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1996 | 63 | 2 | 155-165
Tytuł artykułu

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

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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 $x = (x₁,..., x_m) ∈ G ⊂ ℝ^m$, G is an open and bounded domain with $C^{2+α}$ (0 < α ≤ 1) boundary, the operator
    Az := ∑_{j,k=1}^m a_{jk}(x) (∂²z/(∂x_j ∂x_k))

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.
Twórcy
  • Institute of Mathematics, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Kraków, Poland
Bibliografia
  • [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  • [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.
  • [3] H. Amman, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146.
  • [4] H. Amman, Supersolutions, monotone iterations and stability, J. Differential Equations 21 (1976), 363-377.
  • [5] S. Brzychczy, Approximate iterative method and the existence of solutions of nonlinear parabolic differential-functional equations, Ann. Polon. Math. 42 (1983), 37-43.
  • [6] S. Brzychczy, Chaplygin's method for a system of nonlinear parabolic differential-functional equations, Differentsial'nye Uravneniya 22 (1986), 705-708 (in Russian).
  • [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.
  • [8] O. Diekmann and N. M. Temme, Nonlinear Diffusion Problems, MC Syllabus 28, Mathematisch Centrum, Amsterdam, 1982.
  • [9] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.
  • [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983.
  • [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] 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] 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] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000.
  • [15] D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Math. 309, Springer, Berlin, 1973.
  • [16] J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), 257-282.
  • [17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.
  • [18] J. Szarski, Strong maximum principle for nonlinear parabolic differential-functional inequalities, Ann. Polon. Math. 29 (1974), 207-214.
  • [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] J. Wloka, Funktionalanalysis und Anwendungen, de Gruyter, Berlin, 1971.
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Bibliografia
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