On the stability of solutions of nonlinear parabolic differential-functional equations

Stanisław Brzychczy

Annales Polonici Mathematici (1996)

  • Volume: 63, Issue: 2, page 155-165
  • ISSN: 0066-2216

Abstract

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We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x)     for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x)     for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0  for x ∈ G, ⎨z(x) = h(x)    for x ∈ ∂G ⎩ where x = ( x ₁ , . . . , x m ) ∈ G ⊂ ℝ m , G is an open and bounded domain with C 2 + α (0 < α ≤ 1) boundary, the operator     Az := ∑j,k=1m ajk(x) (∂²z/(∂xj ∂xk)) is uniformly elliptic in G̅ and z is a real L p ( G ) function. The purpose of this paper is to give some conditions which will guarantee that the parabolic problem has a stable solution. Basing on the results obtained in [7] and [5, 6], we prove that the limit of the solution of the parabolic problem (1) as t → ∞ is the solution of the associated elliptic problem (2), obtained by the monotone iterative method. The problem of stability of solutions of the parabolic differential equation has been studied by D. H. Sattinger [14, 15], H. Amann [3, 4], O. Diekmann and N. M. Temme [8], and J. Smoller [17]. Our results generalize these papers to encompass the case of differential-functional equations. Differential-functional equations arise frequently in applied mathematics. For example, equations of this type describe the heat transfer processes and the prediction of ground water level.

How to cite

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Stanisław Brzychczy. "On the stability of solutions of nonlinear parabolic differential-functional equations." Annales Polonici Mathematici 63.2 (1996): 155-165. <http://eudml.org/doc/262858>.

@article{StanisławBrzychczy1996,
abstract = {We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x)     for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x)     for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0  for x ∈ G, ⎨z(x) = h(x)    for x ∈ ∂G ⎩ where $x = (x₁,..., x_m) ∈ G ⊂ ℝ^m$, G is an open and bounded domain with $C^\{2+α\}$ (0 < α ≤ 1) boundary, the operator     Az := ∑j,k=1m ajk(x) (∂²z/(∂xj ∂xk)) is uniformly elliptic in G̅ and z is a real $L^p(G)$ function. The purpose of this paper is to give some conditions which will guarantee that the parabolic problem has a stable solution. Basing on the results obtained in [7] and [5, 6], we prove that the limit of the solution of the parabolic problem (1) as t → ∞ is the solution of the associated elliptic problem (2), obtained by the monotone iterative method. The problem of stability of solutions of the parabolic differential equation has been studied by D. H. Sattinger [14, 15], H. Amann [3, 4], O. Diekmann and N. M. Temme [8], and J. Smoller [17]. Our results generalize these papers to encompass the case of differential-functional equations. Differential-functional equations arise frequently in applied mathematics. For example, equations of this type describe the heat transfer processes and the prediction of ground water level.},
author = {Stanisław Brzychczy},
journal = {Annales Polonici Mathematici},
keywords = {nonlinear differential-functional equations of parabolic and elliptic type; monotone iterative method; method of lower and upper functions; stability of solutions; nonlinear parabolic differential-functional equations},
language = {eng},
number = {2},
pages = {155-165},
title = {On the stability of solutions of nonlinear parabolic differential-functional equations},
url = {http://eudml.org/doc/262858},
volume = {63},
year = {1996},
}

TY - JOUR
AU - Stanisław Brzychczy
TI - On the stability of solutions of nonlinear parabolic differential-functional equations
JO - Annales Polonici Mathematici
PY - 1996
VL - 63
IS - 2
SP - 155
EP - 165
AB - We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x)     for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x)     for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0  for x ∈ G, ⎨z(x) = h(x)    for x ∈ ∂G ⎩ where $x = (x₁,..., x_m) ∈ G ⊂ ℝ^m$, G is an open and bounded domain with $C^{2+α}$ (0 < α ≤ 1) boundary, the operator     Az := ∑j,k=1m ajk(x) (∂²z/(∂xj ∂xk)) is uniformly elliptic in G̅ and z is a real $L^p(G)$ function. The purpose of this paper is to give some conditions which will guarantee that the parabolic problem has a stable solution. Basing on the results obtained in [7] and [5, 6], we prove that the limit of the solution of the parabolic problem (1) as t → ∞ is the solution of the associated elliptic problem (2), obtained by the monotone iterative method. The problem of stability of solutions of the parabolic differential equation has been studied by D. H. Sattinger [14, 15], H. Amann [3, 4], O. Diekmann and N. M. Temme [8], and J. Smoller [17]. Our results generalize these papers to encompass the case of differential-functional equations. Differential-functional equations arise frequently in applied mathematics. For example, equations of this type describe the heat transfer processes and the prediction of ground water level.
LA - eng
KW - nonlinear differential-functional equations of parabolic and elliptic type; monotone iterative method; method of lower and upper functions; stability of solutions; nonlinear parabolic differential-functional equations
UR - http://eudml.org/doc/262858
ER -

References

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  1. [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
  2. [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. Zbl0093.10401
  3. [3] H. Amman, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146. 
  4. [4] H. Amman, Supersolutions, monotone iterations and stability, J. Differential Equations 21 (1976), 363-377. Zbl0319.35039
  5. [5] S. Brzychczy, Approximate iterative method and the existence of solutions of nonlinear parabolic differential-functional equations, Ann. Polon. Math. 42 (1983), 37-43. Zbl0537.35044
  6. [6] S. Brzychczy, Chaplygin's method for a system of nonlinear parabolic differential-functional equations, Differentsial'nye Uravneniya 22 (1986), 705-708 (in Russian). Zbl0613.35041
  7. [7] S. Brzychczy, Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type, Ann. Polon. Math. 58 (1993), 139-146. Zbl0787.35114
  8. [8] O. Diekmann and N. M. Temme, Nonlinear Diffusion Problems, MC Syllabus 28, Mathematisch Centrum, Amsterdam, 1982. 
  9. [9] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964. Zbl0144.34903
  10. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. Zbl0562.35001
  11. [11] M. A. Krasnosel'skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, Gostekhizdat, Moscow, 1956 (in Russian); English transl.: Macmillan, New York, 1964. 
  12. [12] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs Adv. Texts and Surveys in Pure and Appl. Math. 27, Pitman, Boston, 1985. 
  13. [13] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1964 (in Russian); English transl.: Academic Press, New York, 1968. 
  14. [14] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000. Zbl0223.35038
  15. [15] D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Math. 309, Springer, Berlin, 1973. Zbl0248.35003
  16. [16] J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), 257-282. Zbl0008.25502
  17. [17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983. Zbl0508.35002
  18. [18] J. Szarski, Strong maximum principle for nonlinear parabolic differential-functional inequalities, Ann. Polon. Math. 29 (1974), 207-214. Zbl0291.35048
  19. [19] M. M. Vaĭnberg, Variational Methods for the Study of Nonlinear Operators, Gostekhizdat, Moscow, 1956 (in Russian); English transl.: Holden-Day, San Francisco, 1964. 
  20. [20] J. Wloka, Funktionalanalysis und Anwendungen, de Gruyter, Berlin, 1971. 

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