Klaczak, J. Stability of a transport equation. (English) Zbl 0673.45009 Ann. Pol. Math. 49, No. 1, 69-80 (1988). 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 Cited in 1 Review MSC: 45K05 Integro-partial differential equations 45M10 Stability theory for integral equations 82C70 Transport processes in time-dependent statistical mechanics Keywords:transport equation; asymptotical stability; semigroup; Lyapunov functions PDFBibTeX XMLCite \textit{J. Klaczak}, Ann. Pol. Math. 49, No. 1, 69--80 (1988; Zbl 0673.45009) Full Text: DOI