Topological Methods in Nonlinear Analysis

Topological structure of solution sets to parabolic problems

Vladimír Ďurikovič and Monika Ďurikovičová

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Abstract

In this paper we deal with the Peano phenomenon for general initial-boundary value problems of quasilinear parabolic equations with arbitrary even order space derivatives.

The nonlinearity is assumed to be a continuous or continuously Fréchet differentiable function. Using a method of transformation to an operator equation and employing the theory of proper, Fredholm (linear and nonlinear) and Nemitskiĭ operators, we study the existence of solution of the given problem and qualitative and quantitative structure of its solution and bifurcation sets. These results can be applied to the different technical and natural science models.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 25, Number 2 (2005), 313-349.

Dates
First available in Project Euclid: 23 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1466705112

Mathematical Reviews number (MathSciNet)
MR2154431

Zentralblatt MATH identifier
1091.35032

Citation

Ďurikovič, Vladimír; Ďurikovičová, Monika. Topological structure of solution sets to parabolic problems. Topol. Methods Nonlinear Anal. 25 (2005), no. 2, 313--349. https://projecteuclid.org/euclid.tmna/1466705112


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References

  • M. S. Agranovič and M. I. Višik, Elliptic problems with a parameter and parabolic general typ problems , Uspekhi Mat. Nuk, 19 (1964), 53–161, (Russian) \ref\key 2
  • A. Ambrosetti, Global inversion theorems and applications to nonlinear problems , Conferenze del Seminario di Matematica dell' Università di Bari, Atti del $3^\circ $ Seminario di Analisi Funzionale ed Applicazioni, A Survey on the Theoretical and Numerical Trends in Nonlinear Analysis, Gius. Laterza et Figli, Bari (1976), 211–232 \ref\key 3
  • N. Aronszajn, Le correspondant topologique de l'unicité dans la théorie des équations différentielles, Ann. of Math., 43(1942), 730–738 \ref\key 6
  • S. Banach and S. Mazur\paper \HUber mehrdeutige stetige Abbildungen, Studia Math., 5(1934), 174–178 \ref\key 7
  • Ju. G. Borisovič, V. G. Zvjagin and Ju. G. Sapronov, Nonlinear Fredholm mappings and the Leray–Schauder theory , Uspekhi Mat. Nauk, XXXII \rom(4) (1977), 3–54, (Russian) \ref\key 8
  • F. E. Browder and Ch. P. Gupta, Topological degree and nonlinear mappings of analytic type in Banach spaces, J. Math. Anal. Appl., 26(1969), 390–402 \ref\key 9
  • R. Cacciopoli, Un principio di inversione per le corrispondenze funzionali e sue applicazioni alle equazioni alle derivate parziali, Rend. Accademia Naz. Lincei, VI \rom(16) (1932) \ref\key 10
  • K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin, Heidelberg (1985) \ref\key 11
  • V. \vDurikovič, An initial-boundary value problem for quasi-linear parabolic systems of higher order , Ann. Polon. Math., XXX (1974), 145–164 \ref\key 12
  • V. \vDurikovič and Ma. \vDurikovičová, Some generic properties of nonlinear second order diffusional type problem , Arch. Math. (Brno), 35 (1999), 229–244 \ref\key 13 ––––, Sets of solutions of nonlinear initial-boundary value problems , Topol Methods Nonlinear Anal., 17(2001), 157–182 \ref\key 14
  • S. D. Eide\v lman, Parabolic Systems, Nauka, Moskva (1964), (Russian) \ref\key 15
  • S. D. Eide\v lman and S. D. Ivasišen, The investigation of the Green's matrix for homogeneous boundary value problems of a parabolic type, Trudy Moskov. Mat. Obshch., 23(1970), 179–234, (Russian) \ref\key 16
  • H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equation, Proc. Sympos. Pure Math., 28 ;, Nonlinear Functional Analysis, Amer. Math. Soc., Providence, R. J. (1970) \ref\key 17
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer–Verlag, Berlin, Heidelberg, New York (1981) \ref\key 19
  • S. D. Ivasišen, Green Matrices of Parabolic Boundary Value Problems, Vyšša Škola, Kijev (1990), (Russian) \ref\key 20
  • M. Marteli and A. Vignoli, A generalized Leray–Schauder condition , Lincei Rend. Sci. Fis. Mat. Nat., LVII(1974), 374–379 \ref\key 21
  • J. Mawhin, Generic properties of nonlinear boundary value problems , Differential Equations and Mathematical Physics, Academic Press Inc., New York (1992), 217–234 \ref\key 22
  • R. A. Plastock, Nonlinear Fredholm maps of index zero and their singularities, Proc. Amer. Math. Soc., 68(1978), 317–322 \ref\key 23
  • F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. (Global Analysis), 15(1970), 213–223 \ref\key 24
  • R. S. Sadyrkhanov, Selected Questions of Nonlinear Functional Analysis, Publishers ELM, Baku (1989), (Russian) \ref\key 25
  • S. Smale, An infinite dimensional version of Sard's theorem , Amer. J. Math., 87(1965), 861–866 \ref\key 26
  • V. A. Solonikov, Estimations of $L_p$ solutions for elliptic and parabolic systems, Trudy Math. Inst. Steklov., 102(1969), 446–472 \ref\key 27 ––––, On Boundary value problem for linear parabolic differential systems of the general type, Trudy Math. Inst. Steklov., 83(1965), 3–162, (Russian) \ref\key 28
  • V. Šeda, Fredholm mappings and the generalized boundary value problem , Differential Integral Equations, 8(1995), 19–40 \ref\key 29
  • A. E. Taylor, Introduction of Functional Analysis, John Wiley and Sons, Inc., New York (1967) \ref\key 30
  • V. A. Trenogin, Functional Analysis, Nauka, Moscow(1980), (Russian) \ref\key 31
  • K. Yosida, Functional Analysis, Springer–Verlag, Berlin, Heidelberg, New York (1966) \ref\key 32
  • E. Zeidler, Nonlinear Functional Analysis and its Applications \romI, Fixed-Point Theorems, Springer–Verlag, Berlin, Heidelberg, Tokyo (1986) \ref\key 33 ––––, Nonlinear Functional Analysis and its Applications \romII/B, Nonlinear Monoton Operators, Springer–Verlag, Berlin, Heidelberg, London, Paris, Tokyo (1990)