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Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : Δ u ε = u ε ( n + 2 ) / ( n 2 ) , u ε > 0 in A ε , u ε = 0 on A ε , where A ε is an annulus-shaped domain of n , n 3 , which becomes thinner as ε 0 . We show that for every solution u ε , the energy A ε | u | 2 as well as the Morse index tend to infinity as ε 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Entire solutions in 2 for a class of Allen-Cahn equations

Francesca Alessio, Piero Montecchiari (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of semilinear elliptic equations of the form - ε 2 Δ u ( x , y ) + a ( x ) W ' ( u ( x , y ) ) = 0 , ( x , y ) 2 where ε > 0 , a : is a periodic, positive function and W : is modeled on the classical two well Ginzburg-Landau potential W ( s ) = ( s 2 - 1 ) 2 . We look for solutions to (1) which verify the asymptotic conditions u ( x , y ) ± 1 as x ± uniformly with respect to y . We show via variational methods that if ε is sufficiently small and a is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

Entire solutions in 2 for a class of Allen-Cahn equations

Francesca Alessio, Piero Montecchiari (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of semilinear elliptic equations of the form 15.7cm - ε 2 Δ u ( x , y ) + a ( x ) W ' ( u ( x , y ) ) = 0 , ( x , y ) 2 where ε > 0 , a : is a periodic, positive function and W : is modeled on the classical two well Ginzburg-Landau potential W ( s ) = ( s 2 - 1 ) 2 . We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions u ( x , y ) ± 1 as x ± uniformly with respect to y . We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct...

Entropy of scalar reaction-diffusion equations

Siniša Slijepčević (2014)

Mathematica Bohemica

We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms....

Équation des ondes amorties dans un domaine extérieur

Moez Khenissi (2003)

Bulletin de la Société Mathématique de France

On étudie la position des pôles de diffusion du problème de Dirichlet pour l’équation des ondes amorties du type t 2 - Δ + a ( x ) t dans un domaine extérieur. Sous la condition du « contrôle géométrique extérieur », on déduit alors le comportement des solutions en grand temps. On calcule en particulier le meilleur taux de décroissance de l’énergie locale en dimension impaire d’espace.

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