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Semilinear parabolic problems define semiflows on $ C\sp{k}$ spaces


Author: Xavier Mora
Journal: Trans. Amer. Math. Soc. 278 (1983), 21-55
MSC: Primary 35K55; Secondary 35K22, 47D05
DOI: https://doi.org/10.1090/S0002-9947-1983-0697059-8
MathSciNet review: 697059
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Abstract: Linear parabolic problems of a general class are proved to determine analytic semigroups on certain closed subspaces of $ {C^k}(\overline \Omega )$ ($ k$ integer); $ {C^k}(\overline \Omega )$ denotes the space of functions whose derivatives or order $ \leqslant k$ are bounded and uniformly continuous, with the usual supremum norm; the closed subspaces where the semigroups are obtained, denoted by $ {\hat C^k}(\overline \Omega )$, are determined by the boundary conditions and a possible condition at infinity. One also obtains certain embedding relations concerning the fractional power spaces associated to these semigroups. Usually, results of this type are based upon the theory of solution of elliptic problems, while this work uses the corresponding theory for parabolic problems. The preceding results are applied to show that certain semilinear parabolic problems define semiflows on spaces of the type $ {\hat C^k}(\overline \Omega )$.


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  • [1] V. S. Belonosov, Estimates of solutions of parabolic systems in weighted Hölder classes and some of their applications, Mat. Sb. 110 (1979), 163-188 = Math. USSR Sb. 38 (1979), 151-173. MR 552110 (80m:35045)
  • [2] J. Bergh and J. Löfström, Interpolation spaces, Springer, Berlin, 1976.
  • [3] N. P. Bhatia and O. Hájek, Local semi-dynamical systems, Lecture Notes in Math., vol. 90, Springer-Verlag, Berlin and New York, 1969. MR 0325187 (48:3536)
  • [4] S. D. Éĭdel'man and S. D. Ivasishen, Investigation of the Green matrix for a homogeneous parabolic boundary value problem, Trudy Moskov. Mat. Obšč. 23 179-234 = Trans. Moscow Math. Soc. 23 (1970), 179-242. MR 0367455 (51:3697)
  • [5] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin and New York, 1981. MR 610244 (83j:35084)
  • [6] M. R. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183-192. MR 0003434 (2:219c)
  • [7] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 0089373 (19:664d)
  • [8] H. Kielhöfer, Halbgruppen und semilinears Anfangs-Randwertprobleme, Manuscripta Math. 12 (1974), 121-152. MR 0344681 (49:9420)
  • [9] -, Existenz und Regularität von Lösungen semilinearer parabolischer Anfangs-Randwertprobleme, Math. Z. 142 (1975), 131-160. MR 0393854 (52:14662)
  • [10] H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1967), 285-346. MR 0201985 (34:1862)
  • [11] S. G. Kreĭn, Linear differential equations in Banach space, "Nauka", Moskow, 1963; Transl. Math. Monographs, vol. 28, Amer. Math. Soc., Providence, R. I., 1971.
  • [12] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, "Nauka", Moskow, 1967; Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R. I., 1968.
  • [13] C. Miranda, Partial differential equations of elliptic type, Springer, Berlin, 1970. MR 0284700 (44:1924)
  • [14] X. Mora, Ph. D. Dissertation, Univ. Autònoma de Barcelona, Fac. Ciències, Sec. Matemàtiques, 1982.
  • [15] J. D. Murray, Lectures on nonlinear differential equation models in biology, Clarendon Press, Oxford, 1977.
  • [16] G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems, Wiley, New York, 1977. MR 0522141 (58:25436)
  • [17] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Dept. of Math., Univ. of Maryland, Lecture Notes no. 10, 1974.
  • [18] J. A. Smoller, Shock waves and reaction-diffusion equations, Springer, New York, 1982.
  • [19] P. E. Sobolevskii, Estimates of the Green function for second-order parabolic partial differential equations, Dokl. Akad. Nauk SSSR 138 (1961), 313-316 = Soviet Math. Dokl. 2 (1961), 617-620. MR 0126083 (23:A3379)
  • [20] -, Green's function for arbitrary (in particular, integral) powers of elliptic operators, Dokl. Akad. Nauk SSSR 142 (1961), 804-807 = Soviet Math. Dokl. 3 (1961), 183-187.
  • [21] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov. 83 (1965), 3-163 = Proc. Steklov Inst. Math. 83 (1965), 1-184. MR 0211083 (35:1965)
  • [22] -, Green matrices for parabolic boundary value problems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 14 (1969), 256-287 = Sem. Math. V. A. Steklov Math. Inst. Leningrad 14 (1969), 132-150. MR 0296527 (45:5587)
  • [23] V. A. Solonnikov and A. G. Khachatryan, Estimates for solutions of parabolic initial-boundary value problems in weighted Hölder norms, Trudy Mat. Inst. Steklov. 147 (1980), 147-155 = Proc. Steklov. Math. Inst. 147 (1980), 153-162. MR 573905 (81f:35060)
  • [24] H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc. 199 (1974), 141-162. MR 0358067 (50:10532)
  • [25] -, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Soc. 259 (1980), 299-310. MR 561838 (82h:35048)
  • [26] H. Tanabe, Evolution equations, Iwanami Shoten, Tokyo, 1975; English transl., Pitman, London, 1979.
  • [27] H. Triebel, A remark on embedding theorems for Banach spaces of distributions, Ark. Mat. 11 (1973), 65-74. MR 0348484 (50:982)
  • [28] -, Interpolation theory, function spaces, differential operators, Akademie Verlag, Berlin; North-Holland, Amsterdam, 1978. MR 503903 (80i:46032b)
  • [29] W. von Wähl, Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 11 (1972), 231-258. See also Manuscripta Math. 11 (1974), 199-201. MR 0340821 (49:5571)
  • [30] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $ {L^p}$, Indiana Univ. Math. J. 29 (1980), 79-102. MR 554819 (81c:35072)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0697059-8
Keywords: Parabolic semigroups in $ {C^k}$ spaces, abstract evolution equations
Article copyright: © Copyright 1983 American Mathematical Society

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