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Large time behaviour of solutions to nonhomogeneous diffusion equations

Jean Dolbeault, Grzegorz Karch (2006)

Banach Center Publications

This note is devoted to the study of the long time behaviour of solutions to the heat and the porous medium equations in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csiszár-Kullback type estimates.

L’existence et le comportement asymptotique des solutions d’ondes progressives pour une équation fortement non linéaire

Ahmed Hamydy (2008)

Annales mathématiques Blaise Pascal

Dans ce papier on étudie l’existence et le comportement asymptotique des solutions de type ondes progressives à propagations finies de l’équation U t = A U x p - 2 U x x + K U q . On prouve que ces solutions existent si et seulement si q < 1 et c < 0 ou bien q ≤ p - 1 et c > 0 . On donne aussi le comportement asymptotique de ces solutions.

Lifshitz tails for some non monotonous random models

Frédéric Klopp, Shu Nakamura (2007/2008)

Séminaire Équations aux dérivées partielles

In this talk, we describe some recent results on the Lifshitz behavior of the density of states for non monotonous random models. Non monotonous means that the random operator is not a monotonous function of the random variables. The models we consider will mainly be of alloy type but in some cases we also can apply our methods to random displacement models.

Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

Thomas Bartsch, Peter Poláčik, Pavol Quittner (2011)

Journal of the European Mathematical Society

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation u t = Δ u + u p - 1 u . We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.

Local attractivity in nonautonomous semilinear evolution equations

Joël Blot, Constantin Buşe, Philippe Cieutat (2014)

Nonautonomous Dynamical Systems

We study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity...

Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

Let  P be a long range metric perturbation of the Euclidean Laplacian on  ℝ d , d ≥ 2 . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to  P . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group e i t f ( P ) where f has a suitable development at zero (resp. infinity).

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