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

100%
Brzychczy S.
Annales Polonici Mathematici
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1996
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tom 63
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nr 2
155-165
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.
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Existence of solutions and monotone iterative method for infinite systems of parabolic differential-functional equations

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Brzychczy S.
Annales Polonici Mathematici
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1999
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tom 72
|
nr 1
15-24
EN
We consider the Fourier first boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations. To prove the existence and uniqueness of solution, we apply a monotone iterative method using J. Szarski's results on differential-functional inequalities and a comparison theorem for infinite systems.
3
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Impulsive differential equations with initial data difference

88%
Skóra L.
Commentationes Mathematicae
|
2011
|
tom 51
|
nr 1
EN
In this paper, we present some results on impulsive differential inequalities and equations with initial and impulsive data difference.
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