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

• Autorzy: Stanisław Brzychczy
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

nonlinear differential-functional equations of parabolic and elliptic type; monotone iterative method; method of lower and upper functions; stability of solutions

Kategorie tematyczne

• 35J65: Nonlinear boundary value problems for linear elliptic equations
• 35B35: Stability
• 35K60: Nonlinear initial value problems for linear parabolic equations
• 35B40: Asymptotic behavior of solutions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1996
• Tom: 63
• Numer: 2
• Strony: 155-165

Daty

wydano

1996

otrzymano

1994-10-27

poprawiono

1995-01-20

Twórcy

autor

Stanisław Brzychczy

• Institute of Mathematics, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Kraków, Poland

Bibliografia

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• [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.
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• [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.
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