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Bifurcations of invariant measures in discrete-time parameter dependent cocycles

Anastasia Maltseva, Volker Reitmann (2015)

Mathematica Bohemica

We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent...

Bi-spaces global attractors in abstract parabolic equations

J. W. Cholewa, T. Dłotko (2003)

Banach Center Publications

An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on X α . This semigroup possesses an ( X α - Z ) -global attractor that is closed, bounded, invariant in X α , and attracts bounded subsets of X α in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan Martel, Frank Merle, Pierre Raphaël (2015)

Journal of the European Mathematical Society

We consider the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

Cauchy problems in weighted Lebesgue spaces

Jan W. Cholewa, Tomasz Dłotko (2004)

Czechoslovak Mathematical Journal

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ℝ n are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ℝ n , n ∈ ℕ . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Comportement asymptotique des solutions d’un système d’équations de Schrödinger-Poisson sur un domaine borné de ℝ 3

Amna Dabaa (2010)

Annales mathématiques Blaise Pascal

Nous étudions le comportement pour les grands temps de l’équation de Schrödinger-Poisson (NLSP) avec un terme de force extérieure supplémentaire et un terme de dissipation d’ordre zéro, la variable d’espace x étant dans un domaine borné Ω de ℝ 3 . Nous démontrons que ce comportement est décrit par un attracteur global de dimension de Hausdorff finie pour la topologie forte de H 0 1 ( Ω ) .

Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, K. P. Rybakowski (2002)

Studia Mathematica

Consider the family uₜ = Δu + G(u), t > 0, x ∈ Ω ε , ∂ ν ε u = 0 , t > 0, x ∈ ∂ Ω ε , ( E ε ) of semilinear Neumann boundary value problems, where, for ε > 0 small, the set Ω ε is a thin domain in ℝ l , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ℝ l . If G is dissipative, then equation ( E ε ) has a global attractor ε . We identify a “limit” equation for the family ( E ε ) , prove convergence of trajectories and establish an upper semicontinuity result for the family ε as ε → 0⁺.

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