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

Stanisław Brzychczy (1996)

Annales Polonici Mathematici

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...

On the Stefan problem with a small parameter

Galina I. Bizhanova (2008)

Banach Center Publications

We consider the multidimensional two-phase Stefan problem with a small parameter κ in the Stefan condition, due to which the problem becomes singularly perturbed. We prove unique solvability and a coercive uniform (with respect to κ) estimate of the solution of the Stefan problem for t ≤ T₀, T₀ independent of κ, and the existence and estimate of the solution of the Florin problem (Stefan problem with κ = 0) in Hölder spaces.

On the vanishing viscosity method for first order differential-functional IBVP

Krzysztof A. Topolski (2008)

Czechoslovak Mathematical Journal

We consider the initial-boundary value problem for first order differential-functional equations. We present the `vanishing viscosity' method in order to obtain viscosity solutions. Our formulation includes problems with a retarded and deviated argument and differential-integral equations.

Parabolic equations involving 0th and 1st order terms with L1 data.

Thierry Goudon, Mazen Saad (2001)

Revista Matemática Iberoamericana

This paper is devoted to general parabolic equations involving 0th and 1st order terms, in linear and nonlinear expressions, while the data only belong to L1. Existence and entropic-uniqueness of solutions are proved.

Periodic parabolic problems with nonlinearities indefinite in sign.

Tomás Godoy, Uriel Kaufmann (2007)

Publicacions Matemàtiques

Let Ω ⊂ RN be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) up - b (x, t) uq in Ω × R, where 0 &lt; p, q &lt; 1 and λ &gt; 0. In some cases we also show the existence of solutions uλ in the interior of the positive cone and that uλ can...

Perona-Malik equation: properties of explicit finite volume scheme

Angela Handlovičová (2007)

Kybernetika

The Perona–Malik nonlinear parabolic problem, which is widely used in image processing, is investigated in this paper from the numerical point of view. An explicit finite volume numerical scheme for this problem is presented and consistency property is proved.

Currently displaying 281 – 300 of 454