [4] H. Amann, Periodic solutions of semilinear parabolic equations, in « Nonlinear Analysis: A Collection of Papers in Honour of Erich H. Rothe », edited by L. Cesari, R. Kannan, H. F. Weinberger, Academic Press, New York, 1978, pp. 1-29. Zbl0464.35050MR499089
  5. [5] H. Amann, Gewöhnliche Differentialgleichungen, W. de Gruyter, Berlin, 1983. MR713040
  6. [6] H. Amann, Parabolic evolution equations with nonlinear boundary conditions, Proc. AMS Summer Inst. Nonl. Funct. Anal., Berkeley, 1983, to appear. Zbl0611.35043MR932367
  7. [7] P. Acquistapace - B. Terreni, On the abstract nonautonomous Cauchy problem in the case of constant domains, Ann. Mat. Pura Appl., to appear. Zbl0579.34001MR815156
  8. [8] P. Acquistapace - B. Terreni, Some existence and regularity results for abstract nonautonomous parabolic equations, J. Math. Anal. Appl., 99 (1984), pp. 9-64. Zbl0555.34051MR732703
  9. [9] J. Bergh - J. Loefstroem, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976. Zbl0344.46071MR482275
  10. [10] N.P. Bhatia - O. Hajek, Local Semi-Dynamical Systems, Lecture Notes in Mathematics, no. 90, Springer-Verlag, Berlin, 1969. Zbl0176.39102MR251328
  11. [11] F.E. Browder, On the spectral theory of elliptic operators, I, Math. Ann., 142 (1961), pp. 22-130. Zbl0104.07502MR209909
  12. [12] P.L. Butzer - H. Berens, Semi-Groups of Operators and Approximation, Springer-Verlag, Berlin, 1967. Zbl0164.43702MR230022
  13. [13] S. Campanato, Generation of analytic semigroups by elliptic operators of second order in Hölder spaces, Ann. Scuola Norm. Sup. Pisa, 8 (1981), pp. 495-512. Zbl0475.35039MR634859
  14. [14] G. Da Prato, Abstract differential equations, maximal regularity and linearization, preprint, Scuola Norm. Sup. Pisa, 1983. MR843572
  15. [15] G. Da Prato - P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationelles, J. Math. Pures Appl., 54 (1975), pp. 305-387. Zbl0315.47009MR442749
  16. [16] G. Da Prato - P. Grisvard, Equations d'évolution abstraites non lineaires de type parabolic, Ann. Mat. Pura Appl., (IV), 120 (1979), pp. 329-396. Zbl0471.35036MR551075
  17. [17] G. Da Prato - P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, preprint, Scuola Norm. Sup. Pisa, 1982. Zbl0593.47041MR757990
  18. [18] G. Da Prato - E. Sinestrari, Hölder regularity for nonautonomous abstract parabolic equations, Israel J. Math., 42 (1982), pp. 1-19. Zbl0495.47031MR687930
  19. [19] E.B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030MR591851
  20. [20] H.O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, Mass., 1983. Zbl0493.34005MR692768
  21. [21] A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, New York, 1969. Zbl0224.35002MR445088
  22. [22] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964. Zbl0144.34903MR181836
  23. [23] P. Grisvard, Équations différentielles abstraites, Ann. Scient. Éc. Norm. Sup., Sér. 4, 2 (1969), pp. 311-395. Zbl0193.43502MR270209
  24. [24] A. Haraux, Nonlinear Evolution Equations. Global Behaviour of Solutions, Lecture Notes in Math., no. 841, Springer-Verlag, Berlin, 1981. Zbl0461.35002MR610796
  25. [25] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., no. 840, Springer-Verlag, Berlin, 1981. Zbl0456.35001MR610244
  26. [26] T. Kato - H. Tanabe, On the abstract evolution equation, Osaka Math. J., 14 (1962), pp. 107-133. Zbl0106.09302MR140954
  27. [27] H. Kielhoefer, Halbgruppen und semilineare Anfangs-Randwertprobleme, Manuscripta Math., 12 (1974), pp. 121-152. Zbl0276.35059MR344681
  28. [28] H. Kielhoefer, Existenz und Regularität von Lösungen semilinearer parabolischer Anfangs-Bandwertprobleme, Math. J., 142 (1975), pp. 131-160. Zbl0324.35047MR393854
  29. [29] M.A. Krasnoselskii - P.P. Zabreiko - E.I. Pustylnik - P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976. MR385645
  30. [30] O.A. Lady - V.A. Solonnikov - N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968. MR241822
