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Front propagation for nonlinear diffusion equations on the hyperbolic space

Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)

Journal of the European Mathematical Society

We study the Cauchy problem in the hyperbolic space n ( n 2 ) for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space n new phenomena arise, which depend on the properties of the diffusion process in n . We also investigate a family of travelling wave solutions, named...

Global Attractor for the Convective Cahn-Hilliard Equation

Xiaopeng Zhao, Changchun Liu (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

This paper is concerned with the convective Cahn-Hilliard equation. We use a classical theorem on existence of a global attractor to derive that the convective Cahn-Hilliard equation possesses a global attractor on some subset of H².

Global Attractor for the Convective Cahn-Hilliard Equation in H k

Xiaopeng Zhao, Ning Duan (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in H k .

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on N

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove the existence of global attractors for the following semilinear degenerate parabolic equation on N : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method.

Homogenization of a carcinogenesis model with different scalings with the homogenization parameter

Isabell Graf, Malte A. Peter (2014)

Mathematica Bohemica

In the context of periodic homogenization based on two-scale convergence, we homogenize a linear system of four coupled reaction-diffusion equations, two of which are defined on a manifold. The system describes the most important subprocesses modeling the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum, which constitutes a fine structure inside the cell....

Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach

Abdellatif Benchérif-Madani, Étienne Pardoux (2007)

ESAIM: Probability and Statistics

In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.

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...

Long time behaviour of a Cahn-Hilliard system coupled with viscoelasticity

Irena Pawłow, Wojciech M. Zajączkowski (2010)

Annales Polonici Mathematici

The long-time behaviour of a unique regular solution to the Cahn-Hilliard system coupled with viscoelasticity is studied. The system arises as a model of the phase separation process in a binary deformable alloy. It is proved that for a sufficiently regular initial data the trajectory of the solution converges to the ω-limit set of these data. Moreover, it is shown that every element of the ω-limit set is a solution of the corresponding stationary problem.

Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

Mark O. Gluzman, Nataliia V. Gorban, Pavlo O. Kasyanov (2015)

Nonautonomous Dynamical Systems

In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction...

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