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A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding.

Julián Fernández Bonder;  Julio D. Rossi — 2002

Publicacions Matemàtiques

In this paper we study the Sobolev trace embedding W(Ω) → L (∂Ω), where V is an indefinite weight. This embedding leads to a nonlinear eigenvalue problem where the eigenvalue appears at the (nonlinear) boundary condition. We prove that there exists a sequence of variational eigenvalues λ / +∞ and then show that the first eigenvalue is isolated, simple and monotone with respect to the weight. Then we prove a nonexistence result related to the first eigenvalue and we end this article...

Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

Gabriel Acosta;  Julián Fernández Bonder;  Pablo Groisman;  Julio Daniel Rossi — 2002

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations u t = Δ u , v t = Δ v in Ω × ( 0 , T ) ; fully coupled by the boundary conditions ∂ u ∂ η = u p 11 v p 12 , ∂ v ∂ η = u p 21 v p 22 on ∂ Ω × ( 0 , T ) , where Ω is a bounded smooth domain in ℝ d . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation ( U , V ) . We prove that if U blows up in finite time then V can fail to blow up if and only if p 11 > 1 and p 21 < 2 ( p 11 - 1 ) , which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover,...

Simultaneous non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

Gabriel Acosta;  Julián Fernández Bonder;  Pablo Groisman;  Julio Daniel Rossi — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations , in Ω x (0,); fully coupled by the boundary conditions ∂ u ∂ η = u p 11 v p 12 , ∂ v ∂ η = u p 21 v p 22 on ∂Ω x (0,), where is a bounded smooth domain in ℝ d . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation . We prove that if blows up in finite time then can fail to blow up if and only if > 1 and < 2( - 1) , which is the same...

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