  31. [31] J.-L. Lions - E. Magenes, Problemi ai limiti non omogenei (V), Ann. Scuola Norm. Sup. Pisa, 16 (1962), pp. 1-44. Zbl0115.31401MR146527
  32. [32] J.L. Lions - E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, I, II, Springer-Verlag, Berlin, 1972. Zbl0223.35039
  33. [33] A. Lunardi, Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations, Math. Nachr., to appear. Zbl0568.47035MR809327
  34. [34] A. Lunardi, Abstract quasilinear parabolic equations, Math. Ann., 267 (1984), pp. 395-415. Zbl0547.35054MR738260
  35. [35] X. Mora, Semilinear parabolic problems define semiflows on Ck spaces, Trans. Amer. Math. Soc., 278 (1983), pp. 21-55. Zbl0525.35044MR697059
  36. [36] Ch B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. Zbl0142.38701MR202511
  37. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Zbl0516.47023MR710486
  38. [38] R. Seeley, Interpolation in Lp with boundary conditions, Studia Math., 44 (1972), pp. 47-60. Zbl0237.46041MR315432
  39. [39] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., to appear. Zbl0589.47042MR786012
  40. [40] E. Sinestrari - P. Vernole, Semi-linear evolution equations in interpolation spaces, Nonlinear Analysis, Theory, Meth. & Appl., 1 (1977), pp. 249-261. Zbl0357.34061MR637091
  41. [41] P.E. Sobolevskii, Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl., Ser. 2, 49 (1966), pp. 1-62. 
  42. [42] V.A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Proc. Steklov Inst. Math., 83 (1965), pp. 1-184. Zbl0164.12502MR211083
  43. [43] H. Tanabe, Equations of Evolution, Pitman, London, 1979. Zbl0417.35003MR533824
  44. [44] H. Tanabe, On the equation of evolution in a Banach space, Osaka Math. J., 12 (1960), pp. 363-376. Zbl0098.31301MR125455
  45. [45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978. Zbl0387.46032MR503903
  46. [46] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. Zbl0546.46027MR781540
  47. [47] W. Von Wahl, Gebrochene Potenzen eines elliptischen Operators und parabolische Differentiatgteichungen in Räumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen, II. math.-phys. Kl., 11 (1972), pp. 231-258. Zbl0251.35052MR656525
  48. [48] W. Von Wahl, Einige Bemerkungen zu meiner Arbeit « Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen », Manuseripta math., 11 (1974), pp. 199-201. Zbl0285.35039MR340821
  49. [49] W. Von Wahl, Lineare und semilineare parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen, Abh. Math. Sem. Univ. Hamburg, 43 (1975), pp. 234-262. Zbl0324.35044MR473528
  50. [50] J. Wloka, Partielle Differentialgleichungen, Teubner, Stuttgart, 1981. Zbl0482.35001MR652934
  51. [51] A. Yagi., On the abstract linear evolution equations in Banach spaces, J. Math. Soc. Japan, 28 (1976), pp. 290-303. Zbl0318.34068MR397478
  52. [52] K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965. 
  53. [53] V.A. Solonnikov - A.G. Ha, On the question of the solvability of initial-boundary value problems for quasilinear parabolic systems, Zap. Nauc. Sem. LOMI., 110 (1981), pp. 225-234 (1981) (in Russian). Zbl0485.35056MR643987
  54. [54] V.A. Solonnikov - A. G. HAčATRJAN, Estimates for sotutions of parabolic initial-boundary value problems in weighted Hölder norms, Proc. Steklov Inst. Math., 2 (1981), pp. 153-162. Zbl0464.35011MR573905

Citations in EuDML Documents

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  1. Stefano Cardanobile, Delio Mugnolo, Qualitative properties of coupled parabolic systems of evolution equations
  2. Juraj Földes, Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems
  3. Sahbi Boussandel, Ralph Chill, Eva Fašangová, Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow
  4. Philippe Souplet, Fred B. Weissler, Poincaré's inequality and global solutions of a nonlinear parabolic equation

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