scholarly journals Application of the mean ergodic theorem to certain semigroups

2008 ◽  
Vol 83 (97) ◽  
pp. 49-55
Author(s):  
Gerd Herzog ◽  
Christoph Schmoeger
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behaviour ◽  
Ergodic Theorem ◽  
Mean Ergodic Theorem ◽  
Asymptotic Behaviour Of Solutions ◽  
The Cauchy Problem ◽  
The Mean

We study the asymptotic behaviour of solutions of the Cauchy problem u' = ?n j=1(Aj + A-1 j ) - 2nI_u, u(0) = x as t??, for invertible isometries A1, . . . , An.

Self-similar behaviour for the equation of fast nonlinear diffusion

1993 ◽  
Vol 343 (1668) ◽  
pp. 337-375 ◽  
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behaviour ◽  
Similarity Solution ◽  
Finite Time ◽  
Nonlinear Diffusion ◽  
Extinction Time ◽  
Similar Form ◽  
Finite Mass ◽  
The Cauchy Problem ◽  
Self Similar

We consider the Cauchy problem for the equation u t = ▿ ( u -n ▿ u ) in [ R N for the cases N > 2 with 2 / N ⩽ n < 1 and N = 2 with n =1 for which the time-asymptotic behaviour of finite mass solutions has not previously been established. For N > 2 with n = 2 / N the behaviour as t → + ∞ is shown to take an unusual self-similar form. For N > 2 with 2/ N < n < 1 and N = 2 with n = 1 solutions extinguish in finite time. In the former case we show that the behaviour close to the extinction time is given by a similarity solution of the second kind and we derive a number of results for the similarity exponent. For N = 2 with n = 1 the solution to the Cauchy problem is not uniquely specified, and we characterize the possible types of solution and establish their behaviour close to extinction. We also indicate how physical considerations can lead to a unique selection from among the available solutions. The limit N→∞is also analysed, illustrating how the various types of asymptotic behaviour arise from the evolution over earlier times.

Equations de Schrödinger non linéaires en dimension deux

1979 ◽  
Vol 84 (3-4) ◽  
pp. 327-346 ◽  
Author(s):  
Thierry Cazenave
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Asymptotic Behaviour ◽  
Blow Up ◽  
Global Solutions ◽  
Two Dimensions ◽  
Linear Optics ◽  
Non Linear Optics ◽  
Non Linear ◽  
The Cauchy Problem

SynopsisThis paper is devoted to the study of some non linear Schrödinger equations in two dimensions, arising in non linear optics; in particular, it is concerned with solutions to the Cauchy problem. The problem of global existence and regularity of the solutions, the asymptotic behaviour of global solutions, and the blow-up of non global solutions are studied.

Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Zsolt Biró
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behaviour ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
The Cauchy Problem ◽  
Nonlinear Degenerate

AbstractThe aim of this paper is to investigate the asymptotic behaviour as t → ∞ of the solutions to the Cauchy problem for the nonlinear degenerate KPP-type diffusion-reaction equation u

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