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Stability of a transport equation. (English) Zbl 0673.45009

It is shown that, under certain conditions, a semigroup \(\{T_ 0(t)\}_{t\geq 0}\) of linear operators on \(L^ 1=L^ 1(G)\) (where G is an unbounded, Lebesgue measurable subset of \({\mathbb{R}}^ n)\) associated with the integro-differential equation \[ \partial u(t,x)/\partial t+\sum^{n}_{i=1}\partial /\partial x_ i(a_ i(x)u(t,x))+u(t,x)=\int_{\Omega}\quad k(x;y)u(t,y)dy, \] with \(u(t,x_ 1,...,x_{i-1},0,x_{i+1},...,x_ n)=0\) \((i=1,...,n\); \(t\geq 0)\), and \(u(0,x)=f(x)\) (x\(\in \Omega)\), is asymptotically stable. Here, the kernel k is assumed to be measurable and stochastic, i.e. \(\int_{\Omega}k(x;y)dx=1\) and k(x;y)\(\geq 0\) for \(x,y\in \Omega\). The proof is based on a result regarding lower bounds for a class of certain Lyapunov functions. Three examples are used to illustrate the theory.
Reviewer: H.Brunner

MSC:

45K05 Integro-partial differential equations
45M10 Stability theory for integral equations
82C70 Transport processes in time-dependent statistical mechanics
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