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On the existence of an invariant measure for the dynamical system generated by partial differential equation. (English) Zbl 0572.35015

Let \({\mathcal X}\) be the space of functions \(v: [0,1]\to {\mathbb{R}}\), \(v(0)=0\), satisfying a Lipschitz condition. The space \({\mathcal X}\) is provided with a certain topology. Consider the boundary value problem \((1)\quad u_ t=\lambda u-xu_ x\) in the domain \(t\geq 0\), \(0\leq x\leq 1\), with boundary conditions \(u(t,0)=0\) for \(t\geq 0\), \(u(0,x)=v(x)\), where \(v\in {\mathcal X}\) is a given function.
In this paper the semi-dynamical system \({\mathcal T}_ t: {\mathcal X}\to {\mathcal X}\) is studied, which is generated by this boundary value problem. Especially, for \(\lambda >1\) in (1), the existence of a probabilistic ergodic \({\mathcal T}_ t\)-invariant measure \(\mu\) on \({\mathcal X}\) (such that \(\mu\) (\({\mathcal E})>0\) for each open non-empty set \({\mathcal E}\) and \(\mu\) (\({\mathcal E}_ 0)=0\), where \({\mathcal E}_ 0\) is the set of periodic points) is proved.
Reviewer: A.Klič

MSC:

35F15 Boundary value problems for linear first-order PDEs
58D07 Groups and semigroups of nonlinear operators
37C80 Symmetries, equivariant dynamical systems (MSC2010)
28D10 One-parameter continuous families of measure-preserving transformations
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