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1996 | 63 | 2 | 155-165
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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
⎧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
⎧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.
  • Institute of Mathematics, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Kraków, Poland
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